Nondegenerate Symmetric Bilinear Form Research Articles

Maximal orthogonality and pseudo-orthogonality with applications to generalized inverses

Equip a finite-dimensional vector space V over a field F with a nondegenerate symmetric bilinear, alternating bilinear, or hermitian form. Fix some subspace U of V. The concept of a maximally orthogonal complementary subspace for U is presented and shown to be related to the concept of a pseudo-orthogonal complementary subspace from an earlier paper. When F = GF( q), the number of maximally orthogonal (pseudo-orthogonal) complements is computed. Also, both types of complements are characterized as orthogonal complements with respect to related forms. This is used to relate certain reflexive generalized inverses of linear transformations to Moore–Penrose inverses. Techniques are presented for deriving the related forms, which, together with standard techniques for computing Moore–Penrose inverses, can be used to compute the desired generalized inverses. Lastly, vectors of V that occur in such complementary subspaces of U are characterized.

Center and universal R-matrix for quantized Borcherds superalgebras

We construct a nondegenerate symmetric bilinear form on quantized enveloping algebras associated to Borcherds superalgebras. With this, we study its center and its universal R matrix.

Systems of Correlators and Solutions of the WDVV Equations

We construct new solutions of the Witten–Dijkgraaf–E. Verlinde–H. Verlinde equations. Introduction In 1990 E. Witten [6] and R. Dijkgraaf, E. Verlinde, H. Verlinde [1] introduced a remarkable system of partial differential equations (the WDVV equations) determining deformations of 2-dimensional topological field theories. The WDVV equations consist of three groups of differential equations called the associativity equations, the normalisation equations, and the quasihomogeneity equations. B. Dubrovin [2] observed that solutions of these equations lead to new deep differential-geometrical structures on manifolds. He found that such structures naturally arise in various branches of mathematics including algebraic geometry, integrable systems, Coxeter groups, quantum cohomology, etc. The theory of the WDVV equations is closely related to the theory of Frobenius algebras. Namely, every solution of the associativity equations gives rise to a family of Frobenius algebras. In this paper we show that conversely every Frobenius algebra A over C gives rise in a purely algebraic way to a family of holomorphic solutions of the associativity equations. More precisely, we construct an infinite family of holomorphic functions A → C satisfying the associativity equations with respect to any choice of linear coordinates on A. If A has a unity 1 = 1A, then this family is parametrised by infinite sequences of elements of A. If A has no unity, then this family is parametrised by infinite sequences of A-linear homomorphisms A → A. The functions A → C constructed here in general do not satisfy the normalisation equations. We show how to modify them in order to satisfy these equations as well. We show finally that if the Frobenius algebra is graded, then the functions constructed in this paper are homogeneous and thus satisfy all the WDVV equations. Examples of 400 S. Natanzon, V. Turaev graded Frobenius algebras are provided by cohomology of closed oriented manifolds; in this case our methods yield polynomial solutions of the WDVV equations. Note that our solutions are in general different from those appearing in the theory of quantum cohomology, see [4]. It should be stressed that our aim here is not to recover the quantum cohomology but rather to study solutions of WDVV arising in an algebraic way via the theory of Frobenius algebras. The constructions of this paper can be easily generalised to produce solutions of the super-WDVV equations from Frobenius super-algebras [3]. We study solutions of WDVV in terms of the coefficients of their Taylor expansions, called correlators (cf. [2]). Section 1 is concerned with preliminaries on the associativity equations and correlators. In Sect. 2 we derive from a Frobenius algebra A a family of holomorphic solutions of the associativity equations. In Sect. 3 we show how to modify these solutions in order to satisfy the normalisation equations. In Sect. 4 we discuss the quasihomogeneity equations and consider examples of normalized quasihomogeneous systems of correlators related to the ordinary and quantum cohomology of manifolds. 1. Frobenius Algebras, Associativity Equations and Correlators Frobenius algebras. By a Frobenius algebra over C we mean a commutative associative finite dimensional algebra A over C provided with a non-degenerate symmetric bilinear form η : A × A → C such that η(ab, c) = η(a, bc), for any a, b, c ∈ A. The form η is called the inner product on A. If A has a unity 1 = 1A, then the inner product is determined by the linear functional a 7→ η(a, 1) : A → C because η(a, b) = η(ab, 1). The associativity equations. Let (η)λ,μ=1 be a non-degenerate symmetric N × N matrix over C. Let F = F (t1, ..., tN ) be a C-valued function on C where t1, ..., tN are the coordinates on C . The associativity equations (associated with (η)) are the following nonlinear partial differential equations:

