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This gives the following quadratic program (QP) in the variables y ^ 1 , … , y ^ n {\displaystyle {\hat {y}}_{1},\ldots ,{\hat {y}}_{n}}: min ∑ i = 1 n w i ( y ^ i − y i ) 2 {\displaystyle \min \sum _{i=1}^{n}w_{i}({\hat {y}}_{i}-y_{i})^{2}} subject to y ^ i ≤ y ^ j for all ( i , j ) ∈ E {\displaystyle {\hat {y}}_{i}\l... | Wikipedia - Isotonic regression | null | null | null |
Let ( Ω , F , { F t } t ≥ 0 , P ) {\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)} be a filtered probability space. A random variable τ: Ω → {\displaystyle \tau :\Omega \rightarrow } is a stopping time with respect to the filtration { F t } t ≥ 0 {\displaysty... | Wikipedia - Filtration (mathematics) | null | null | null |
In particular, if the underlying probability space is finite (i.e. F {\displaystyle {\mathcal {F}}} is finite), the minimal sets of F τ {\displaystyle {\mathcal {F}}_{\tau }} (with respect to set inclusion) are given by the union over all t ≥ 0 {\displaystyle t\geq 0} of the sets of minimal sets of F t {\displaystyle {... | Wikipedia - Filtration (mathematics) | null | null | null |
Let (A, B, C) mean (Rank, Name, ID) in the Enterprise relation and let (D, E, F) mean (Name, DeptName, ID) in the Department relation All captains of the starship USS Enterprise: In this example, A, B, C denotes both the result set and a set in the table Enterprise. Names of Enterprise crew members who are in Stellar C... | Wikipedia - Domain relational calculus | null | null | null |
Let (Bt)t ≥ 0 be a one-dimensional Brownian motion started from B0 = a > 0, and (Wt)t≥0 be a standard two-dimensional Brownian motion started from W0 = 0 ∈ R2. Define the stopping time at which B first hits the origin, T = inf { t ≥ 0: B t = 0 } {\displaystyle T=\inf\{t\geq 0\colon B_{t}=0\}} . Ray and Knight (independ... | Wikipedia - Local time (mathematics) | null | null | null |
Let (Bt)t ≥ 0 be a standard one-dimensional Brownian motion B0 = 0 ∈ R, and let (Lt)t ≥ 0 be the associated field of local times. Let Ta be the first time at which the local time at zero exceeds a > 0 T a = inf { t ≥ 0: L t 0 > a } . {\displaystyle T_{a}=\inf\{t\geq 0\colon L_{t}^{0}>a\}.} Let (Wt)t ≥ 0 be an independe... | Wikipedia - Local time (mathematics) | null | null | null |
Let (G,K) be a pair consisting of a unimodular locally compact topological group G and a closed subgroup K of G. Then the space of bi-K-invariant continuous functions of compact support Ccan be endowed with a structure of an associative algebra under the operation of convolution. This algebra is denoted H(G//K)and call... | Wikipedia - Hecke algebra of a locally compact group | null | null | null |
Let (M, d) be a metric space, and let B(M) be the Borel σ-algebra on M, the σ-algebra generated by the d-open subsets of M. (For technical reasons, it is also convenient to assume that M is a separable space with respect to the metric d.) Let 1 ≤ p ≤ ∞. The p-th central moment of a measure μ on the measurable space (M,... | Wikipedia - Statistical moment | null | null | null |
Let (M, d) be a metric space, namely a set M with a metric (distance function) d. The open (metric) ball of radius r > 0 centered at a point p in M, usually denoted by Br(p) or B(p; r), is defined by The closed (metric) ball, which may be denoted by Br or B, is defined by Note in particular that a ball (open or closed)... | Wikipedia - Metric ball | null | null | null |
The open balls of a metric space can serve as a base, giving this space a topology, the open sets of which are all possible unions of open balls. This topology on a metric space is called the topology induced by the metric d. Let Br(p) denote the closure of the open ball Br(p) in this topology. While it is always the c... | Wikipedia - Metric ball | null | null | null |
Let (M, g) and (N, h) be Riemannian manifolds. Given a smooth map f from M to N, one can consider its differential df as a section of the vector bundle T *M ⊗ f *TN over M; this is to say that for each p in M, one has a linear map dfp between tangent spaces TpM → Tf(p)N. The vector bundle T *M ⊗ f *TN has a connection ... | Wikipedia - Harmonic map | null | null | null |
