Nondegenerate Symmetric Bilinear Form Research Articles

Pseudo-Euclidean Novikov algebras of arbitrary signature

A pseudo-Euclidean Novikov algebra (g,•,〈,〉) is a Novikov algebra (g,•) endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. If 〈,〉 is of signature (1,n−1) then (g,•,〈,〉) is called a Lorentzian Novikov algebra. In (H. Lebzioui, 2020 [11]), the author studied Lorentzian Novikov algebras and showed that a Lorentzian Novikov algebra is transitive. In this paper, we study pseudo-Euclidean Novikov algebras in the general case, where 〈,〉 is of arbitrary signature. We show that a pseudo-Euclidean Novikov algebra of arbitrary signature must be transitive and the associated Lie algebra is two-solvable. This implies that a flat left-invariant pseudo-Riemannian metric on a corresponding Lie group is geodesically complete. We show that if (g,•,〈,〉) is a pseudo-Euclidean Novikov algebra such that g•g is non-degenerate then the underlying Lie algebra is a Milnor Lie algebra; that is g=b⊕b⊥, where b is a sub-Lie algebra, b⊥ is a sub-Lie ideal and adb is 〈,〉-skew symmetric for any b∈b. If g•g is degenerate, then we show that we can obtain the pseudo-Euclidean Novikov algebra through a double extension process starting from a Milnor Lie algebra. Finally, as applications, we classify all pseudo-Euclidean Novikov algebras of dimension ≤5 such that g•g is degenerate.

Products of unipotent elements of index 2 in orthogonal and symplectic groups

An automorphism u of a vector space is called unipotent of index 2 whenever (u−id)2=0. Let b be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space V over a field F of characteristic different from 2.Here, we characterize the elements of the isometry group of b that are the product of two unipotent isometries of index 2. In particular, if b is skewsymmetric and nondegenerate we prove that an element of the symplectic group of b is the product of two unipotent isometries of index 2 if and only if it has no Jordan cell of odd size for the eigenvalue −1. As an application, we prove that every element of a symplectic group is the product of three unipotent elements of index 2 (and no fewer in general).For orthogonal groups, the classification closely matches the classification of sums of two square-zero skewselfadjoint operators that was obtained in a recent article [7].

Complete descriptions of pseudo-Euclidean left-symmetric L-algebras and their pseudo-Euclidean modules

A pseudo-Euclidean left-symmetric algebra (A,.,〈,〉) is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form 〈,〉 such that left multiplications by any element of A are skew-symmetric with respect to 〈,〉. We recall that a pseudo-Euclidean Lie algebra (g,[,],〈,〉) is flat if and only if (g,.,〈,〉), its underlying vector space endowed with the Levi-Civita product associated with 〈,〉, is a pseudo-Euclidean left-symmetric algebra. In this paper, we study pseudo-Euclidean left-symmetric algebras (A,.,〈,〉) such that commutators of all elements of A are contained in the left annihilator of (A,.), these algebras will be called pseudo-Euclidean left-symmetric L-algebras. Next, we introduce and study pseudo-Euclidean modules of pseudo-Euclidean left-symmetric L-algebras. We develop double extension processes that allow us to have inductive descriptions of all pseudo-Euclidean left-symmetric L-algebras and of all its pseudo-Euclidean modules of any signature. This, in particular, improves some of the results obtained in [11].

Sets, groups, and fields definable in vector spaces with a bilinear form

We study definable sets, groups, and fields in the theory $T_\infty$ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an ($\mathbb{N}\times \mathbb{Z},\leq_{lex}$)-valued dimension on definable sets in $T_\infty$ enjoying many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all groups definable in $T_\infty$ are (algebraic-by-abelian)-by-algebraic, which, in particular, answers a question of Granger. We conclude that every infinite field definable in $T_\infty$ is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension in $T_\infty$, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component. We also consider the theory $T^{RCF}_\infty$ of vector spaces over a real closed field equipped with a nondegenerate alternating bilinear form or a nondegenerate symmetric positive-definite bilinear form. Using the same construction as in the case of $T_\infty$, we define a dimension on sets definable in $T^{RCF}_\infty$, and using it we prove analogous results about definable groups and fields: every group definable in $T^{RCF}_{\infty}$ is (semialgebraic-by-abelian)-by-semialgebraic (in particular, it is (Lie-by-abelian)-by-Lie), and every field definable in $T^{RCF}_{\infty}$ is definable in the field of scalars, hence it is either real closed or algebraically closed.