Finite dimensional imbeddings of harmonic spaces

For a noncompact harmonic manifoldM we establish finite dimensionality of the eigensubspacesV γ generated by radial eigenfunctions of the form coshr+c. As a consequence, for such harmonic manifolds, we give an isometric imbedding ofM into (V γ,B), whereB is a nondegenerate symmetric bilinear indefinite form onV γ (analogous to the imbedding of the real hyperbolic spaceH n into ℝn+1 with the indefinite formQ(x,x)=−x 20 +Σx 2i ). This imbedding is minimal in a ‘sphere’ in (V γ,B). Finally we give certain conditions under whichM is symmetric.

Finitely based ideals of weak polynomial identities

Let K be a field, char K≠2, and let Vk be a k-dimensional vector space over K equipped with a nondegenerate symmetric bilinear form. Denote Ck the Clifford algebra of Vk . We study the polynomial identities for the pair (Ck ,Vk ). A basis of the identities for this pair is found. It is proved that they are consequences of the single identity [x 2 ,y] = 0 when k = ∝ It is shown that when k < ∝ the identities for (Ck ,Vk ) follow from[x 2,y]=0 and Wk+1 = 0 where Wk+1 is an analog of the standard polynomial Stk+1 Denote M 2(K) the matrix algebra of order two over K, and let sl 2(K) be the Lie algebra of all traceless 2 × 2 matrices over K. As an application, new proof of the fact that the identity [x 2,y] = 0 is a basis of the weak Lie identities for the pair (M 2(K),sl 2(K) is given.

Totally isotropic subspaces, complementary subspaces, and generalized inverses

Let us fix a field F, a finite-dimensional F-vector space V, and a nondegenerate symmetric bilinear form on V, subject to the following restriction. If char( F) = 2, then the bilinear form must be selected so that the space of all isotropic vectors in V is nondegenerate. Let N be the set of all totally isotropic subspaces of V. There exists a mapping p: N → N( U → U p such that U + U p is nondegenerate for all U ϵ N. From such, a construction is given for obtaining a “pseudoorthogonal” complementary subspace for any subspace of V. Based on this construction, it is shown how to construct generalized inverses of linear transformations on V whose associated projection maps are normal linear transformations. The resulting operation for obtaining a generalized inverse has the additional property that it commutes with the operation of taking adjoints. When char( F) ≠ 2, it is shown that p can be selected so as to be an involution. For this case, constructions of such p are presented. The constructions which are derived from these, as outlined above, are then also involutory. Moreover, when F is an ordered field, p may be selected so as to be an involutory automorphism of the partially ordered set ( N, ⊆).

Manin triples for Lie bialgebroids

In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket in the definition of a Courant algebroid. This structure on a vector bundle $E\rightarrow M$, consists of an antisymmetric bracket on the sections of $E$ whose ``Jacobi anomaly'' has an explicit expression in terms of a bundle map $E\rightarrow TM$ and a field of symmetric bilinear forms on $E$. When $M$ is a point, the definition reduces to that of a Lie algebra carrying an invariant nondegenerate symmetric bilinear form. For any Lie bialgebroid $(A,A^{*})$ over $M$ (a notion defined by Mackenzie and Xu), there is a natural Courant algebroid structure on $A\oplus A^{*}$ which is the Drinfel'd double of a Lie bialgebra when $M$ is a point. Conversely, if $A$ and $A^*$ are complementary isotropic subbundles of a Courant algebroid $E$, closed under the bracket (such a bundle, with dimension half that of $E$, is called a Dirac structure), there is a natural Lie bialgebroid structure on $(A,A^{*})$ whose double is isomorphic to $E$. The theory of Manin triples is thereby extended from Lie algebras to Lie algebroids. Our work gives a new approach to bihamiltonian structures and a new way of combining two Poisson structures to obtain a third one. We also take some tentative steps toward generalizing Drinfel'd's theory of Poisson homogeneous spaces from groups to groupoids.

On the structure of symmetric self-dual Lie algebras

A finite-dimensional Lie algebra is called (symmetric) self-dual, if it possesses an invariant nondegenerate (symmetric) bilinear form. Symmetric self-dual Lie algebras have been studied by Medina and Revoy, who have proven a very useful theorem about their structure. In this paper we prove a refinement of their theorem that has wide applicability in conformal field theory, where symmetric self-dual Lie algebras start to play an important role due to the fact that they are precisely the Lie algebras that admit a Sugawara construction. We also prove a few corollaries that are important in conformal field theory.