Let (S, U) be an instance of the set cover problem with the universe U and the family of subsets S = {Si: i ∈ I}; we assume that U and the index set I are disjoint. Construct a graph G = (V, E) as follows: the set of vertices is V = I ∪ U, there is an edge {i, j} ∈ E between each pair i, j ∈ I, and there is also an edg... | Wikipedia - Total dominating set | null | null | null |
Let (X, T) be a topological space, and let V and W be subsets of X. We say that V is compactly embedded in W, and write V ⊂⊂ W, if V ⊆ Cl(V) ⊆ Int(W), where Cl(V) denotes the closure of V, and Int(W) denotes the interior of W; and Cl(V) is compact. | Wikipedia - Compact embedding | null | null | null |
Let (X, Δ, ∇, ε, η) be a bialgebra with comultiplication Δ, multiplication ∇, unit η, and counit ε. The convolution is a product defined on the endomorphism algebra End(X) as follows. Let φ, ψ ∈ End(X), that is, φ, ψ: X → X are functions that respect all algebraic structure of X, then the convolution φ∗ψ is defined as ... | Wikipedia - Convolution | null | null | null |
The convolution appears notably in the definition of Hopf algebras (Kassel 1995, §III.3). A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism S such that S ∗ id X = id X ∗ S = η ∘ ε . {\displaystyle S*\operatorname {id} _{X}=\operatorname {id} _{X}*S=\eta \circ \varepsilon .} | Wikipedia - Convolution | null | null | null |
Let (b1, b2, b3) be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal. However, they are required to be linearly independent. Then a vector v can be expressed as: 27 The components vk are the contravariant components of the vector v.... | Wikipedia - Tensors in curvilinear coordinates | null | null | null |
Let (e1, e2, e3) be the usual Cartesian basis vectors for the Euclidean space of interest and let where Fi is a second-order transformation tensor that maps ei to bi. Then, From this relation we can show that Let J := det F {\displaystyle J:=\det {\boldsymbol {F}}} be the Jacobian of the transformation. Then, from the ... | Wikipedia - Tensors in curvilinear coordinates | null | null | null |
Recall that where A is a, yet undetermined, constant. Then This observation leads to the relations In index notation, where ε i j k {\displaystyle \varepsilon _{ijk}} is the usual permutation symbol. We have not identified an explicit expression for the transformation tensor F because an alternative form of the mapping... | Wikipedia - Tensors in curvilinear coordinates | null | null | null |
Let 1 {\displaystyle {\boldsymbol {\mathit {1}}}} be the second order identity tensor. Then the derivative of this tensor with respect to a second order tensor A {\displaystyle {\boldsymbol {A}}} is given by This is because 1 {\displaystyle {\boldsymbol {\mathit {1}}}} is independent of A {\displaystyle {\boldsymbol {A... | Wikipedia - Tensor derivative (continuum mechanics) | null | null | null |
Let 6 , 3 , 3 , 3 , 3 , 2 , 2 , 2 , 2 , 1 , 1 {\displaystyle 6,3,3,3,3,2,2,2,2,1,1} be a nonincreasing, finite degree sequence of nonnegative integers. To test whether this degree sequence is graphic, we apply the Havel-Hakimi algorithm: First, we remove the vertex with the highest degree — in this case, 6 {\displaysty... | Wikipedia - Havel–Hakimi algorithm | null | null | null |
We continue this removal to get 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 {\displaystyle 1,1,1,1,1,1,1,1} , and then 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 {\displaystyle 0,0,0,0,0,0,0,0} . This sequence is clearly graphic, as it is the simple graph of 8 {\displaystyle 8} isolated vertices. To show an example of a non-graphic sequence, let... | Wikipedia - Havel–Hakimi algorithm | null | null | null |
Applying the algorithm, we first remove the degree 6 {\displaystyle 6} vertex and all its incident edges to get 4 , 4 , 3 , 2 , 1 , 0 {\displaystyle 4,4,3,2,1,0} . Already, we know this degree sequence is not graphic, since it claims to have 6 {\displaystyle 6} vertices with one vertex not adjacent to any of the other ... | Wikipedia - Havel–Hakimi algorithm | null | null | null |
