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searching for Symmetric bilinear form 39 found (100 total)

Quasi-Frobenius Lie algebra (314 words) [view diff] exact match in snippet view article find links to article

({\mathfrak {g}},[\,\,\,,\,\,\,])} equipped with a nondegenerate skew-symmetric bilinear form β : g × g → k {\displaystyle \beta :{\mathfrak {g}}\times {\mathfrak

Symplectic representation (283 words) [view diff] exact match in snippet view article find links to article

preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form ω : V × V → F {\displaystyle \omega \colon V\times V\to \mathbb

Polarization (Lie algebra) (803 words) [view diff] exact match in snippet view article

polarization is the maximal totally isotropic subspace of a certain skew-symmetric bilinear form on a Lie algebra. The notion of polarization plays an important

Symplectic matrix (2,558 words) [view diff] exact match in snippet view article find links to article

space V {\displaystyle V} equipped with a nondegenerate, skew-symmetric bilinear form ω {\displaystyle \omega } called the symplectic form. A symplectic

Surgery obstruction (1,052 words) [view diff] exact match in snippet view article find links to article

K_{k}({\tilde {M}})} . This defines a symmetric bilinear form in case k = 2 l {\displaystyle k=2l} and a skew-symmetric bilinear form in case k = 2 l + 1 {\displaystyle

Invariant convex cone (3,569 words) [view diff] exact match in snippet view article find links to article

R2n diagonalizing the original positive symmetric bilinear form. Thus every positive symmetric bilinear form lies in the orbit of a diagonal form under

Yang–Mills–Higgs equations (468 words) [view diff] exact match in snippet view article find links to article

{\displaystyle \langle \cdot ,\cdot \rangle } is an invariant symmetric bilinear form on the adjoint bundle. This is sometimes written as tr {\displaystyle

Witt's theorem (564 words) [view diff] exact match in snippet view article find links to article

different from 2 together with a non-degenerate symmetric or skew-symmetric bilinear form. If f : U → U' is an isometry between two subspaces of V then f

Symplectic group (3,109 words) [view diff] exact match in snippet view article find links to article

vector space over the field F which preserve a non-degenerate skew-symmetric bilinear form. Such a vector space is called a symplectic vector space, and the

Perfect obstruction theory (596 words) [view diff] exact match in snippet view article find links to article

theory is a perfect obstruction theory together with nondegenerate symmetric bilinear form. Example: Let f be a regular function on a smooth variety (or stack)

Outermorphism (1,653 words) [view diff] exact match in snippet view article find links to article

{\displaystyle b} , where ⋅ {\displaystyle \cdot } is the nondegenerate symmetric bilinear form (scalar product of vectors). This results in the property that

Skew-Hamiltonian matrix (311 words) [view diff] exact match in snippet view article find links to article

matrix is a specific type of matrix that corresponds to a skew-symmetric bilinear form on a symplectic vector space. Let V {\displaystyle V} be a vector

Hermitian manifold (1,548 words) [view diff] exact match in snippet view article find links to article

{\displaystyle g={1 \over 2}\left(h+{\bar {h}}\right).} The form g is a symmetric bilinear form on TMC, the complexified tangent bundle. Since g is equal to its

Donaldson's theorem (1,244 words) [view diff] exact match in snippet view article find links to article

diagonalizable. Michael Freedman had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifold

Finsler manifold (1,952 words) [view diff] exact match in snippet view article find links to article

at v is positive definite. Here the Hessian of F2 at v is the symmetric bilinear form g v ( X , Y ) := 1 2 ∂ 2 ∂ s ∂ t [ F ( v + s X + t Y ) 2 ] | s

Coxeter complex (1,258 words) [view diff] exact match in snippet view article find links to article

{\displaystyle (e_{s})_{s\in S}} , which is equipped with the symmetric bilinear form B ( e s , e t ) = − cos ⁡ ( π m ( s , t ) ) {\displaystyle B(e_{s}

Vector (mathematics and physics) (2,684 words) [view diff] no match in snippet view article

vector space, a vector space V equipped with a non-degenerate, skew-symmetric, bilinear form Topological vector space, a blend of topological structure with

Fourier transform on finite groups (2,052 words) [view diff] exact match in snippet view article find links to article

{\displaystyle U} associated to a G {\displaystyle G} -invariant symmetric bilinear form as U = A A ∗ {\displaystyle U=AA^{*}} , where ∗ {\displaystyle

Topological Yang–Mills theory (672 words) [view diff] exact match in snippet view article find links to article

real parameter. tr {\displaystyle {\text{tr}}} is an invariant, symmetric bilinear form on g {\displaystyle {\mathfrak {g}}} . It is denoted tr {\displaystyle

