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text_id stringlengths 22 22 | page_url stringlengths 31 389 | page_title stringlengths 1 250 | section_title stringlengths 0 4.67k | context_page_description stringlengths 0 108k | context_section_description stringlengths 1 187k | media list | hierachy list | category list |
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projected-00310918-003 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | Geography | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | The neighboring municipalities of Joensuu are Liperi, Kontiolahti, Lieksa, Ilomantsi, Tohmajärvi and Rääkkylä. In addition, the city is part of the Joensuu sub-region, which currently also includes the municipalities of Heinävesi, Ilomantsi, Juuka, Kontiolahti, Liperi and Polvijärvi, as well as the town of Outokumpu. | [] | [
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projected-00310918-004 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | Climate | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | Joensuu has a subarctic climate (Köppen: Dfc) due to its high latitude and inland position. Being quite far inland, Joensuu has a more continental climate than most of Finland. As a result, Joensuu can be prone to temperature extremes both in winter and summer. For example, Joensuu is on average warmer than Dublin or M... | [] | [
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"Joensuu",
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projected-00310918-005 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | Economy | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | Joensuu is a growing provincial center with a service-oriented business life. The concentration of information and communication technology companies has taken place in the premises offered by the Joensuu Science Park. Major industrial companies include lock manufacturer Abloy Oy and forest machine manufacturer John De... | [
"Sokos Joensuu.jpg"
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projected-00310918-006 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | Transport | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | Joensuu has a railway station and a bus station, which offers intercity connections to Helsinki and local connections to several other places. Numbered bus service is available to all parts of Joensuu (Route maps, Timetables). Joensuu also has an airport (located in nearby Liperi), with flights to Helsinki.
Joensuu is... | [
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projected-00310918-007 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | Sports | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | The city is known for its basketball club Kataja, which plays in the Finnish first-tier league Korisliiga. Other championship level clubs of Joensuu include Josba (floorball), Mutalan Riento (volleyball), the world leading orienteering club Kalevan Rasti (orienteering) and Joensuun Prihat (women's volleyball). The ice... | [] | [
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projected-00310918-008 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | Education | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | Joensuu is a city of students. The University of Eastern Finland (UEF) has one of its two main campuses in Joensuu and the University of Applied Sciences Karelia has two Joensuu campuses. There are also five high schools in Joensuu: Lyceum High School, Upper Secondary Normal School, Joensuu Coeducational High School, F... | [
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projected-00310918-009 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | Notable people | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | Ismo Alanko (born 1960), musician
Aku Korhonen (1892−1960), actor
Suvi Lindén (born 1962), politician and former Minister of Culture
Kaisa Mäkäräinen (born 1983), 3-time world-cup-winning biathlete
Sini Manninen (1944–2012), painter
Taneli Mustonen (born 1978), film director and screenwriter
Esa Pakarinen (1911–1... | [] | [
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projected-00310918-010 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | Friendship cities | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | Joensuu is twinned with:
Linköping, Sweden (since 1940)
Ísafjörður, Iceland (since 1948)
Tønsberg, Norway (1948)
Hof, Germany (since 1970)
Vilnius, Lithuania (since 1970)
Petrozavodsk, Russia (cooperation agreement) (since 1994) | [] | [
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"1848 establishments in Finland",
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"Grand Duchy of Finland",
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"Populated lakeshore places in Finland"
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projected-00310918-011 | https://en.wikipedia.org/wiki/Joensuu | Joensuu | See also | Joensuu (; ; ) is a city and municipality in North Karelia, Finland, located on the northern shore of Lake Pyhäselkä (northern part of Lake Saimaa) at the mouth of the Pielinen River (Pielisjoki). It was founded in 1848. The population of Joensuu is (), and the economic region of Joensuu has a population of 115,000. I... | Eno, Finland
Finnish Lake Road
Ilosaarirock
Öllölä
Pielisjoki Castle | [] | [
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"Joensuu",
"1848 establishments in Finland",
"Cities and towns in Finland",
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projected-00310919-000 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Introduction | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | [] | [
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projected-00310919-001 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Background | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | By 1212, Innocent III had been pope for 14 years and had faced the disappointment of the Fourth Crusade and its inability to recover Jerusalem, the on-going Albigensian Crusade, begun in 1209, and the popular fervor of the Children's Crusade of 1212. The Latin Empire of Constantinople was established, with the emperor ... | [] | [
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projected-00310919-002 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Preparations for the Crusade | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | In April 1213, Innocent III issued his papal bull Quia maior, calling all of Christendom to join a new Crusade. This was followed by a conciliar decree, the Ad Liberandam, in 1215. The attendant papal instructions engaged a new enterprise to recover Jerusalem while establishing Crusading norms that were to last nearly ... | [] | [
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projected-00310919-003 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | In Iberia and the Levant | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | The departure of the Crusaders began finally in early July 1217. Many of the Crusaders decided to go to the Holy Land by their traditional sea journey. The fleet made their first stop at Dartmouth on the southern coast of England. There they elected their leaders and the laws by which they would organize their venture.... | [] | [
