Invariant Bilinear Form Research Articles

Invariant bilinear forms under the operator group of order p³ with odd prime p

For an odd prime p, we formulate the number of all degree n representations of a group of order p3. And calculating the dimension of space of invariant bilinear forms corresponding to degree n representation over a field F which contains a primitive p3 root of unity. Here we also explicitly discussed the existence of a non-degenerate invariant bilinear form of the same space.

A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie Poisson bialgebras

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and [Formula: see text]-operators are studied.

Metrics related to oscillator algebras

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable non-abelian metric example. This algebra is the first step of a countable series of solvable Lie algebras that support invariant Lorentzian forms. Generalizing this situation, in this paper, we arrive at the oscillator Lie -algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics, and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.

Non-relativistic and ultra-relativistic expansions of three-dimensional spin-3 gravity theories

In this paper, we present novel and known non-relativistic and ultra-relativistic spin-3 algebras, by considering the Lie algebra expansion method. We start by applying the expansion procedure using different semigroups to the spin-3 extension of the AdS algebra, leading to spin-3 extensions of known non-relativistic and ultra-relativistic algebras. We then generalize the procedure considering an infinite-dimensional semigroup, which allows to obtain a spin-3 extension of two new infinite families of the Newton-Hooke type and AdS Carroll type. We also present the construction of the gravity theories based on the aforementioned algebras. In particular, the expansion method based on semigroups also allows to derive the (non-degenerate) invariant bilinear forms, ensuring the proper construction of the Chern-Simons gravity actions. Interestingly, in the vanishing cosmological constant limit we recover the spin-3 extensions of the infinite-dimensional Galilean and infinite-dimensional Carroll gravity theories.

A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of B_n

Generalizing a construction presented in Arsie and Lorenzoni (Lett Math Phys 107:1919–1961, 2017), we show that the orbit space of B_2 less the image of the coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schrödinger (NLS) equations. Motivated by this example, we study the case of B_n and we show that the defocusing case can be generalized to arbitrary n leading to a Dubrovin-Frobenius manifold structure on the orbit space of the group. The construction is based on the existence of a non-degenerate and non-constant invariant bilinear form that plays the role of the Euclidean metric in the Dubrovin–Saito standard setting. Up to n=4 the prepotentials we get coincide with those associated with the constrained KP equations discussed in Liu et al. (J Geom Phys 97:177–189, 2015).

Manin triples and non-degenerate anti-symmetric bilinear forms on Lie superalgebras in characteristic 2

In this paper, we introduce and develop the notion of a Manin triple for a Lie superalgebra g defined over a field of characteristic p=2. We find cohomological necessary conditions for the pair (g,g⁎) to form a Manin triple. We introduce the concept of Lie bi-superalgebras for p=2 and establish a link between Manin triples and Lie bi-superalgebras. In particular, we study Manin triples defined by a classical r-matrix with an extra condition (called an admissible classical r-matrix). A particular case is examined where g has an even invariant non-degenerate bilinear form. In this case, admissible r-matrices can be obtained inductively through the process of double extensions. In addition, we introduce the notion of double extensions of Manin triples, and show how to get a new Manin triple from an existing one.

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.

Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras

In this paper, first we show that the invariant bilinear form in a quadratic Lie-Yamaguti algebra induces an isomorphism between the adjoint representation and the coadjoint representation. Then we introduce the notions of relative Rota-Baxter operators on Lie-Yamaguti algebras and pre-Lie-Yamaguti algebras. We prove that a pre-Lie-Yamaguti algebra gives rise to a Lie-Yamaguti algebra naturally and a relative Rota-Baxter operator induces a pre-Lie-Yamaguti algebra. Finally, we study symplectic structures on Lie-Yamaguti algebra, which give rise to relative Rota-Baxter operators as well as pre-Lie-Yamaguti algebras. As applications, we study phase spaces of Lie-Yamaguti algebras, and show that there is a one-to-one correspondence between phase spaces of Lie-Yamaguti algebras and Manin triples of pre-Lie-Yamaguti algebras.

A Screw Theory Approach to Computing the Instantaneous Rotation Centers of Indeterminate Planar Linkages

This paper presents a screw theory approach for the computation of the instantaneous rotation centers of indeterminate planar linkages. Since the end of the 19th century, the determination of the instantaneous rotation, or velocity centers of planar mechanisms has been an important topic in kinematics that has led to the well-known Aronhold–Kennedy theorem. At the beginning of the 20th century, it was found that there were planar mechanisms for which the application of the Aronhold–Kennedy theorem was unable to find all the instantaneous rotation centers (IRCs). These mechanisms were denominated complex or indeterminate. The beginning of this century saw a renewed interest in complex or indeterminate planar mechanisms. In this contribution, a new and simpler screw theory approach for the determination of indeterminate rotation centers of planar linkages is presented. The new approach provides a simpler method for setting up the equations. Furthermore, the algebraic equations to be solved are simpler than the ones published to date. The method is based on the systematic application of screw theory, isomorphic to the Lie algebra, se(3), of the Euclidean group, SE(3), and the invariant symmetric bilinear forms defined on se(3).