Quadratic Lie algebras with 2-plectic structures

We study the existence of 2-plectic structures on Lie algebras which admit an ad-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.

Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

We provide a proof of Jacobson’s theorem on derivations of primitive rings with nonzero socle. Both Jacobson’s theorem and its formulation (in terms of the so-called differential operators on left vector spaces over a division ring) underlie our paper. We apply Jacobson’s theorem to describe derivations of standard operator rings on real, complex, or quaternionic left normed spaces. Indeed, when the space is infinite-dimensional, every derivation of such a standard operator ring is of the form Arightarrow AB-BA for some continuous linear operator B on the space. Our quaternionic approach allows us to generalize Rickart’s theorem on representation of primitive complete normed associative complex algebras with nonzero socle to the case of primitive real or complex associative normed Q-algebras with nonzero socle. We prove that additive derivations of the Jordan algebra of a continuous nondegenerate symmetric bilinear form on any infinite-dimensional real or complex Banach space are in a one-to-one natural correspondence with those continuous linear operators on the space which are skew-adjoint relative to the form. Finally we prove that additive derivations of a real or complex (possibly non-associative) H^*-algebra with no nonzero finite-dimensional direct summand are linear and continuous.

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8—most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold),—we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras—the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.

Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

Let G be a Lie group and g its Lie algebra. The aim here is to develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in g⊗g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor in the tensor square of the Lie algebra g measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence, in terms of explicit algebraic expressions, between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures arising from the data including the symmetric 2-tensor in g⊗g. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. Along the way, a side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.

On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-32 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V(32) is naturally a V(2)-module equipped with a V(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A,U) and a simple vertex operator superalgebra LL(A,U)(ℓ,0) for every nonzero number ℓ such that LL(A,U)(ℓ,0)(2)=A and LL(A,U)(ℓ,0)(32)=U.

Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2

As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2.We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0|1, or 1|1; it is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic ≠2, see arXiv:1806.05505).We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2.We also recall examples of multiple NISes on simple Lie algebras over non-closed fields.

On the Cohomology of Lie Algebras with an Invariant Inner Product

In this work we consider the moduli space of all noncommutative metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over \(\mathbb C\) and \(\mathbb R\) up to dimension 5 and all metric Lie algebras over \(\mathbb C\) in dimension 6. We introduce cyclic and reduced cyclic cohomology to identify their metric deformations.

On pseudo-Euclidean Novikov algebras

A flat pseudo-Euclidean Lie algebra is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We observe that, in many classes of flat pseudo-Euclidean Lie algebras, the left-symmetric product is actually of Novikov. This leads us to study pseudo-Euclidean Novikov algebras (A,〈,〉), that is, Novikov algebra A endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. We show that a Lorentzian Novikov algebra (A,〈,〉) must be transitive. This implies that the underlying Lie algebra AL must be unimodular. In geometrical terms, the left-invariant metric on a corresponding Lie group is geodesically complete. In this case, we show that the restriction of the product to [AL,AL]⊥ is trivial. Using the double extension process, we prove that (A,〈,〉) is a pseudo-Euclidean Novikov algebra such that [AL,AL] is Lorentzian if and only if AL splits as AL=[AL,AL]⊥⊕[AL,AL] where [AL,AL] and [AL,AL]⊥ are abelian, [AL,AL] is Lorentzian and adx is skew-symmetric for any x∈[AL,AL]⊥. This also implies, in this case, that A is transitive, AL is unimodular and two-solvable. This result solves the problem for Lorentzian Novikov algebras such that [AL,AL] is non-degenerate. If [AL,AL] is degenerate, we show that such Lorentzian Novikov algebra is obtained by the double extension process from a flat Euclidean algebra, and we give some applications in low dimensions.