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projected-00310919-004 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | The situation in the Holy Land | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | Saladin had died in 1193 and was succeeded in most of his domain by his brother al-Adil, who was the patriarch of all successive Ayyubid sultans of Egypt. Saladin's son az-Zahir Ghazi retained his leadership in Aleppo. An exceptionally low Nile River resulted in a failure of the crops in 1201–1202, and famine and pest... | [] | [
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projected-00310919-005 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Crusade of Andrew II of Hungary | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | Andrew II had been called on by the pope in July 1216 to fulfill his father Béla III's vow to lead a crusade, and finally agreed, having postponed three times earlier. Andrew, who was reputed to have designs on becoming Latin emperor, mortgaged his estates to finance the Crusade. In July 1217, he departed from Zagreb, ... | [
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projected-00310919-006 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | The campaign in Egypt | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | On 27 May 1218, the first of the Crusader's fleet arrived at the harbor of Damietta, on the right bank of the Nile. Simon III of Sarrebrück was chosen as temporary leader pending the arrival of the rest of the fleet. Within a few days, the remaining ships arrived, carrying John of Brienne, Leopold VI of Austria and mas... | [
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projected-00310919-007 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | The Tower of Damietta | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | The fortifications of Damietta were impressive, consisting of three walls of varying heights, with dozens of towers on the interior, and were enhanced to repel the invaders. Situated on an island in the Nile was the Burj al-Silsilah––the chain tower––called so because of the massive iron chains that could stretch acros... | [] | [
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] |
projected-00310919-008 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Preparation for the Siege | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | The Crusaders did not press their advantage, and many prepared to return home, regarding their crusading vows satisfied. Further offensive action would nevertheless have to wait until the Nile was more favourable and the arrival of additional forces. Among them were papal legate Pelagius Galvani and his aide Robert of ... | [] | [
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projected-00310919-009 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Saint Francis in Egypt | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | In September 1219, Francis of Assisi arrived in the Crusader camp seeking permission from Pelagius to visit sultan al-Kamil. Francis had a long history with the Crusades. In 1205, Francis prepared to enlist in the army of Walter III of Brienne (brother of John), diverted from the Fourth Crusade to fight in Italy. He re... | [
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projected-00310919-010 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | The Siege of Damietta | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | With the negotiations with the Crusaders stalled and Damietta isolated, on 3 November 1219 al-Kamil sent a resupply convoy through the sector manned by the troops of the Frenchman Hervé IV of Donzy. The Egyptians were by and large stopped, some getting through to the city, resulting in the expulsion of Hervé. The intru... | [] | [
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projected-00310919-011 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | John of Brienne returns to Jerusalem | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | The father-in-law of John of Brienne, Leo I of Armenia, died on 2 May 1219, leaving his succession in doubt. John's claim to the Armenian throne was through his wife Stephanie of Armenia and their infant son, and Leo I had instead left the kingdom to his infant daughter Isabella of Armenia. The pope decreed in February... | [] | [
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projected-00310919-012 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Disaster at Mansurah | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | The situation in Damietta after the February 1220 celebration was one of inactivity and discontent. The army lacked discipline despite Pelagius' draconian rule. His extensive regulations prevented adequate protection of the shipping lanes from Cyprus, and several ships carrying pilgrims were sunk. Many Crusaders depart... | [] | [
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projected-00310919-013 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Aftermath | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | The Fifth Crusade ended with nothing gained for the West, with much loss of life, resources and reputations. Most were bitter that offensive operations were begun prior to the arrival of the emperor's forces, and had opposed the treaty. Walter of Palearia was stripped of his possessions and sent into exile. Admiral Hen... | [] | [
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projected-00310919-014 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Participants | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | A partial list of those that participated in the Fifth Crusade can be found in the category collections of Christians of the Fifth Crusade and Muslims of the Fifth Crusade. | [] | [
"Participants"
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projected-00310919-015 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Historiography | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | The historiography of the Fifth Crusade is concerned with the "history of the histories" of the military campaigns discussed herein as well as biographies of the important figures of the period. The primary sources include works written in the medieval period, generally by participants in the Crusade or written contemp... | [] | [
"Historiography"
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] |
projected-00310919-017 | https://en.wikipedia.org/wiki/Fifth%20Crusade | Fifth Crusade | Bibliography | The Fifth Crusade (1217–1221) was a campaign in a series of Crusades by Western Europeans to reacquire Jerusalem and the rest of the Holy Land by first conquering Egypt, ruled by the powerful Ayyubid sultanate, led by al-Adil, brother of Saladin.
