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id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
c56797569ef15e25b02ef169da449267c342619a	invariant theory studies actions on algebraic varieties from	invariant theory studies actions on algebraic varieties from the point of view of their effect on functions, which form representations of the group. classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a giv...	wikipedia
8b8b0fe61c53475b65e19f2b41d37e74f91131c3	linear algebraic groups (or more generally, affine group	linear algebraic groups (or more generally, affine group schemes) are analogues in algebraic geometry of lie groups, but over more general fields than just r or c. in particular, over finite fields, they give rise to finite groups of lie type. although linear algebraic groups have a classification that is very similar ...	wikipedia
28b8d3503ff337c6ade8ff7484167d534dabf219	lie superalgebras are generalizations of lie algebras in	lie superalgebras are generalizations of lie algebras in which the underlying vector space has a z -grading, and skew-symmetry and jacobi identity properties of the lie bracket are modified by signs. their representation theory is similar to the representation theory of lie algebras.	wikipedia
e76ea32fcbcf780f5311c27ba6a461cba5e961a4	affine lie algebras are a special case of	affine lie algebras are a special case of kac–moody algebras, which have particular importance in mathematics and theoretical physics, especially conformal field theory and the theory of exactly solvable models. kac discovered an elegant proof of certain combinatorial identities, macdonald identities, which is based on...	wikipedia
a622396b153efc680f326e5358ef6fd90a60f446	there are many classes of infinite-dimensional lie algebras	there are many classes of infinite-dimensional lie algebras whose representations have been studied. among these, an important class are the kac–moody algebras. they are named after victor kac and robert moody, who independently discovered them. these algebras form a generalization of finite-dimensional semisimple lie ...	wikipedia
284da71dde795f4fbba3c3667870ee3e31e35875	lie algebras, like lie groups, have a levi	lie algebras, like lie groups, have a levi decomposition into semisimple and solvable parts, with the representation theory of solvable lie algebras being intractable in general. in contrast, the finite-dimensional representations of semisimple lie algebras are completely understood, after work of Élie cartan. a repres...	wikipedia
78432771cb03d44e6ef13ae01e2a8d3fd862e950	a lie algebra over a field f is	a lie algebra over a field f is a vector space over f equipped with a skew-symmetric bilinear operation called the lie bracket, which satisfies the jacobi identity. lie algebras arise in particular as tangent spaces to lie groups at the identity element, leading to their interpretation as "infinitesimal symmetries". an...	wikipedia
71cbee5b9efc173d333c12f6ea36031afdaae046	a general lie group is a semidirect product	a general lie group is a semidirect product of a solvable lie group and a semisimple lie group (the levi decomposition). the classification of representations of solvable lie groups is intractable in general, but often easy in practical cases. representations of semidirect products can then be analysed by means of gene...	wikipedia
3cddcf55611f39d24b2336619bfb70b9dd852d8e	the representation theory of lie groups can be	the representation theory of lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply. this theory can be extended to finite-dimensional representations of semisimple lie groups using weyl's unitary trick: each semisimple real lie group g has a complex...	wikipedia
de4af669603a228534839cb4f3e860d6012858dd	a lie group is a group that is	a lie group is a group that is also a smooth manifold. many classical groups of matrices over the real or complex numbers are lie groups. many of the groups important in physics and chemistry are lie groups, and their representation theory is crucial to the application of group theory in those fields.	wikipedia
234151943200f200635349393f83fc8f49f3e504	if the group is neither abelian nor compact,	if the group is neither abelian nor compact, no general theory is known with an analogue of the plancherel theorem or fourier inversion, although alexander grothendieck extended tannaka–krein duality to a relationship between linear algebraic groups and tannakian categories.	wikipedia
ed3c4e937ccc07f82e93e0116e3fb61d6405860f	a major goal is to provide a general	a major goal is to provide a general form of the fourier transform and the plancherel theorem. this is done by constructing a measure on the unitary dual and an isomorphism between the regular representation of g on the space l (g) of square integrable functions on g and its representation on the space of l functions o...	wikipedia
4ab8c08f4204bca17619a127b324a8a50187cd1a	the duality between the circle group s and	the duality between the circle group s and the integers z, or more generally, between a torus t and z is well known in analysis as the theory of fourier series, and the fourier transform similarly expresses the fact that the space of characters on a real vector space is the dual vector space. thus unitary representatio...	wikipedia