Thus, the sequence is not graphic. For the sake of the algorithm, if we were to reiterate the process, we would get 3 , 2 , 1 , 0 , 0 {\displaystyle 3,2,1,0,0} which is yet more clearly not graphic. One vertex claims to have a degree of 3 {\displaystyle 3} , and yet only two other vertices have neighbors. Thus the sequ... | Wikipedia - Havel–Hakimi algorithm | null | null | null |
Let A = (a1,a2) and B = (b1,b2) be quaternion algebras over F. The Albert form for A, B is q = ⟨ − a 1 , − a 2 , a 1 a 2 , b 1 , b 2 , − b 1 b 2 ⟩ . {\displaystyle q=\left\langle {-a_{1},-a_{2},a_{1}a_{2},b_{1},b_{2},-b_{1}b_{2}}\right\rangle \ .} It can be regarded as the difference in the Witt ring of the ternary for... | Wikipedia - Biquaternion algebra | null | null | null |
Let A = K be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings; see Hilbert's syzygy theorem. More generally, if R is a Noetherian ring of finite global dimensio... | Wikipedia - Global homological dimension | null | null | null |
In particular if a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same. The triangular matrix ring {\displaystyle {\begin{bmatrix}\mathbb {Z} &\mathbb {Q} \\0&\mathbb {Q} \end{bmatrix}}} has right global dimension 1, weak global dimension 1, but le... | Wikipedia - Global homological dimension | null | null | null |
Let A = a 1 a 2 . . . | Wikipedia - Smith-Waterman algorithm | null | null | null |
a n {\displaystyle A=a_{1}a_{2}...a_{n}} and B = b 1 b 2 . . . | Wikipedia - Smith-Waterman algorithm | null | null | null |
b m {\displaystyle B=b_{1}b_{2}...b_{m}} be the sequences to be aligned, where n {\displaystyle n} and m {\displaystyle m} are the lengths of A {\displaystyle A} and B {\displaystyle B} respectively. Determine the substitution matrix and the gap penalty scheme. s ( a , b ) {\displaystyle s(a,b)} - Similarity score of t... | Wikipedia - Smith-Waterman algorithm | null | null | null |
The size of the scoring matrix is ( n + 1 ) ∗ ( m + 1 ) {\displaystyle (n+1)*(m+1)} . The matrix uses 0-based indexing. H k 0 = H 0 l = 0 f o r 0 ≤ k ≤ n a n d 0 ≤ l ≤ m {\displaystyle H_{k0}=H_{0l}=0\quad for\quad 0\leq k\leq n\quad and\quad 0\leq l\leq m} Fill the scoring matrix using the equation below. | Wikipedia - Smith-Waterman algorithm | null | null | null |
H i j = max { H i − 1 , j − 1 + s ( a i , b j ) , max k ≥ 1 { H i − k , j − W k } , max l ≥ 1 { H i , j − l − W l } , 0 ( 1 ≤ i ≤ n , 1 ≤ j ≤ m ) {\displaystyle H_{ij}=\max {\begin{cases}H_{i-1,j-1}+s(a_{i},b_{j}),\\\max _{k\geq 1}\{H_{i-k,j}-W_{k}\},\\\max _{l\geq 1}\{H_{i,j-l}-W_{l}\},\\0\end{cases}}\qquad (1\leq i\l... | Wikipedia - Smith-Waterman algorithm | null | null | null |
Let A B C D {\displaystyle ABCD} be a Saccheri quadrilateral having base A B , {\displaystyle AB,} summit C D , {\displaystyle CD,} and legs C A {\displaystyle CA} and D B . {\displaystyle DB.} The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry: The summit angles C {\displaystyle C}... | Wikipedia - Saccheri quadrilateral | null | null | null |
Two Saccheri quadrilaterals are congruent if: the base segments and summit angles are congruent the summit segments and summit angles are congruent. The line segment joining the midpoint of the base and the midpoint of the summit: Is perpendicular to the base and the summit, is the only line of symmetry of the quadrila... | Wikipedia - Saccheri quadrilateral | null | null | null |
Let A and B be algebraic structures of a given type and let f be a homomorphism of that type from A to B. Then the kernel of f is the subset of the direct product A × A consisting of all those ordered pairs of elements of A whose components are both mapped by f to the same element in B. The kernel is usually denoted ke... | Wikipedia - Kernel (group theory) | null | null | null |
Since f is a function, the elements of the form (a, a) must belong to the kernel. The homomorphism f is injective if and only if its kernel is exactly the diagonal set {(a, a): a ∈ A}. It is easy to see that ker f is an equivalence relation on A, and in fact a congruence relation. | Wikipedia - Kernel (group theory) | null | null | null |