After the failure of the Fourth Crusade, Innocent III again called for a... | Category:Fifth Crusade
Category:13th century in the Ayyubid Sultanate
Category:13th-century crusades
Category:Wars involving the Ayyubid Sultanate
Category:1210s
Category:1220s
Category:1217 in Asia
Category:13th-century military alliances | [] | [
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projected-00310921-000 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Introduction | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | [] | [
"Introduction"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
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projected-00310921-001 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Definition and basic concepts | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | A Lie algebroid is a triple consisting of
a vector bundle over a manifold
a Lie bracket on its space of sections
a morphism of vector bundles , called the anchor, where is the tangent bundle of
such that the anchor and the bracket satisfy the following Leibniz rule:
where and is the derivative of along ... | [] | [
"Definition and basic concepts"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
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"Vector bundles"
] |
projected-00310921-002 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | First properties | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | It follows from the definition that
for every , the kernel is a Lie algebra, called the isotropy Lie algebra at
the kernel is a (not necessarily locally trivial) bundle of Lie algebras, called the isotropy Lie algebra bundle
the image is a singular distribution which is integrable, i.e. its admits maximal immers... | [] | [
"Definition and basic concepts",
"First properties"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-003 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Subalgebroids and ideals | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | A Lie subalgebroid of a Lie algebroid is a vector subbundle of the restriction such that takes values in and is a Lie subalgebra of . Clearly, admits a unique Lie algebroid structure such that is a Lie algebra morphism. With the language introduced below, the inclusion is a Lie algebroid morphism.
A Lie subal... | [] | [
"Definition and basic concepts",
"Subalgebroids and ideals"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-004 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Morphisms | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | A Lie algebroid morphism between two Lie algebroids and with the same base is a vector bundle morphism which is compatible with the Lie brackets, i.e. for every , and with the anchors, i.e. .
A similar notion can be formulated for morphisms with different bases, but the compatibility with the Lie brackets becomes... | [] | [
"Definition and basic concepts",
"Morphisms"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-006 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Trivial and extreme cases | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Given any manifold , its tangent Lie algebroid is the tangent bundle together with the Lie bracket of vector fields and the identity of as an anchor.
Given any manifold , the zero vector bundle is a Lie algebroid with zero bracket and anchor.
Lie algebroids over a point are the same thing as Lie algebras.
More gen... | [] | [
"Examples",
"Trivial and extreme cases"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-007 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Examples from differential geometry | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Given a foliation on , its foliation algebroid is the associated involutive subbundle , with brackets and anchor induced from the tangent Lie algebroid.
Given the action of a Lie algebra on a manifold , its action algebroid is the trivial vector bundle , with anchor given by the Lie algebra action and brackets uniqu... | [] | [
"Examples",
"Examples from differential geometry"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-008 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Constructions from other Lie algebroids | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Given any Lie algebroid , there is a Lie algebroid , called its tangent algebroid, obtained by considering the tangent bundle of and and the differential of the anchor.
Given any Lie algebroid , there is a Lie algebroid , called its k-jet algebroid, obtained by considering the k-jet bundle of , with Lie bracket uniq... | [] | [
"Examples",
"Constructions from other Lie algebroids"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-010 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Totally intransitive Lie algebroids | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | A Lie algebroid is called totally intransitive if the anchor map is zero.
Bundle of Lie algebras (hence also Lie algebras) are totally intransitive. This actually exhaust completely the list of totally intransitive Lie algebroids: indeed, if is totally intransitive, it must coincide with its isotropy Lie algebra bun... | [] | [
"Important classes of Lie algebroids",
"Totally intransitive Lie algebroids"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-011 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Transitive Lie algebroids | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | A Lie algebroid is called transitive if the anchor map is surjective. As a consequence:
there is a short exact sequence
right-splitting of defines a principal bundle connections on ;
the isotropy bundle is locally trivial (as bundle of Lie algebras);
the pullback of exist for every .
The prototypical examples... | [] | [
"Important classes of Lie algebroids",
"Transitive Lie algebroids"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-012 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Regular Lie algebroids | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | A Lie algebroid is called regular if the anchor map is of constant rank. As a consequence
the image of defines a regular foliation on ;
the restriction of over each leaf is a transitive Lie algebroid.