Schur's lemma (Riemannian geometry) (2,544 words) [view diff] exact match in snippet view article

d\kappa ={\frac {n}{2}}d\kappa .} Let B {\displaystyle B} be a symmetric bilinear form on an n {\displaystyle n} -dimensional inner product space ( V

Building (mathematics) (3,216 words) [view diff] exact match in snippet view article

with automorphisms of the Dynkin diagram. Taking the standard symmetric bilinear form with orthonormal basis vi, the map sending a lattice to its dual

Virasoro algebra (4,140 words) [view diff] exact match in snippet view article find links to article

universal enveloping algebra. Then the Shapovalov form is the symmetric bilinear form on the Verma module V c , h {\displaystyle {\mathcal {V}}_{c,h}}

Dirac structure (940 words) [view diff] exact match in snippet view article find links to article

{\displaystyle D=D^{\perp }} , where orthogonality is with respect to the symmetric bilinear form on V × V ∗ {\displaystyle V\times V^{*}} given by ⟨ ( u , α )

Simple Lie group (2,384 words) [view diff] exact match in snippet view article find links to article

exceptional groups, the exponent −26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup. It is

Schwarz triangle (10,933 words) [view diff] exact match in snippet view article find links to article

es, et be a basis for a 3-dimensional real vector space V with symmetric bilinear form Λ such that Λ ( e s , e t ) = − A , Λ ( e t , e r ) = − B , Λ (

Multilinear form (4,632 words) [view diff] no match in snippet view article find links to article

to as a bilinear form. A familiar and important example of a (symmetric) bilinear form is the standard inner product (dot product) of vectors. An important

Inner product space (7,357 words) [view diff] exact match in snippet view article find links to article

an inner product on a real vector space is a positive-definite symmetric bilinear form. The binomial expansion of a square becomes ⟨ x + y , x + y ⟩ =

Heisenberg group (5,924 words) [view diff] exact match in snippet view article find links to article

finite-dimensional real symplectic vector space (so ω is a nondegenerate skew symmetric bilinear form on V). The Heisenberg group H(V) on (V, ω) (or simply V for brevity)

Covariance and contravariance of vectors (7,132 words) [view diff] exact match in snippet view article find links to article

finite-dimensional vector space V over a field K with a non-degenerate symmetric bilinear form g : V × V → K (which may be referred to as the metric tensor),

Glossary of classical algebraic geometry (11,193 words) [view diff] exact match in snippet view article find links to article

projective space of a vector space is essentially a non-degenerate skew-symmetric bilinear form, up to multiplication by scalars. See also polarity. (Semple &

Mutation (Jordan algebra) (15,817 words) [view diff] exact match in snippet view article

b_{2})+(b_{1},a_{2})+\beta (T_{1},T_{2}),}} where β(T1,T2) is the symmetric bilinear form defined by β ( R ( a , b ) , R ( c , d ) ) = ( R ( a , b ) c ,

Grassmannian (8,402 words) [view diff] exact match in snippet view article find links to article

\operatorname {SO} (n-k)).} Given a real or complex nondegenerate symmetric bilinear form Q {\displaystyle Q} on the n {\displaystyle n} -dimensional space

Spinor (9,924 words) [view diff] exact match in snippet view article find links to article

be a finite-dimensional complex vector space with nondegenerate symmetric bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V

W-algebra (5,488 words) [view diff] exact match in snippet view article find links to article

} denotes the Killing form. This induces a non-degenerate anti-symmetric bilinear form on the −1 graded piece by the rule: ω χ ( x , y ) = χ ( [ x , y

Symmetric cone (16,613 words) [view diff] exact match in snippet view article find links to article

adjoint relation L(a*) = L(a)* holds for a in EC. Similarly the symmetric bilinear form β(a,b) = (a,b*) satisfies β(ab,c) = β(b,ac). If the inner product

Derivations of the Lorentz transformations (11,515 words) [view diff] exact match in snippet view article find links to article

p ) {\displaystyle (n,p)} . Suppose g {\displaystyle g} is a symmetric bilinear form on V {\displaystyle V} such that the null set of the associated

Differential geometry of surfaces (17,641 words) [view diff] exact match in snippet view article find links to article

with a Riemannian metric. Note that in some more recent texts the symmetric bilinear form on the right hand side is referred to as the second fundamental

Oscillator representation (21,532 words) [view diff] exact match in snippet view article find links to article

Q ( a + b ) − 1 {\displaystyle (a,b)=Q(a)Q(b)Q(a+b)^{-1}} is a symmetric bilinear form on A that is non-degenerate, so permits an identification between

Lie algebra extension (17,708 words) [view diff] exact match in snippet view article find links to article

model with Lie algebra u(1) ⊕ su(2) ⊕ su(3). The Killing form is a symmetric bilinear form on g defined by K ( G 1 , G 2 ) = t r a c e ( a d G 1 a d G 2 )