Lattice basis reduction for indefinite forms and an application

In this paper we present an analogue of the lattice basis reduction algorithm of A.K. Lenstra, H.W. Lenstra and L. Lovász for the case of an indefinite non-degenerate symmetric bilinear form. The algorithm produces a reduced basis with similar size properties as in the Euclidean case. As an application, we present an algorithm, which finds zero divisors in rings isomorphic to M 2( Z) in polynomial time.

Orthogonal lie algebras with cone potential

We say that a Lie algebra g is orthogonal if it admits a non-degenerate invariant symmetric bilinear form. In this paper we give a complete decription of all orthogonal Lie algebras with cone potential, a property defined in terms of the root decomposition with respect to a compactly embedded Cartan subalgebra. The class of these Lie algebras arises naturally in the study of convexity properties of Hamiltonian torus actions which are related to decompositions of groups similar to Iwasawa decompositions G = KAN of real semisimple Lie groups. In [HiNe94] we have generalized Kostant’s nonlinear convexity theorem for the Iwasawa decomposition for a complex Lie group to the class of complex groups whose Lie algebras have a real form which is an orthogonal Lie algebra with cone potential. We even obtained a generalization to the class of real semisimple Lie algebras having a solvable minimal parabolic subalgebra. This work generalized results of Lu and Ratiu ([LR91]) and aims at a unified symplectic framework for the convexity theorem in [Ne94] and the linear convexity theorems for coadjoint orbits of convexity type in [HNP93]. The motivation for this paper is to clarify the scope of the convexity theorems proved in [HiNe94]. Our description gives a rather clear picture of the situation. Every orthognal Lie algebra with cone potential is the ideal direct sum of a solvable Lie algebra and a semisimple Lie algebra of the same type. A semisimple Lie algebras has cone potential if and only if has a compactly embedded Cartan subalgebra. The orthogonal solvable Lie algebras with cone potential also decompose into certain indecomposable pieces which can be described by a root decomposition.

Four-dimensional BF theory as a topological quantum field theory

Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G=GL(4,ℝ) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.

Infinite families of isomorphic nonconjugate finitely generated subgroups

Let ⟨ , ⟩ : L × L → Z \langle \;,\;\rangle :L \times L \to \mathbb {Z} be a nondegenerate symmetric bilinear form on a finitely generated free abelian group L which splits as an orthogonal direct sum ( L , ⟨ , ⟩ ) ≅ ( L 1 , ⟨ , ⟩ ) ⊥ ( L 2 , ⟨ , ⟩ ) ⊥ ( L 3 , ⟨ , ⟩ ) (L,\;\langle \;,\;\rangle ) \cong ({L_1},\;\langle \;,\;\rangle ) \bot ({L_2},\;\langle \;,\;\rangle ) \bot ({L_3},\;\langle \;,\;\rangle ) in which ( L 1 , ⟨ , ⟩ ) ({L_1},\;\langle \;,\;\rangle ) has signature (2, 1), ( L 2 , ⟨ , ⟩ ) ({L_2},\;\langle \;,\;\rangle ) has signature (n, 1) with n ≥ 2 n \geq 2 , and ( L 3 , ⟨ , ⟩ ) ({L_3},\;\langle \;,\;\rangle ) is either zero or indefinite with rk Z ( L 3 ) ≥ 3 {\text {rk}}_\mathbb {Z}({L_3}) \geq 3 . We show that the integral automorphism group Aut Z ( L , ⟨ , ⟩ ) {\operatorname {Aut} _\mathbb {Z}}(L,\;\langle \;,\;\rangle ) contains an infinite family of mutually isomorphic finitely generated subgroups ( Γ σ ) σ ∈ Σ {({\Gamma _\sigma })_{\sigma \in \Sigma }} , no two of which are conjugate. In the simplest case, when L 3 = 0 {L_3} = 0 , the groups Γ σ {\Gamma _\sigma } are all normal subdirect products in a product of free groups or surface groups. The result can be seen as a failure of the rigidity property for subgroups of infinite covolume within the corresponding Lie group Aut Z ( L ⊗ Z R , ⟨ , ⟩ ⊗ 1 ) {\operatorname {Aut} _\mathbb {Z}}(L{ \otimes _\mathbb {Z}}\mathbb {R},\;\langle \;,\;\rangle \otimes 1) .