Thus, it makes sense to speak of the quotient algebra A/(ker f). The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). Note that the definition of kernel here (as in the monoid example) doesn't depend on the a... | Wikipedia - Kernel (group theory) | null | null | null |
Let A and B be different sets and B ( A , B ) {\displaystyle {\mathcal {B}}(A,B)} the collection of heterogeneous relations between them. For p , q , r ∈ B ( A , B ) {\displaystyle p,q,r\in {\mathcal {B}}(A,B)} define the ternary operator = p q T r {\displaystyle =pq^{T}r} where qT is the converse relation of q. The r... | Wikipedia - Heap (mathematics) | null | null | null |
Let A and B be two types of data item. Then the index of dissimilarity is D = 1 2 ∑ i = 1 K | A i A − B i B | {\displaystyle D={\frac {1}{2}}\sum _{i=1}^{K}\left|{\frac {A_{i}}{A}}-{\frac {B_{i}}{B}}\right|} where A = ∑ i = 1 K A i {\displaystyle A=\sum _{i=1}^{K}A_{i}} B = ∑ i = 1 K B i {\displaystyle B=\sum _{i=1}^{K... | Wikipedia - Lieberson's isolation index | null | null | null |
This index is biased as its expectation under a uniform distribution is > 0. A modification of this index has been proposed by Gorard and Taylor. Their index (GT) is G T = D ( 1 − A A + B ) {\displaystyle GT=D\left(1-{\frac {A}{A+B}}\right)} | Wikipedia - Lieberson's isolation index | null | null | null |
Let A be a Banach space, X be a topological Hausdorff space. Define B := A × X {\displaystyle B:=A\times X} and π: B → X {\displaystyle \pi \colon B\to X} by π ( a , x ) := x {\displaystyle \pi (a,x):=x} . Then ( B , π ) {\displaystyle (B,\pi )} is a Banach bundle, called the trivial bundle | Wikipedia - Banach bundle (non-commutative geometry) | null | null | null |
Let A be a C*-algebra acting on a Hilbert space H. For ρ in A* and S in Φ(A)−, let Sρ in A* be defined by Sρ(a) = ρ∘Φ−1(Φ(a)S) for all a in A. If P is the projection in the above commutative diagram when π:A → B(H) is the inclusion mapping, then ρ in A* is ultraweakly continuous if and only if ρ = Pρ. A functional ρ in... | Wikipedia - Universal representation (C*-algebra) | null | null | null |
Let A be a C*-algebra and πU be its universal representation, acting on Hilbert space HU. The image of πU, πU(A), is a C*-subalgebra of bounded operators on HU. The enveloping von Neumann algebra of A is the closure of πU(A) in the weak operator topology. It is sometimes denoted by A′′. | Wikipedia - Enveloping von Neumann algebra | null | null | null |
Let A be a Euclidean Hurwitz algebra and let Mn(A) be the algebra of n-by-n matrices over A. It is a unital nonassociative algebra with an involution given by ( x i j ) ∗ = ( x j i ∗ ) . {\displaystyle \displaystyle {(x_{ij})^{*}=(x_{ji}^{*}).}} The trace Tr(X) is defined as the sum of the diagonal elements of X and th... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
{\displaystyle \operatorname {Tr} _{\mathbf {R} }XY=\operatorname {Tr} _{\mathbf {R} }YX,\qquad \operatorname {Tr} _{\mathbf {R} }(XY)Z=\operatorname {Tr} _{\mathbf {R} }X(YZ).} These are immediate consequences of the known identities for n = 1. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
In A define the associator by = a ( b c ) − ( a b ) c . {\displaystyle \displaystyle {=a(bc)-(ab)c.}} It is trilinear and vanishes identically if A is associative. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Since A is an alternative algebra = 0 and = 0. Polarizing it follows that the associator is antisymmetric in its three entries. Furthermore, if a, b or c lie in R then = 0. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
These facts imply that M3(A) has certain commutation properties. In fact if X is a matrix in M3(A) with real entries on the diagonal then = a I , {\displaystyle \displaystyle {=aI,}} with a in A. In fact if Y = , then y i j = ∑ k , ℓ . {\displaystyle \displaystyle {y_{ij}=\sum _{k,\ell }.}} | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Since the diagonal entries of X are real, the off diagonal entries of Y vanish. Each diagonal entry of Y is a sum of two associators involving only off diagonal terms of X. Since the associators are invariant under cyclic permutations, the diagonal entries of Y are all equal. Let Hn(A) be the space of self-adjoint elem... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Theorem. Hn(A) is a Euclidean Jordan algebra if A is associative (the real numbers, complex numbers or quaternions) and n ≥ 3 or if A is nonassociative (the octonions) and n = 3. The exceptional Jordan algebra H3(O) is called the Albert algebra after A.A. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Albert. To check that Hn(A) satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a symmetric bilinear form with (X, X) = Σ ‖ xij ‖2. So it is an inner product. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