052f107e211e82d2c8fae7672b97b3e9b3961a56	for non-compact g, the question of which representations	for non-compact g, the question of which representations are unitary is a subtle one. although irreducible unitary representations must be "admissible" (as harish-chandra modules) and it is easy to detect which admissible representations have a nondegenerate invariant sesquilinear form, it is hard to determine when thi...	wikipedia
929118298450ae680f6c01b704bc492881f74dcf	a major goal is to describe the "	a major goal is to describe the " unitary dual ", the space of irreducible unitary representations of g. the theory is most well-developed in the case that g is a locally compact (hausdorff) topological group and the representations are strongly continuous. for g abelian, the unitary dual is just the space of character...	wikipedia
ad8f933efd75b334ee34f93eba4259406e7c8512	a unitary representation of a group g is	a unitary representation of a group g is a linear representation φ of g on a real or (usually) complex hilbert space v such that φ (g) is a unitary operator for every g ∈ g. such representations have been widely applied in quantum mechanics since the 1920s, thanks in particular to the influence of hermann weyl, and thi...	wikipedia
ae67781939172217bcef232362bf9c36848f821b	modular representations of a finite group g are	modular representations of a finite group g are representations over a field whose characteristic is not coprime to \| g \|, so that maschke's theorem no longer holds (because \| g \| is not invertible in f and so one cannot divide by it). nevertheless, richard brauer extended much of character theory to modular representa...	wikipedia
b7368a8dc2109d203089686328e6932c9657c669	representations of a finite group g are also	representations of a finite group g are also linked directly to algebra representations via the group algebra f, which is a vector space over f with the elements of g as a basis, equipped with the multiplication operation defined by the group operation, linearity, and the requirement that the group operation and scalar...	wikipedia
36334c7414746c8cfc71d5a63b3cdef09f118f85	over arbitrary fields, another class of finite groups	over arbitrary fields, another class of finite groups that have a good representation theory are the finite groups of lie type. important examples are linear algebraic groups over finite fields. the representation theory of linear algebraic groups and lie groups extends these examples to infinite-dimensional groups, th...	wikipedia
c959bc0d7e5659ba62989ea8154a8affb3477b8d	results such as maschke's theorem and the unitary	results such as maschke's theorem and the unitary property that rely on averaging can be generalized to more general groups by replacing the average with an integral, provided that a suitable notion of integral can be defined. this can be done for compact topological groups (including compact lie groups), using haar me...	wikipedia
b265d64483a28ea3e010a94b786dd850d4522dbd	unitary representations are automatically semisimple, since maschke's result	unitary representations are automatically semisimple, since maschke's result can be proven by taking the orthogonal complement of a subrepresentation. when studying representations of groups that are not finite, the unitary representations provide a good generalization of the real and complex representations of a finit...	wikipedia
a43d19b621fd4efe69aa479aca96d0100ed2cd64	averaging techniques also show that if f is	averaging techniques also show that if f is the real or complex numbers, then any g -representation preserves an inner product ⟨ ⋅, ⋅ ⟩ {\displaystyle \langle \cdot,\cdot \rangle } on v in the sense that	wikipedia
455c0918103dae1d7559da6464f61748598a3962	maschke's theorem holds more generally for fields of	maschke's theorem holds more generally for fields of positive characteristic p, such as the finite fields, as long as the prime p is coprime to the order of g. when p and \| g \| have a common factor, there are g -representations that are not semisimple, which are studied in a subbranch called modular representation theo...	wikipedia
0eec23ba2bf86acdec8093a4d1e54a297a958420	over a field of characteristic zero, the representation	over a field of characteristic zero, the representation of a finite group g has a number of convenient properties. first, the representations of g are semisimple (completely reducible). this is a consequence of maschke's theorem, which states that any subrepresentation v of a g -representation w has a g -invariant comp...	wikipedia
cfb85af63969742466b2e9f3a08f7790eba30e3e	group representations are a very important tool in	group representations are a very important tool in the study of finite groups. they also arise in the applications of finite group theory to geometry and crystallography. representations of finite groups exhibit many of the features of the general theory and point the way to other branches and topics in representation ...	wikipedia
083aa3c8d819bd825bb5903ff6f0b4f8c37fe450	representation theory is notable for the number of	representation theory is notable for the number of branches it has, and the diversity of the approaches to studying representations of groups and algebras. although, all the theories have in common the basic concepts discussed already, they differ considerably in detail. the differences are at least 3-fold:	wikipedia
1eda98f6cac3bbae346d2e8f36ebb3e16f3d7d62	in the case of the representation theory of	in the case of the representation theory of the group su(2) (or equivalently, of its complexified lie algebra s l (2; c) {\displaystyle \mathrm {sl} (2;\mathbb {c})}), the decomposition is easy to work out. the irreducible representations are labeled by a parameter l {\displaystyle l} that is a non-negative integer or ...	wikipedia
