Nondegenerate Bilinear Form Research Articles

On a new class of holonomy groups in pseudo-Riemannian geometry

We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, let $g$ be a nondegenerate bilinear form on a vector space $V$, and $L : V \to V$ a $g$-symmetric operator. Then the identity component of the centraliser of $L$ in $\mathrm{SO}(g)$ is a holonomy group for a suitable Levi-Civita connection.

Nonlinear bi-integrable couplings with Hamiltonian structures

Bi-integrable couplings of soliton equations are presented through introducing non-semisimple matrix Lie algebras on which there exist non-degenerate, symmetric and ad-invariant bilinear forms. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. An application to the AKNS spectral problem gives bi-integrable couplings with Hamiltonian structures for the AKNS equations.

Quantum groups of GL(2) representation type

We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear form. A detailed study of these Hopf algebras gives us an isomorphic classification and the description of their corepresentation categories.

Uniqueness of decomposition of pseudo-Riemannian superalgebras

This paper is primarily concerned with pseudo-Riemannian superalgebras, which are superalgebras endowed with pseudo-Riemannian non-degenerate supersymmetric consistent bilinear forms. Decompositions of pseudo-Riemannian superalgebras whose left centers ar

Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

Let Jn be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G-gradings on Jn where G is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1, where n is the dimension of the vector space V defining Jn. We prove that in this case the algebra Jn is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.

Hom–Lie algebras with symmetric invariant nondegenerate bilinear forms

The aim of this paper is to introduce and study quadratic Hom–Lie algebras, which are Hom–Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom–Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom–Lie algebras. Centerless involutive quadratic Hom–Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom–Lie algebras and quadratic simple Lie algebras with symmetric involution.

Self-dual representations with vectors fixed under an Iwahori subgroup

Let G be the group of F-points of a split connected reductive F-group over a non-Archimedean local field F of characteristic 0. Let π be an irreducible smooth self-dual representation of G. The space W of π carries a non-degenerate G-invariant bilinear form (,) which is unique up to scaling. The form is easily seen to be symmetric or skew-symmetric and we set ε(π)=±1 accordingly. In this article, we show that ε(π)=1 when π is a generic representation of G with non-zero vectors fixed under an Iwahori subgroup I.

On Generalized Intersection Matrix Algebras Arising from 2-Fold Affinization of Cartan Matrices

Let 𝔤𝔦𝔪(A [2]) be the generalized intersection matrix algebra associated to the 2-fold affinization A [2] of a simply-laced Cartan matrix . The root system, nondegenerate invariant bilinear form, and Weyl group of 𝔤𝔦𝔪(A [2]) are discussed.

Solvable Pseudo-Euclidean Jordan Superalgebras

Jordan superalgebras which are endowed with an even nondegenerate supersymmetric associative bilinear form are called pseudo-Euclidean Jordan superalgebras. In this work, we introduce the notion of T*-extension of Jordan superalgebras to give some examples of such superalgebras. The main result of this article is to give an inductive description of solvable pseudo-Euclidean Jordan superalgebras. We obtain this description by introducing the notions of double extension and generalized double extension of Jordan superalgebras.

Induced representations and invariant forms

We give an elementary proof that if H is a subgroup of a finite group G and M is a simple ℂH-module that admits a nondegenerate H-invariant bilinear form such that the induced module M G is simple, then M G admits a form of the same type. This situation may occur, in particular, if one is given an imprimitive ℂG-module.

The Crepant Resolution Conjecture for Three-Dimensional Flags Modulo an Involution

After fixing a nondegenerate bilinear form on a vector space V, we define a ℤ2-action on the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov–Witten potential function of [F/ℤ2] agrees (up to unstable terms) with the genus zero Gromov–Witten potential function of a crepant resolution Y of the quotient scheme F/ℤ2, after setting a quantum parameter to −1, making a linear change of variables, and analytically continuing coefficients. We explicitly compute several invariants for the orbifold and the resolution, then argue that these determine the others via basic properties of Gromov–Witten invariants.

NAKAYAMA AUTOMORPHISMS OF FROBENIUS CELLULAR ALGEBRAS

AbstractLet A be a finite-dimensional Frobenius cellular algebra with cell datum (Λ,M,C,i). Take a nondegenerate bilinear form f on A. In this paper, we study the relationship among i, f and a certain Nakayama automorphism α. In particular, we prove that the matrix associated with α with respect to the cellular basis is uni-triangular under a certain condition.

Semi-invariants of Symmetric Quivers of Finite Type

Let $(Q,\sigma)$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ on a representation $V$ of $Q$. We shall call the representation orthogonal if $<,>$ is symmetric and symplectic if $<,>$ is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type $c^V$ and, in the case when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $pf^V$.

Octo-bialgebras

The notion of octo-algebra was introduced by Leroux as a Loday algebra with 8 operations. In this paper, we introduce a notion of octo-bialgebra as a bialgebra theory of octo-algebras, which is equivalent to a double construction of a quadri-algebra with a nondegenerate 2-cocycle or a double construction of an octo-algebra with a nondegenerate invariant bilinear form. Some properties of octo-bialgebras are given, including the study of the coboundary cases which leads to a construction from an analogue of the classical Yang-Baxter equation in an octo-algebra.