It satisfies the associativity property (Z∘X, Y) = (X, Z∘Y) because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators L(X) defined by L(X)Y = X∘Y: = 0. {\displaystyle \displaystyle {=0.}} | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
This is easy to check when A is associative, since Mn(A) is an associative algebra so a Jordan algebra with X∘Y = 1/2(X Y + Y X). When A = O and n = 3 a special argument is required, one of the shortest being due to Freudenthal (1951).In fact if T is in H3(O) with Tr T = 0, then D ( X ) = T X − X T {\displaystyle \disp... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
{\displaystyle (D(X),X^{2})=0.} Polarizing yields: ( D ( X ) , Y ∘ Z ) + ( D ( Y ) , Z ∘ X ) + ( D ( Z ) , X ∘ Y ) = 0. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
{\displaystyle (D(X),Y\circ Z)+(D(Y),Z\circ X)+(D(Z),X\circ Y)=0.} Setting Z = 1, shows that D is skew-adjoint. The derivation property D(X∘Y) = D(X)∘Y + X∘D(Y) follows by this and the associativity property of the inner product in the identity above. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
With A and n as in the statement of the theorem, let K be the group of automorphisms of E = Hn(A) leaving invariant the inner product. It is a closed subgroup of O(E) so a compact Lie group. Its Lie algebra consists of skew-adjoint derivations. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Freudenthal (1951) showed that given X in E there is an automorphism k in K such that k(X) is a diagonal matrix. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies the Jordan condition, since Jordan products by real diagonal matrices commute on Mn(A) for a... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Since K preserves the sums of all the squares, this is equivalent to maximizing the sums of the squares of the norms of the diagonal terms of k(X). Replacing X by k X, it can be assumed that the maximum is attained at X. Since the symmetric group Sn, acting by permuting the coordinates, lies in K, if X is not diagonal,... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
The derivative of x11(t) at t = 0 is the (1, 1) coordinate of , i.e. a* x21 + x12 a = 2(x21, a). This derivative is non-zero if a = x21. On the other hand, the group kt preserves the real-valued trace. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Since it can only change x11 and x22, it preserves their sum. However, on the line x + y =constant, x2 + y2 has no local maximum (only a global minimum), a contradiction. Hence X must be diagonal. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Let A be a Hopf algebra, and let M and N be A-modules. Then, M ⊗ N is also an A-module, with a ( m ⊗ n ) := Δ ( a ) ( m ⊗ n ) = ( a 1 ⊗ a 2 ) ( m ⊗ n ) = ( a 1 m ⊗ a 2 n ) {\displaystyle a(m\otimes n):=\Delta (a)(m\otimes n)=(a_{1}\otimes a_{2})(m\otimes n)=(a_{1}m\otimes a_{2}n)} for m ∈ M, n ∈ N and Δ(a) = (a1, a2). ... | Wikipedia - Hopf algebras | null | null | null |
Let A be a commutative C*-algebra and let X be the spectrum of A. Let γ: A → C 0 ( X ) {\displaystyle \gamma :A\to C_{0}(X)} be the Gelfand representation defined above. Theorem. The Gelfand map γ is an isometric *-isomorphism from A onto C0(X). See the Arveson reference below. | Wikipedia - Gelfand representation | null | null | null |
The spectrum of a commutative C*-algebra can also be viewed as the set of all maximal ideals m of A, with the hull-kernel topology. (See the earlier remarks for the general, commutative Banach algebra case.) For any such m the quotient algebra A/m is one-dimensional (by the Gelfand-Mazur theorem), and therefore any a i... | Wikipedia - Gelfand representation | null | null | null |