c4c592a34f1e03a71ff4e536a8718a565778757a	this product can be recognized as the coproduct	this product can be recognized as the coproduct on a coalgebra. in general, the tensor product of irreducible representations is not irreducible; the process of decomposing a tensor product as a direct sum of irreducible representations is known as clebsch–gordan theory.	wikipedia
879e0369a3e656a7cfc54d91f64b7804405ef5af	suppose ϕ 1: g → g l (v	suppose ϕ 1: g → g l (v 1) {\displaystyle \phi _{1}:g\rightarrow \mathrm {gl} (v_{1})} and ϕ 2: g → g l (v 2) {\displaystyle \phi _{2}:g\rightarrow \mathrm {gl} (v_{2})} are representations of a group g {\displaystyle g}. then we can form a representation ϕ 1 ⊗ ϕ 2 {\displaystyle \phi _{1}\otimes \phi _{2}} of g acting...	wikipedia
af7fd3656aa85657cc6eb11f499297136a247627	in favorable circumstances, every finite-dimensional representation is a	in favorable circumstances, every finite-dimensional representation is a direct sum of irreducible representations: such representations are said to be semisimple. in this case, it suffices to understand only the irreducible representations. examples where this " complete reducibility " phenomenon occur include finite ...	wikipedia
363d06ec95fbba2a62782dbf99a59130086f43ab	the direct sum of two representations carries no	the direct sum of two representations carries no more information about the group g than the two representations do individually. if a representation is the direct sum of two proper nontrivial subrepresentations, it is said to be decomposable. otherwise, it is said to be indecomposable.	wikipedia
0a7ce3b04a152cc46283aeed263352d02c3ab9f8	irreducible representations are the building blocks of representation	irreducible representations are the building blocks of representation theory for many groups: if a representation v {\displaystyle v} is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if v {\displaystyle v} is finite-dimensional, then both t...	wikipedia
b367400920c00ebeea8910eaace2725d051e39b2	the definition of an irreducible representation implies schur's	the definition of an irreducible representation implies schur's lemma: an equivariant map α: (v, ψ) → (v ′, ψ ′) {\displaystyle \alpha :(v,\psi)\to (v',\psi ')} between irreducible representations is either the zero map or an isomorphism, since its kernel and image are subrepresentations. in particular, when v = v ′ {\...	wikipedia
e704faedd17b64924ecee436c731f1829809b284	if (v, ψ) {\displaystyle (v,\psi)} is a representation	if (v, ψ) {\displaystyle (v,\psi)} is a representation of (say) a group g {\displaystyle g}, and w {\displaystyle w} is a linear subspace of v {\displaystyle v} that is preserved by the action of g {\displaystyle g} in the sense that for all w ∈ w {\displaystyle w\in w} and g ∈ g {\displaystyle g\in g}, g ⋅ w ∈ w {\dis...	wikipedia
a4ba951447d279a50864e20324f518ce167e2c54	equivariant maps for representations of an associative or	equivariant maps for representations of an associative or lie algebra are defined similarly. if α is invertible, then it is said to be an isomorphism, in which case v and w (or, more precisely, φ and ψ) are isomorphic representations, also phrased as equivalent representations. an equivariant map is often called an int...	wikipedia
2a2d99aeb605dbeda507d1c2347e880088d43633	the vector space v is called the representation	the vector space v is called the representation space of φ and its dimension (if finite) is called the dimension of the representation (sometimes degree, as in). it is also common practice to refer to v itself as the representation when the homomorphism φ is clear from the context; otherwise the notation (v, φ) can be ...	wikipedia
2ad2931ddd8401bfd4b78efe5aa38a89e1802c34	the definition for associative algebras is analogous, except	the definition for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (2.1) is omitted. equation (2.2) is an abstract expression of the associativity of matrix multiplication. this doesn't hold for the matrix commutator and also there is no...	wikipedia
5feec4764ec30f22cbea7748bc3ce35c2ca9f371	a representation of a group g {\displaystyle g}	a representation of a group g {\displaystyle g} or (associative or lie) algebra a {\displaystyle a} on a vector space v {\displaystyle v} is a map Φ: g × v → v or Φ: a × v → v {\displaystyle \phi \colon g\times v\to v\quad {\text{or}}\quad \phi \colon a\times v\to v} with two properties.	wikipedia
50f28bbaa2eb091e660e5836eecdea42518edad9	this generalizes to any field f {\displaystyle \mathbb	this generalizes to any field f {\displaystyle \mathbb {f} } and any vector space v {\displaystyle v} over f {\displaystyle \mathbb {f} }, with linear maps replacing matrices and composition replacing matrix multiplication: there is a group gl (v, f) {\displaystyle {\text{gl}}(v,\mathbb {f})} of automorphisms of v {\di...	wikipedia
f7fbef54ba19827593378d4342305858236d5b09	let v {\displaystyle v} be a vector space	let v {\displaystyle v} be a vector space over a field f {\displaystyle \mathbb {f} }. for instance, suppose v {\displaystyle v} is r n {\displaystyle \mathbb {r} ^{n}} or c n {\displaystyle \mathbb {c} ^{n}}, the standard n -dimensional space of column vectors over the real or complex numbers, respectively. in this ca...	wikipedia