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searching for Symmetric bilinear form 40 found (98 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

Intersection form of a 4-manifold (966 words) [view diff] exact match in snippet view article find links to article

intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd (co)homology group of the 4-manifold. It reflects much

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) (802 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 (2,297 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

Dirac structure (912 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 )

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 (469 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 (3,076 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

Okubo algebra (830 words) [view diff] exact match in snippet view article find links to article

x|x\rangle y\ .} Conversely, an algebra with a non-degenerate symmetric bilinear form satisfying this equation is either a para-Hurwitz algebra or an

Donaldson's theorem (1,240 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

Hermitian manifold (1,507 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

Finsler manifold (1,942 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

Vector (mathematics and physics) (2,015 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

Coxeter complex (1,253 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}

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) (2,522 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

Raising and lowering indices (3,706 words) [view diff] exact match in snippet view article find links to article

metric to be indefinite). Formally, this is a non-degenerate, symmetric bilinear form g : V × V → K a bilinear form {\displaystyle g:V\times V\rightarrow

Building (mathematics) (3,170 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

Simple Lie group (2,262 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

Hurwitz's theorem (composition algebras) (3,684 words) [view diff] exact match in snippet view article

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. It satisfies

Schwarz triangle (10,972 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 (

Inner product space (7,305 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 ⟩ =

Covariance and contravariance of vectors (5,574 words) [view diff] exact match in snippet view article find links to article

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

Multilinear form (4,599 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

Heisenberg group (5,894 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)

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 ,

Glossary of classical algebraic geometry (11,125 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 &

Grassmannian (8,384 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 (9,919 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 (5,418 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 (16,607 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 (11,482 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 (17,463 words) [view diff] exact match in snippet view article find links to article

Tangent vector Zoll surface 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 (21,523 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 (17,698 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 )