Rational Hilbert series of relatively free Jordan algebras

Let K be a fixed field of characteristic zero. Assume V, is a k-dimensional vector space with a non-degenerate symmetric bilinear form over K and let Gk = V, + K be the Jordan algebra of this form [6]. Let V= V, be the vector space of countable dimension over K and set G = G,. Let 2Bk = var Gk and m = var G be the varieties of unitary Jordan algebras determined by the identities of the algebras Gk and G, respectively. The main result of this paper states that for every subvariety U of !IB the relatively free algebra FJ,(U) has a Hilbert (or Poincare) series which is a rational function. (This gives an affirmative answer to Problem No. 8.6 stated in [2].) As a consequence of the method of proof, an analogous result is established for the variety of pairs generated by the pair (C, V), where C denotes the Clifford algebra of I/.

Nondegenerate symmetric bilinear forms on finite abelian 2-groups

Let B 2 {\mathcal {B}_2} be the semigroup of isomorphism classes of finite abelian 2 2 -groups with a nondegenerate symmetric bilinear form having values in Q / Z Q/{\mathbf {Z}} . Generators for B 2 {\mathcal {B}_2} were given by C. T. C. Wall and the known relations among these generators were proved to be complete by A. Kawauchi and S. Kojima. In this article we describe a normal form for such bilinear forms, expressed in terms of Wall’s generators, and as a by-product we obtain a simpler proof of the completeness of the known relations.

Symmetric bilinear forms, quadratic forms and MilnorK-theory in characteristic two

is bijective for any field F of characteristic two and for any n>0. Here, K,(F) is the K-group of Milnor defined in [11] and I is the unique maximal ideal of the Witt ring W(F) of non-degenerate symmetric bilinear forms over F. On the other hand, Sah studied quadratic forms in characteristic two in [15]. Let F be a field of characteristic two, Wq(F) the Witt group of non-degenerate quadratic forms over F, and define I'Wq(F) (n>O) by the canonical W(F)-module structure on Wq(F). Then, he proved that Wq(F)/IWq(F) is isomorphic to the group of quadratic cyclic extensions of F and IWq(F)/I2Wq(F) is isomorphic to the subgroup Br(F)2 of the Brauer group Br(F) of F consisting of elements annihilated by 2. In this paper, we prove the above Milnor's conjecture, and describe the precise structures of W(F) and Wq(F) generalizing the result of Sah. For any field F of characteristic p>0, let ~ be the n-th exterior power over F of the absolute differential module n t _g21 Let d(~2~ 1) be the .~F/Z-F/FP" image of the exterior derivation d: f2~ t --~ f2~, and let v(n)v (resp. H"~+1(F)) be the kernel (resp. cokernel) of the homomorphism

Algebras with nondegenerate associative symmetric bilinear forms permitting composition, II

(1981). Algebras with nondegenerate associative symmetric bilinear forms permitting composition. Communications in Algebra: Vol. 9, No. 12, pp. 1233-1261.

Algebras with nondegenerate associative symmetric bilinear forms permitting composition

Algebras with nondegenerate associative symmetric bilinear forms permitting composition

The Space Groups of Two Dimensional Minkowski Space

LetEbe an n-dimensional real affine space,Vits vector space of translations andA(E)the affine group ofE.Suppose that (. , .) is a nondegenerate symmetric bilinear form on F of signature(n —1, 1), O(V) its orthogonal group andS(V)its group of similarities.

Signatures and Semi Signatures of Abstract Witt Rings and Witt Rings of Semilocal Rings

This paper originated in an attempt to carry over the results of [3] from the case of a field of characteristic different from two to that of semilocal rings. To carry this out, we reverse the point of view of [3] and do assume a full knowledge of the theory of Witt rings of classes of nondegenerate symmetric bilinear forms over semilocal rings as given, for example, in [10; 11]. It turns out that the rings WT of [3] are just the residue class rings of W(C), the Witt ring of a semilocal ring C, modulo certain intersections of prime ideals.

Witts Theorem for Quadratic Forms Over Non-Dyadic Discrete Valuation Rings

Let R be a discrete valuation ring, with maximal ideal pR, such that ½ ϵ R. Let L be a finitely generated R-module and B : L × L → R a non-degenerate symmetric bilinear form. The module L is called a quadratic module. For notational convenience we shall write xy = B(x, y). Let O(L) be the group of isometries, i.e. all R-linear isomorphisms φ : L → L such that B((φ(x), (φ(y)) = B(x, y).