Let A be a finite-dimensional algebra over a field k. Then A is an Artinian ring. | Wikipedia - Linear associative algebra | null | null | null |
Let A be a finite-dimensional central simple algebra over a field F. Then A is said to be cyclic if it contains a strictly maximal subfield E such that E/F is a cyclic field extension (i.e., the Galois group is a cyclic group). | Wikipedia - Cyclic algebra | null | null | null |
Let A be a finite-dimensional complex semisimple unital Jordan algebra. If T is an operator on A, let Tt be its transpose with respect to the trace form. Thus L(a)t = L(a), Q(a)t = Q(a), R(a,b)t = R(b,a) and B(a,b)t = B(b,a). | Wikipedia - Mutation (Jordan algebra) | null | null | null |
The structure group of A consists of g in GL(A) such that Q ( g a ) = g Q ( a ) g t . {\displaystyle \displaystyle {Q(ga)=gQ(a)g^{t}.}} They form a group Γ(A). | Wikipedia - Mutation (Jordan algebra) | null | null | null |
The automorphism group Aut A of A consists of invertible complex linear operators g such that L(ga) = gL(a)g−1 and g1 = 1. Since an automorphism g preserves the trace form, g−1 = gt. The structure group is closed under taking transposes g ↦ gt and adjoints g ↦ g*. | Wikipedia - Mutation (Jordan algebra) | null | null | null |
The structure group contains the automorphism group. The automorphism group can be identified with the stabilizer of 1 in the structure group. If a is invertible, Q(a) lies in the structure group. | Wikipedia - Mutation (Jordan algebra) | null | null | null |
If g is in the structure group and a is invertible, ga is also invertible with (ga)−1 = (gt)−1a−1. The structure group Γ(A) acts transitively on the set of invertible elements in A. Every g in Γ(A) has the form g = h Q(a) with h an automorphism and a invertible.The complex Jordan algebra A is the complexification of a ... | Wikipedia - Mutation (Jordan algebra) | null | null | null |
The identity component of Γu(A) is denoted by K. It is a connected closed subgroup of U(A). The stabilizer of 1 in Γu(A) is Aut E. Every g in Γu(A) has the form g = h Q(u) with h in Aut E and u invertible in A with u* = u−1. Γ(A) is the complexification of Γu(A). | Wikipedia - Mutation (Jordan algebra) | null | null | null |
The set S of invertible elements u in A such that u* = u−1 can be characterized equivalently either as those u for which L(u) is a normal operator with uu* = 1 or as those u of the form exp ia for some a in E. In particular S is connected. The identity component of Γu(A) acts transitively on S Given a Jordan frame (ei)... | Wikipedia - Mutation (Jordan algebra) | null | null | null |
{\displaystyle \displaystyle {g(a,b)=(ga,(g^{t})^{-1}b).}} Then (x,y) is quasi-invertible if and only if (gx,(gt)−1y) is quasi-invertible and g ( x y ) = ( g x ) ( g t ) − 1 y . {\displaystyle \displaystyle {g(x^{y})=(gx)^{(g^{t})^{-1}y}.}} | Wikipedia - Mutation (Jordan algebra) | null | null | null |
In fact the covariance relations for g with Q and the inverse imply that g B ( x , y ) g − 1 = B ( g x , ( g t ) − 1 y ) {\displaystyle \displaystyle {gB(x,y)g^{-1}=B(gx,(g^{t})^{-1}y)}} if x is invertible and so everywhere by density. In turn this implies the relation for the quasi-inverse. If a is invertible then Q(a... | Wikipedia - Mutation (Jordan algebra) | null | null | null |
So both types of operators act on X. The defining relations for the structure group show that it is a closed subgroup of g 0 {\displaystyle {\mathfrak {g}}_{0}} of GL(A). Since Q(ea) = e2L(a), the corresponding complex Lie algebra contains the operators L(a). The commutators span the complex Lie algebra of derivations... | Wikipedia - Mutation (Jordan algebra) | null | null | null |
Let A be a finite-dimensional complex semisimple unital Jordan algebra. The group SL(2,C) acts by Möbius transformation on the Riemann sphere C ∪ {∞}, the one-point compactification of C. If g in SL(2,C) is given by the matrix g = ( α β γ δ ) , {\displaystyle \displaystyle {g={\begin{pmatrix}\alpha &\beta \\\gamma &\de... | Wikipedia - Isotope (Jordan algebra) | null | null | null |