For instance:
any transitive Lie algebroid is regular (the anchor has maximal rank);
any totally intransitiv... | [] | [
"Important classes of Lie algebroids",
"Regular Lie algebroids"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-014 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Actions | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | An action of a Lie algebroid on a manifold P along a smooth map consists of a Lie algebra morphismsuch that, for every ,Of course, when , both the anchor and the map must be trivial, therefore both conditions are empty, and we recover the standard notion of action of a Lie algebra on a manifold. | [] | [
"Further related concepts",
"Actions"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-015 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Connections | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Given a Lie algebroid , an A-connection on a vector bundle consists of an -bilinear mapwhich is -linear in the first factor and satisfies the following Leibniz rule:for every , where denotes the Lie derivative with respect to the vector field .
The curvature of an A-connection is the -bilinear mapand is called fla... | [] | [
"Further related concepts",
"Connections"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-016 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Representations | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | A representation of a Lie algebroid is a vector bundle together with a flat A-connection . Equivalently, a representation is a Lie algebroid morphism .
The set of isomorphism classes of representations of a Lie algebroid has a natural structure of semiring, with direct sums and tensor products of vector bundles.
... | [] | [
"Further related concepts",
"Representations"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-017 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Lie algebroid cohomology | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Consider a Lie algebroid and a representation . Denoting by the space of -differential forms on with values in the vector bundle , one can define a differential with the following Koszul-like formula:Thanks to the flatness of , becomes a cochain complex and its cohomology, denoted by , is called the Lie algebroid ... | [] | [
"Further related concepts",
"Lie algebroid cohomology"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-018 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Lie groupoid-Lie algebroid correspondence | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | The standard construction which associates a Lie algebra to a Lie group generalises to this setting: to every Lie groupoid one can canonically associate a Lie algebroid defined as follows:
the vector bundle is , where is the vertical bundle of the source fibre and is the groupoid unit map;
the sections of are... | [] | [
"Lie groupoid-Lie algebroid correspondence"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-019 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Lie functor | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | The mapping sending a Lie groupoid to a Lie algebroid is actually part of a categorical construction. Indeed, any Lie groupoid morphism can be differentiated to a morphism between the associated Lie algebroids.
This construction defines a functor from the category of Lie groupoids and their morphisms to the categor... | [] | [
"Lie groupoid-Lie algebroid correspondence",
"Lie functor"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-020 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Structures and properties induced from groupoids to algebroids | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Let be a Lie groupoid and its associated Lie algebroid. Then
The isotropy algebras are the Lie algebras of the isotropy groups
The orbits of coincides with the orbits of
is transitive and is a submersion if and only if is transitive
an action of on induces an action of (called infinitesimal action), ... | [] | [
"Lie groupoid-Lie algebroid correspondence",
"Structures and properties induced from groupoids to algebroids"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-021 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Examples | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | The Lie algebroid of a Lie group is the Lie algebra
The Lie algebroid of both the pair groupoid and the fundamental groupoid is the tangent algebroid
The Lie algebroid of the unit groupoid is the zero algebroid
The Lie algebroid of a Lie group bundle is the Lie algebra bundle
The Lie algebroid of an actio... | [] | [
"Lie groupoid-Lie algebroid correspondence",
"Examples"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-022 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Detailed example 1 | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Let us describe the Lie algebroid associated to the pair groupoid . Since the source map is , the -fibers are of the kind , so that the vertical space is . Using the unit map , one obtain the vector bundle .
The extension of sections to right-invariant vector fields is simply and the extension of a smooth function ... | [] | [
"Lie groupoid-Lie algebroid correspondence",
"Detailed example 1"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-023 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Detailed example 2 | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Consider the (action) Lie groupoid
where the target map (i.e. the right action of on ) is
The -fibre over a point are all copies of , so that is the trivial vector bundle .