Double Extensions of Restricted Lie (Super)Algebras

A double extension ( $${\mathscr {D}}$$ -extension) of a Lie (super)algebra $${\mathfrak {a}}$$ with a non-degenerate invariant symmetric bilinear form $${\mathscr {B}}$$ , briefly, a NIS-(super)algebra, is an enlargement of $${\mathfrak {a}}$$ by means of a central extension and a derivation; the affine Kac–Moody algebras are the best known examples of double extensions of loops algebras. Let $${\mathfrak {a}}$$ be a restricted Lie (super)algebra with a NIS $${\mathscr {B}}$$ . Suppose $${\mathfrak {a}}$$ has a restricted derivation $${\mathscr {D}}$$ such that $${\mathscr {B}}$$ is $${\mathscr {D}}$$ -invariant. We show that the double extension of $${\mathfrak {a}}$$ constructed by means of $${\mathscr {B}}$$ and $${\mathscr {D}}$$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a $${\mathscr {D}}$$ -extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of $${\mathscr {D}}$$ -extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.

English

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we show that fermionic Novikov algebras equipped with invariant non-degenerate symmetric bilinear forms are Novikov algebras.

Another approach to Hom-Lie bialgebras via Manin triples

Motivated by the essential connection between Lie bialgebras and Manin triples, we introduce the notion of a Hom-Lie bialgebra with emphasis on its compatibility with a Manin triple of Hom-Lie algebras associated to a nondegenerate symmetric bilinear form. Then coboundary Hom-Lie bialgebras can be studied without a skew-symmetric condition of naturally leading to the classical Hom-Yang-Baxter equation whose solutions are used to construct coboundary Hom-Lie bialgebras, in particular on the double space of a Hom-Lie bialgebra. We also derive solutions of the classical Hom-Yang-Baxter equation from -operators and Hom-left-symmetric algebras.

A classical proof that the algebraic homotopy class of a rational function is the residue pairing

Cazanave has identified the algebraic homotopy class of a rational function of 1 variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question posed by Eisenbud for polynomial functions in 1 variable. Finally, we announce results answering this question for functions in an arbitrary number of variables.

Computer-Aided Study of Double Extensions of Restricted Lie Superalgebras Preserving the Nondegenerate Closed 2-Forms in Characteristic 2

A Lie (super)algebra with a nondegenerate invariant symmetric bilinear form B is called a nis-(super)algebra. The double extension of a nis-(super)algebra is the result of simultaneous adding to a central element and a derivation so that is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2, the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozman’s Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the nondegenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7—sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times.

Inverses of Cartan matrices of Lie algebras and Lie superalgebras

The inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. This enables one to express the fundamental weights in terms of simple roots corresponding to the Cartan matrix. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the 0s only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra of any dimension may differ (in any characteristic).We interpret most of the results of Wei and Zou (2017) [31] as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990).

The structured Gerstenhaber problem (II)

Let b be a non-degenerate symmetric (respectively, alternating) bilinear form on a finite-dimensional vector space V, over a field with characteristic different from 2. In a previous work [10], we have determined the maximal possible dimension for a linear subspace of b-alternating (respectively, b-symmetric) nilpotent endomorphisms of V. Here, provided that the cardinality of the underlying field is large enough with respect to the Witt index of b, we classify the spaces that have the maximal possible dimension.Our proof is based on a new sufficient condition for the reducibility of a vector space of nilpotent linear operators. To illustrate the power of that new technique, we use it to give a short new proof of the classical Gerstenhaber theorem on large vector spaces of nilpotent matrices (provided, again, that the cardinality of the underlying field is large enough).

Nearly Finite-Dimensional Jordan Algebras

Nearly finite-dimensional Jordan algebras are examined. Analogs of known results are considered. Namely, it is proved that such algebras are prime and nondegenerate. It is shown that the property of being nearly finite-dimensional is preserved in passing from an alternative algebra to an adjoint Jordan algebra. A similar result is established for associative nearly finite-dimensional algebras with involution. It is stated that a nearly finite-dimensional Jordan PI-algebra with unity either is a finite module over a nearly finite-dimensional center or is a central order in an algebra of a nondegenerate symmetric bilinear form. Also the following result holds: if a locally nilpotent ideal has finite codimension in a Jordan algebra with the ascending chain condition on ideals, then that algebra is finite-dimensional. In addition, E. Formanek’s result in [Comm. Alg., 1, No. 1, 79-86 (1974)], which says that associative prime PI-rings with unity are embedded in a free module of finite rank over its center, is generalized to Albert rings.