It is also generated by the lower (or upper) unitriangular matrices, the diagonal matrices and the matrix J = ( 0 1 − 1 0 ) . {\displaystyle \displaystyle {J={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}.}} The matrix J corresponds to the Möbius transformation j(z) = −z−1 and can be written J = ( 1 0 − 1 1 ) ( 1 1 0 1 ) ( 1 ... | Wikipedia - Isotope (Jordan algebra) | null | null | null |
{\displaystyle \displaystyle {J={\begin{pmatrix}1&0\\-1&1\end{pmatrix}}{\begin{pmatrix}1&1\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\-1&1\end{pmatrix}}.}} The Möbius transformations fixing ∞ are just the upper triangular matrices. | Wikipedia - Isotope (Jordan algebra) | null | null | null |
If g does not fix ∞, it sends ∞ to a finite point a. But then g can be composed with an upper unitriangular to send a to 0 and then with J to send 0 to infinity. For an element a of A, the action of g in SL(2,C) is defined by the same formula g ( a ) = ( α a + β 1 ) ( γ a + δ 1 ) − 1 . {\displaystyle \displaystyle {g(a... | Wikipedia - Isotope (Jordan algebra) | null | null | null |
This defines an element of C provided that γa + δ1 is invertible in A. The action is thus defined everywhere on A if g is upper triangular. On the other hand, the action on X is simple to define for lower triangular matrices. For diagonal matrices g with diagonal entries α and α−1, g(a,b) = (α2a, α−2b) is a well-define... | Wikipedia - Isotope (Jordan algebra) | null | null | null |
For lower unitriangular matrices, with off-diagonal parameter γ, define g(a,b) = (a,b − γ1). Again this is holomorphic on A2 and passes to the quotient X. When b = 0 and γ ≠ 0, g ( a: 0 ) = ( a: − γ ) = ( a − γ: 0 ) = ( a ( γ a + 1 ) − 1: 0 ) {\displaystyle \displaystyle {g(a:0)=(a:-\gamma )=(a^{-\gamma }:0)=(a(\gamma ... | Wikipedia - Isotope (Jordan algebra) | null | null | null |
A simpler method is to note that the operator J can be implemented directly using its intertwining relations with the unitary structure group.In fact on the invertible elements in A, the operator j(a) = −a−1 satisfies j(ga) = (gt)−1j(a). To define a biholomorphism j on X such that j ∘ g = (gt)−1 ∘ j, it is enough to de... | Wikipedia - Isotope (Jordan algebra) | null | null | null |
The computation of j in the associative commutative algebra Ae is straightforward since it is a direct product. For c = Σ αi ei and d = Σ βi ei, the Bergman operator on Ae has determinant det B(c,d) = Π(1 − αiβi)2. | Wikipedia - Isotope (Jordan algebra) | null | null | null |
In particular det B(c,d − λ) ≠ 0 for some λ ≠ 0. So that (c,d) is equivalent to (x,λ). Let μ = −λ−1. | Wikipedia - Isotope (Jordan algebra) | null | null | null |
On A, for a dense set of a, the pair (a,λ) is equivalent to (b,0) with b invertible. Then (−b−1,0) is equivalent to (μ − μ2a,μ). Since a ↦ μ − μ2a is holomorphic it follows that j has a unique continuous extension to X such that j ∘ g = (gt)−1 ∘ j for g in Γ(A), the extension is holomorphic and for λ ≠ 0, μ = −λ−1 The ... | Wikipedia - Isotope (Jordan algebra) | null | null | null |
A direct algebraic construction is given in Dineen, Mackey & Mellon (1999). This action of SL(2,C) is compatible with inclusions. More generally if e1, ..., em is a Jordan frame, there is an action of SL(2,C)m on Ae which extends to A. If c = Σ γiei and b = Σ βiei, then S(c) and T(b) give the action of the product of t... | Wikipedia - Isotope (Jordan algebra) | null | null | null |
Let A be a finite-dimensional complex semisimple unital Jordan algebra. There is a transitive holomorphic action of a complex matrix group G on the compact complex manifold X. Koecher (1967) described G analogously to SL(2,C) in terms of generators and relations. G acts on the corresponding finite-dimensional Lie algeb... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
The identity component H of the compact group acts transitively on X, so that X can be identified as a Hermitian symmetric space of compact type.The group G is generated by three types of holomorphic transformation on X: Operators W corresponding to elements W in Γ(A) given by W(a,b) = (Wa, (Wt)−1b). These were already... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