Since its anchor map is given by the differential of the target map, there are two cases for the isotropy Lie algebras, corresponding to the... | [] | [
"Lie groupoid-Lie algebroid correspondence",
"Detailed example 2"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-025 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Lie theorems | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | A Lie algebroid is called integrable if it is isomorphic to for some Lie groupoid . The analogue of the classical Lie I theorem states that:if is an integrable Lie algebroid, then there exists a unique (up to isomorphism) -simply connected Lie groupoid integrating .Similarly, a morphism between integrable Lie algeb... | [] | [
"Integration of a Lie algebroid",
"Lie theorems"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-026 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Weinstein groupoid | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Given any Lie algebroid , the natural candidate for an integration is given by the Weinstein groupoid , where denotes the space of -paths and the relation of -homotopy between them. Indeed, one can show that is an -simply connected topological groupoid, with the multiplication induced by the concatenation of paths. ... | [] | [
"Integration of a Lie algebroid",
"Weinstein groupoid"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-027 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Integrable examples | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Lie algebras are always integrable (by Lie III theorem)
Atiyah algebroids of a principal bundle are always integrable (to the gauge groupoid of that principal bundle)
Lie algebroids with injective anchor (hence foliation algebroids) are alway integrable (by Frobenius theorem)
Lie algebra bundle are always integrable... | [] | [
"Integration of a Lie algebroid",
"Integrable examples"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-028 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | A non-integrable example | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Consider the Lie algebroid associated to a closed 2-form and the group of spherical periods associated to , i.e. the image of the following group homomorphism from the second homotopy group of
Since is transitive, it is integrable if and only if it is the Atyah algebroid of some principal bundle; a careful analys... | [] | [
"Integration of a Lie algebroid",
"A non-integrable example"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-029 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | See also | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | R-algebroid
Lie bialgebroid | [] | [
"See also"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310921-031 | https://en.wikipedia.org/wiki/Lie%20algebroid | Lie algebroid | Books and lecture notes | In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie gro... | Alan Weinstein, Groupoids: unifying internal and external symmetry, AMS Notices, 43 (1996), 744-752. Also available at arXiv:math/9602220.
Kirill Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge U. Press, 1987.
Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambri... | [] | [
"Books and lecture notes"
] | [
"Lie algebras",
"Differential geometry",
"Differential topology",
"Differential operators",
"Generalizations of the derivative",
"Geometry processing",
"Vector bundles"
] |
projected-00310923-000 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Introduction | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | [] | [
"Introduction"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] | |
projected-00310923-001 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Definition and basic concepts | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | A Lie groupoid consists of
two smooth manifolds and
two surjective submersions (called, respectively, source and target projections)
a map (called multiplication or composition map), where we use the notation
a map (called unit map or object inclusion map), where we use the notation
a map (called inversio... | [] | [
"Definition and basic concepts"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-002 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Alternative definitions | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | The original definition by Ehresmann required and to possess a smooth structure such that only is smooth and the maps and are subimmersions (i.e. have locally constant rank). Such definition proved to be too weak and was replaced by Pradines with the one currently used.
While some authors introduced weaker defini... | [] | [
"Definition and basic concepts",
"Alternative definitions"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-003 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | First properties | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | The fact that the source and the target map of a Lie groupoid are smooth submersions has some immediate consequences:
the -fibres , the -fibres , and the set of composable morphisms are submanifolds;
the inversion map is a diffeomorphism;
the unit map is a smooth embedding;
the isotropy groups are Lie groups;
t... | [] | [
"Definition and basic concepts",
"First properties"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-004 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Subobjects and morphisms | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | A Lie subgroupoid of a Lie groupoid is a subgroupoid (i.e. a subcategory of the category ) with the extra requirement that is an immersed submanifold. As for a subcategory, a (Lie) subgroupoid is called wide if . Any Lie groupoid has two canonical wide subgroupoids:
the unit/identity Lie subgroupoid ;
the inner ... | [] | [
"Definition and basic concepts",
"Subobjects and morphisms"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-005 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Bisections | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | A bisection of a Lie groupoid is a smooth map such that and is a diffeomorphism of . In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold such that and are diffeomorphisms; the relation between the two definitions is given by .
The... | [] | [
"Definition and basic concepts",
"Bisections"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-007 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Trivial and extreme cases | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Lie groupoids with one object are the same thing as Lie groups.
Given any manifold , there is a Lie groupoid called the pair groupoid, with precisely one morphism from any object to any other.
The two previous examples are particular cases of the trivial groupoid , with structure maps , , , and .
Given any manifold... | [] | [
"Examples",
"Trivial and extreme cases"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-008 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Constructions from other Lie groupoids | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Given any Lie groupoid and a surjective submersion , there is a Lie groupoid , called its pullback groupoid or induced groupoid, where contains triples such that and , and the multiplication is defined using the multiplication of . For instance, the pullback of the pair groupoid of is the pair groupoid of .
Given ... | [] | [
"Examples",
"Constructions from other Lie groupoids"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-009 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Examples from differential geometry | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Given a submersion , there is a Lie groupoid , called the submersion groupoid or fibred pair groupoid, whose structure maps are induced from the pair groupoid (the condition that is a submersion ensures the smoothness of ). If is a point, one recovers the pair groupoid.