Associative Superalgebras with Homogeneous Symmetric Structures

A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e., even or odd), and associative bilinear form on A. In this article, we show that any associative superalgebra with non-null product cannot admit simultaneously even-symmetric and odd-symmetric structure. We prove that all simple associative superalgebras admit either even-symmetric or odd-symmetric structure, and we give explicitly, in every case, the homogeneous symmetric structures. We introduce some notions of generalized double extensions in order to give inductive descriptions of even-symmetric associative superalgebras and odd-symmetric associative superalgebras. We obtain also an other interesting description of odd-symmetric associative superalgebras whose even parts are semi-simple bimodules without using the notions of double extensions.

Loop algebras and bi-integrable couplings

A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations. The variational identities under non-degenerate, symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings. A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.

Symmetric semisimple modules of group algebras over finite fields and self-dual permutation codes

A module of a finite group over a finite field with a symmetric non-degenerate bilinear form which is invariant by the group action is called a symmetric module. In this paper, a characterization of indecomposable orthogonal decompositions of symmetric semisimple modules and a criterion for the hyperbolic symmetric modules are obtained, and some applications to the self-dual permutation codes are shown.

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra \(\mathcal{G}\) over the complex field ℂ. Let \(\mathcal{G} = \mathcal{S} \oplus \mathcal{R}\) be the Levi decomposition, where \(\mathcal{R}\) is the radical of \(\mathcal{G}\) and \(\mathcal{S}\) is a strong semisimple subalgebra of \(\mathcal{G}\). Denote by \(m\left( \mathcal{G} \right)\) the number of all minimal ideals of an indecomposable metric n-Lie algebra and \(\mathcal{R}^ \bot\) the orthogonal complement of R. We obtain the following results. As \(\mathcal{S}\)-modules, \(\mathcal{R}^ \bot\) is isomorphic to the dual module of \({\mathcal{G} \mathord{\left/ {\vphantom {\mathcal{G} \mathcal{R}}} \right. \kern-\nulldelimiterspace} \mathcal{R}}\). The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on \(\mathcal{G}\) is equal to that of the vector space of certain linear transformations on \(\mathcal{G}\); this dimension is greater than or equal to \(m\left( \mathcal{G} \right) + 1\). The centralizer of \(\mathcal{R}\) in \(\mathcal{G}\) is equal to the sum of all minimal ideals; it is the direct sum of \(\mathcal{R}^ \bot\) and the center of \(\mathcal{G}\). Finally, \(\mathcal{G}\) has no strong semisimple ideals if and only if \(\mathcal{R}^ \bot \subseteq \mathcal{R}\).

Semi-Invariants of Symmetric Quivers of Tame Type

A symmetric quiver (Q, σ) is a finite quiver without oriented cycles Q = (Q 0, Q 1) equipped with a contravariant involution σ on $Q_0\sqcup Q_1$ . The involution allows us to define a nondegenerate bilinear form $\langle -,-\rangle_V$ on a representation V of Q. We shall say that V is orthogonal if $\langle -,-\rangle_V$ is symmetric and symplectic if $\langle -,-\rangle_V$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if (Q, σ) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type c V and, when the matrix defining c V is skew-symmetric, by the Pfaffians pf V . To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector.

Some properties of sharp Hessenberg pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F -linear transformations A : V → V and A ∗ : V → V that satisfy the following conditions: (i) each of A , A ∗ is diagonalizable on V; (ii) there exists an ordering { V i } i = 0 d of the eigenspaces of A such that A ∗ V i ⊆ V 0 + V 1 + ⋯ + V i + 1 for 0 ⩽ i ⩽ d , where V - 1 : = 0 and V d + 1 : = 0 ; (iii) there exists an ordering { V i ∗ } i = 0 δ of the eigenspaces of A ∗ such that AV i ∗ ⊆ V 0 ∗ + V 1 ∗ + ⋯ + V i + 1 ∗ for 0 ⩽ i ⩽ δ , where V - 1 ∗ : = 0 and V δ + 1 ∗ : = 0 . We call such a pair a Hessenberg pair on V. It is known that if the Hessenberg pair A , A ∗ on V is irreducible then d = δ and for 0 ⩽ i ⩽ d the dimensions of V i and V d - i ∗ coincide. We say a Hessenberg pair A , A ∗ on V is sharp whenever it is irreducible and dim V 0 ∗ = dim V d = 1 . In this paper, we give the definitions of a Hessenberg system and a sharp Hessenberg system. We discuss the connection between a Hessenberg pair and a Hessenberg system. We also define a finite sequence of scalars called the parameter array for a sharp Hessenberg system, which consists of the eigenvalue sequence, the dual eigenvalue sequence and the split sequence. We calculate the split sequence of a sharp Hessenberg system. We show that a sharp Hessenberg pair is a tridiagonal pair if and only if there exists a nonzero nondegenerate bilinear form 〈 , 〉 on V that satisfies 〈 Au , v 〉 = 〈 u , Av 〉 and 〈 A ∗ u , v 〉 = 〈 u , A ∗ v 〉 for all u , v ∈ V .