Operators Sc defined by Sc(a,b) = (a,b + c). These are the analogue of lower unitriangular matrices and form a subgroup isomorphic to the additive group of A, with the given parametrization. Again these act holomorphically on A2 and the action passes to the quotient X. On A the action is given by a ↦ ac if (a,c) is qua... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
The transformation j corresponding to J in SL(2,C). It was constructed above as part of the action of PSL(2,C) = SL(2,C)/{±I} on X. On invertible elements in A it is given by a ↦ −a−1.The operators W normalize the group of operators Sc. Similarly the operator j normalizes the structure group, j ∘ W = (Wt)−1 ∘ j. The op... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
This group is normalized by the operators W of the structure group. The operator Tc acts on A as a ↦ a + c. If c is a scalar the operators Sc and Tc coincide with the operators corresponding to lower and upper unitriangular matrices in SL(2,C). Accordingly, there is a relation j = S1 ∘ T1 ∘ S1 and PSL(2,C) is a subgrou... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
Indeed, SbTa(0:0) = (a:b). Let G−1 and G+1 be the complex Abelian groups formed by the symmetries Tc and Sc respectively. | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
Let G0 = Γ(A). The two expressions for G are equivalent as follows by conjugating by j. For a invertible, Hua's identity can be rewritten Q ( a ) = T a ∘ j ∘ T a − 1 ∘ j ∘ T a ∘ j . | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
{\displaystyle \displaystyle {Q(a)=T_{a}\circ j\circ T_{a^{-1}}\circ j\circ T_{a}\circ j.}} Moreover, j = S1 ∘ T1 ∘ S1 and Sc = j ∘ T−c ∘ j.The convariance relations show that the elements of G fall into sets G0G1, G0G1jG1, G0G1jG1jG1, G0G1jG1jG1jG1. ... The first expression for G follows once it is established that no... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
For this it suffices to show that j ∘ G1 ∘j ∘ G1 ∘j ⊆ G0 G1 ∘j ∘ G1 ∘ j ∘ G1.For then if there are three or more occurrences of j, the number can be recursively reduced to two. Given a, b in A, pick λ ≠ 0 so that c = a − λ and d = b − λ−1 are invertible. Then j T a j T b j = j T c T λ j T λ − 1 T d ∘ j = λ 2 j T c j T ... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
Let A be a finite-dimensional complex unital Jordan algebra which is semisimple, i.e. the trace form Tr L(ab) is non-degenerate. Let X be the quotient of A×A by the equivalence relation. Let Xb be the subset of X of classes (a:b). The map φb:Xb → A, (a:b) ↦ a is injective. | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
A subset U of X is defined to be open if and only if U ∩ Xb is open for all b. The transition maps of the atlas with charts φb are given by φ c b = φ c ∘ φ b − 1: φ b ( X b ∩ X c ) → φ c ( X b ∩ X c ) . {\displaystyle \displaystyle {\varphi _{cb}=\varphi _{c}\circ \varphi _{b}^{-1}:\varphi _{b}(X_{b}\cap X_{c})\rightar... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
{\displaystyle \displaystyle {\varphi _{cb}^{\prime }(a)=B(a,b-c)^{-1}.}} This defines the structure of a complex manifold on X because φdc ∘ φcb = φdb on φb(Xb ∩ Xc ∩ Xd). | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
Indeed, all the polynomial functions pi(b) = det B(ai,bi − b) are non-trivial since pi(bi) = 1. Therefore, there is a b such that pi(b) ≠ 0 for all i, which is precisely the criterion for (ai:bi) to lie in Xb. | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
Loos (1977) uses the Bergman operators to construct an explicit biholomorphism between X and a closed smooth algebraic subvariety of complex projective space. This implies in particular that X is compact. There is a more direct proof of compactness using symmetry groups. | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
Given a Jordan frame (ei) in E, for every a in A there is a k in U = Γu(A) such that a=k(Σ αi ei) with αi ≥ 0 (and αi > 0 if a is invertible). In fact, if (a,b) is in X then it is equivalent to k(c,d) with c and d in the unital Jordan subalgebra Ae = ⊕ Cei, which is the complexification of Ee = ⊕ Rei. Let Z be the comp... | Wikipedia - Hua's identity (Jordan algebra) | null | null | null |
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