Given a Lie group acting on a manifold , ther... | [] | [
"Examples",
"Examples from differential geometry"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-010 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Important classes of Lie groupoids | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Note that some of the following classes make sense already in the category of set-theoretical or topological groupoids. | [] | [
"Important classes of Lie groupoids"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-011 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Transitive groupoids | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | A Lie groupoid is transitive (in older literature also called connected) if it satisfies one of the following equivalent conditions:
there is only one orbit;
there is at least a morphism between any two objects;
the map (also known as the anchor of ) is surjective.
Gauge groupoids constitute the prototypical exam... | [] | [
"Important classes of Lie groupoids",
"Transitive groupoids"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-012 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Proper groupoids | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | A Lie groupoid is called proper if is a proper map. As a consequence
all isotropy groups of are compact;
all orbits of are closed submanifolds;
the orbit space is Hausdorff.
For instance:
a Lie group is proper if and only if it is compact;
pair groupoids are always proper;
unit groupoids are always proper;
... | [] | [
"Important classes of Lie groupoids",
"Proper groupoids"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-013 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Étale groupoids | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | A Lie groupoid is called étale if it satisfies one of the following equivalent conditions:
the dimensions of and are equal;
is a local diffeomorphism;
all the -fibres are discrete
As a consequence, also the -fibres, the isotropy groups and the orbits become discrete.
For instance:
a Lie group is étale if and... | [] | [
"Important classes of Lie groupoids",
"Étale groupoids"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-014 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Effective groupoids | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | An étale groupoid is called effective if, for any two local bisections , the condition implies . For instance:
Lie groups are effective if and only if are trivial;
unit groupoids are always effective;
an action groupoid is effective if the -action is free and is discrete.
In general, any effective étale groupoid ar... | [] | [
"Important classes of Lie groupoids",
"Effective groupoids"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-015 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Source-connected groupoids | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | A Lie groupoid is called -connected if all its -fibres are connected. Similarly, one talks about -simply connected groupoids (when the -fibres are simply connected) or source-k-connected groupoids (when the -fibres are k-connected, i.e. the first homotopy groups are trivial).
Note that the entire space of arrows is ... | [] | [
"Important classes of Lie groupoids",
"Source-connected groupoids"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-017 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Actions and principal bundles | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Recall that an action of a groupoid on a set along a function is defined via a collection of maps for each morphism between . Accordingly, an action of a Lie groupoid on a manifold along a smooth map consists of a groupoid action where the maps are smooth. Of course, for every there is an induced smooth actio... | [] | [
"Further related concepts",
"Actions and principal bundles"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-018 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Representations | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | A representation of a Lie groupoid consists of a Lie groupoid action on a vector bundle , such that the action is fibrewise linear, i.e. each bijection is a linear isomorphism. Equivalently, a representation of on can be described as a Lie groupoid morphism from to the general linear groupoid .
Of course, any fib... | [] | [
"Further related concepts",
"Representations"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-019 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Differentiable cohomology | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | The notion of differentiable cohomology for Lie groups generalises naturally also to Lie groupoids: the definition relies on the symplicial structure of the nerve of , viewed as a category.
More precisely, recall that the space consists of strings of composable morphisms, i.e.
and consider the map .
A differentia... | [] | [
"Further related concepts",
"Differentiable cohomology"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-020 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | The Lie algebroid of a Lie groupoid | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Any Lie groupoid has an associated Lie algebroid , obtained with a construction similar to the one which associates a Lie algebra to any Lie groupː
the vector bundle is the vertical bundle with respect to the source map, restricted to the elements tangent to the identities, i.e. ;
the Lie bracket is obtained by id... | [] | [
"Further related concepts",
"The Lie algebroid of a Lie groupoid"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-021 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Morita equivalence | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | As discussed above, the standard notion of (iso)morphism of groupoids (viewed as functors between categories) restricts naturally to Lie groupoids. However, there is a more coarse notation of equivalence, called Morita equivalence, which is more flexible and useful in applications.
First, a Morita map (also known as a... | [] | [
"Morita equivalence"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-022 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Morita invariance | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Many properties of Lie groupoids, e.g. being proper, being Hausdorff or being transitive, are Morita invariant. On the other hand, being étale is not Morita invariant.
In addition, a Morita equivalence between and preserves their transverse geometry, i.e. it induces:
a homeomorphism between the orbit spaces and ;... | [] | [
"Morita equivalence",
"Morita invariance"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-023 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Examples | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Isomorphic Lie groupoids are trivially Morita equivalent.
Two Lie groups are Morita equivalent if and only if they are isomorphic as Lie groups.
Two unit groupoids are Morita equivalent if and only if the base manifolds are diffeomorphic.
Any transitive Lie groupoid is Morita equivalent to its isotropy groups.
Given... | [] | [
"Morita equivalence",
"Examples"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-024 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Smooth stacks | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Investigating the structure of the orbit space of a Lie groupoid leads to the notion of a smooth stack. For instance, the orbit space is a smooth manifold if the isotropy groups are trivial (as in the example of the Čech groupoid), but it is not smooth in general. The solution is to revert the problem and to define a s... | [] | [
"Morita equivalence",
"Smooth stacks"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310923-026 | https://en.wikipedia.org/wiki/Lie%20groupoid | Lie groupoid | Books | In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions.
A Lie groupoid can thus be thought o... | Category:Differential geometry
Category:Lie groups
Category:Manifolds
Category:Symmetry | [] | [
"Books"
] | [
"Differential geometry",
"Lie groups",
"Lie groupoids",
"Manifolds",
"Symmetry"
] |
projected-00310924-000 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Introduction | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | [] | [
"Introduction"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... | |
projected-00310924-001 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Name | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | The band's name is taken from Norse mythology, where the term Einherjar describes the slain warriors who have gone to Valhalla and joined Odin's table. | [] | [
"Name"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-002 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Battered | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | The band was reconstituted in 2004 as "Battered", a thrash band that included Glesnes, Storesund, and Herløe. A new vocalist joined them in 2005, Siggy Olaisen, The band only released one demo, ...Beyond Recognition, and one album—also named Battered—before disbanding. The self-titled debut album was released Februar... | [] | [
"Battered"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-004 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Current lineup | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | Frode Glesnes (previously Grimar) - vocals, guitar and bass (1993 - 2004, 2008–present)
Gerhard Storesund (previously Ulvar) - drums and keys (1993 - 2004, 2008–present)
Ole Sønstabø - Lead guitar (2016–present)
Tom Enge - Guitar & backing vocals (2020–present) | [] | [
"Personnel",
"Current lineup"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-005 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Previous members | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | Aksel Herløe - guitar (1999 - 2004, 2008 - 2020)
Rune Bjelland (previously Nidhogg) - vocals (1993 - 1997)
Audun Wold (previously Thonar) - bass, keyboards, guitar (1993 - 1997)
Stein Sund - bass (1995 - 1997)
Ragnar Vikse - vocals (1997 - 2002)
Erik Elden - bass (1998) | [] | [
"Personnel",
"Previous members"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-006 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Session and touring musicians | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | Tchort - bass (1998 - 1999)
Stein Sund - bass (1999)
Jon Lind - bass (2000 - 2001)
V'Gandr - bass (2001)
Kjell Håvardsholm - bass (2002)
Sigve Husebø - bass (2003)
Eirik Svendsbø - backing vocals and bass (2008 - 2014) | [] | [
"Personnel",
"Session and touring musicians"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-007 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Battered band members | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | Sigurd "Siggy" Olaisen - lead vocals (Headblock)
Frode Glesnes - guitar and vocals (Einherjer)
Aksel Herløe - guitar (ex-Einherjer)
Gerhard Storesund - drums (Einherjer)
Ole Moldesæther - bass (Headblock) | [] | [
"Battered band members"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-010 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Studio albums | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | Dragons of the North (1996)
Odin Owns Ye All (1998)
Norwegian Native Art (2000)
Blot (2003)
Norrøn (2011)
Av Oss For Oss (2014)
Dragons of the North XX (2016)
Norrøne Spor (2018)
North Star (2021) | [] | [
"Discography",
"Studio albums"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-011 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | EPs | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | Leve Vikingånden (1995)
Aurora Borealis (1996)
Far Far North (1997) | [] | [
"Discography",
"EPs"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-012 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Compilation albums | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | Aurora Borealis / Leve vikingånden (2013) | [] | [
"Discography",
"Compilation albums"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310924-013 | https://en.wikipedia.org/wiki/Einherjer | Einherjer | Demos | Einherjer is a Viking metal band from Haugesund, Norway, founded in 1993. Some of the band's albums are heavily folk influenced, while others have a more traditional symphonic black metal sound. Their lyrics retell Norse legends, and each of their albums has its own theme. The band split up in early 2004 after releasin... | Aurora Borealis (1994)
...Beyond Recognition, (Battered) 2004 | [] | [
"Discography",
"Demos"
] | [
"Norwegian thrash metal musical groups",
"Norwegian viking metal musical groups",
"Norwegian symphonic black metal musical groups",
"Musical groups established in 1993",
"Musical groups disestablished in 2004",
"Musical groups reestablished in 2008",
"1993 establishments in Norway",
"2004 disestablish... |
projected-00310928-000 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Introduction | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | [] | [
"Introduction"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] | |
projected-00310928-001 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | History | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | As a senior in high school, Steve Wozniak's electronics teacher arranged for the leading students in the class to have placements at local electronics companies. Wozniak was sent to Sylvania where he programmed in FORTRAN on an IBM 1130. That same year, General Electric placed a terminal in the high school that was con... | [
"Apple 1 Advertisement Oct 1976.jpg"
] | [
"History"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-003 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Program editing | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | Like most BASIC implementations of the era, Integer BASIC acted as both the language interpreter as well as the line editing environment. When BASIC was running, a command prompt was displayed where the user could enter statements. Unlike later home computer platforms, BASIC was not the default environment when the Ap... | [] | [
"Description",
"Program editing"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-004 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Debugging | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | As in most BASICs, programs were started with the command, and as was common, could be directed at a particular line number like . Execution could be stopped at any time using and then restarted with tinue, as opposed to the more typical .
For step-by-step execution, the instruction could be used at the command pro... | [] | [
"Description",
"Debugging"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-005 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Variable names | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | Where Dartmouth BASIC and HP-BASIC limited variable names to at most two characters (either a single letter or a letter followed by one digit), and where MS-BASIC allowed a letter followed by an optional letter or digit (ignoring subsequent characters), Integer BASIC was unusual in supporting any length variable name (... | [] | [
"Description",
"Variable names"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-006 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Mathematics | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | Integer BASIC, as its name implies, uses integers as the basis for its math package. These were stored internally as a 16-bit number, little-endian (as is the 6502). This allowed a maximum value for any calculation between -32767 and 32767; although the format could also store the value -32768, BASIC could not display ... | [] | [
"Description",
"Mathematics"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-007 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Strings | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | Integer BASIC's string handling was based on the system in HP BASIC. This treated string variables as arrays of characters which had to be ed prior to use. This is similar to the model in C or Fortran 77. This is in contrast to MS-like BASICs where strings are an intrinsic variable-length type. Before MS-derived BASICs... | [] | [
"Description",
"Strings"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-008 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Graphics and sound | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | When launched, the only game controller for the Apple was the paddle controller, which had two controllers on a single connector. The position of the controller could be read using the function, passing in the controller number, 0 or 1, like , returning a value between 0 and 255.
The Apple machines did not include de... | [] | [
"Description",
"Graphics and sound"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-009 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Input/output | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | Integer BASIC lacked any custom input/output commands, and also lacked the statement and the associated . To get data into and out of a program, the input/output functionality was redirected to a selected card slot with the and , which redirected output or input (respectively) to the numbered slot. From then on, data... | [] | [
"Description",
"Input/output"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-010 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Other notes | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | Integer BASIC included a feature, which positioned the cursor on a given column from 0 to 39. It differed from the versions found in most BASICs in that it was a command with a following number, as opposed to a function with the value in parenthesis; one would move the cursor to column 10 using in Integer BASIC where... | [] | [
"Description",
"Other notes"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-012 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Apple BASIC | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | Apple BASIC had the following commands:
AUTO val1, val2
CLR [CLEAR]
DEL val1, val2
LIST val1, val2
RUN val1
SCR [SCRATCH / NEW]
HIMEM = expr
LOMEM = expr
(LET) var = expr
INPUT (prompt,) var1, var2 ...
PRINT item(s)
TAB expr
FOR var = expr1 TO expr2 STEP expr3
NEXT var
IF expr THEN statement
IF expr THEN line number
GO... | [] | [
"Description",
"Reserved words",
"Apple BASIC"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |
projected-00310928-013 | https://en.wikipedia.org/wiki/Integer%20BASIC | Integer BASIC | Integer BASIC | Integer BASIC is a BASIC interpreter written by Steve Wozniak for the Apple I and Apple II computers. Originally available on cassette for the Apple I in 1976, then included in ROM on the Apple II from its release in 1977, it was the first version of BASIC used by many early home computer owners.
The only numeric data... | Integer BASIC added the following:
COLOR = expr
CON [CONTINUE]
DSP
GR
HLIN expr1, expr2 AT expr3
MAN
NEW [replaces SCR]
NOTRACE
PLOT expr1, expr2
POP
TEXT
TRACE
VLIN expr1, expr2 AT expr3
Function: ASC(), PDL(), SCRN(X,Y) | [] | [
"Description",
"Reserved words",
"Integer BASIC"
] | [
"Apple II software",
"BASIC interpreters",
"Assembly language software",
"BASIC programming language family"
] |