Modular representations of Lie algebras of reductive groups and Humphreys' conjecture

Let G be connected reductive algebraic group defined over an algebraically closed field of characteristic p>0 and suppose that p is a good prime for the root system of G, the derived subgroup of G is simply connected and the Lie algebra g=Lie(G) admits a non-degenerate (AdG)-invariant symmetric bilinear form. Given a linear function χ on g we denote by Uχ(g) the reduced enveloping algebra of g associated with χ. By the Kac–Weisfeiler conjecture (now a theorem), any irreducible Uχ(g)-module has dimension divisible by pd(χ) where 2d(χ) is the dimension of the coadjoint G-orbit containing χ. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra Uχ(g) admits a module of dimension pd(χ).

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.

Hom Leibniz Superalgebras With Supersymmetric Invariant Nondegenerate Bilinear Forms

In this paper we study Hom Leibniz superalgebras endowed with a supersymmetric, nondegenerate and invariant bilinear form. Such Hom Leibniz superalgebras are called quadratic Hom Leibniz superalgebras. We show that every quadratic Hom Leibniz superalgebra is symmetric. After introducing representation of Hom Leibniz superalgebras, we prove that Hom Leibniz superalgebras are quadratic if and only if the adjoint representation and its dual representation are equivalent. We also extend the notion of T*-extension to Hom Leibniz superalgebras. Finally, by using double extension, we describe inductively quadratic regular Hom Leibniz superalgebras.

Invariant bilinear forms on Leibniz superalgebras

In [S. Benayadi, F. Mhamdi and S. Omri, Quadratic (resp. symmetric) Leibniz superalgebras, Commun. Algebra, doi:10.1080/00927872.2020.1850751], the quadratic Leibniz superalgebras, which are (left or right) Leibniz superalgebras provided with even supersymmetric non-degenerate and associative bilinear forms, were investigated. Besides the notion of associativity, there are other kinds of invariance for bilinear forms on Leibniz superalgebras, namely the left invariance and the right invariance. In this paper, we investigate the Leibniz superalgebras endowed with even, supersymmetric, non-degenerate and left invariant bilinear forms and highlight the links between these superalgebras and some other algebraic structures. More precisely, every symmetric Leibniz superalgebra provided with such a bilinear form gives rise to a new type of superalgebra which we call LS-Lie superalgebra. We study LS-Lie superalgebras and we give some interesting informations on the structure of these superalgebras by using certain extensions introduced in [S. Benayadi and F. Mhamdi, Odd-quadratic Leibniz superalgebras, Adv. Pure Appl. Math. 10(4) (2019) 287–298; S. Benayadi, F. Mhamdi and S. Omri, Quadratic (resp. symmetric) Leibniz superalgebras, Commun. Algebra, doi:10.1080/00927872.2020.1850751]. Further, several nontrivial examples of LS-Lie superalgebras are included. Finally, we give similar results for Leibniz superalgebras with right invariant bilinear forms.

SPACE OF INVARIANT BILINEAR FORMS UNDER REPRESENTATION OF A GROUP OF ORDER 8

Let G be a group of order 8 and F an algebraically closed field with char(F)≠2. In this paper we compute the number of n degree representations of G and subsequent dimensions of the corresponding spaces of invariant bilinear forms over the field F. We explicitly discuss about the existence of non-degenerate invariant bilinear forms.

The space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism: An approach to the problem of Albert

In this article we explore an approach to the problem of Albert described by U. Umirbaev. We characterize the space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism F:Kn→Kn, where K is a field with characteristic different from two. We obtain some conjectures expressed in the language of polynomial endomorphisms, which are equivalent to the existence of invariant bilinear forms in a finite-dimensional commutative algebra. We give a characterization of the space of invariant bilinear forms in terms of differential forms in the ring K[x1,…,xn]. We also introduce a new kind of algebra, we call them totally symmetric algebras, and we establish the relationship between these algebras and the existence of invariant bilinear forms in any commutative algebra.

Matter-free higher spin gravities in 3D: Partially-massless fields and general structure

We study the problem of interacting theories with (partially)-massless and conformal higher spin fields without matter in three dimensions. A new class of theories that have partially-massless fields is found, which significantly extends the well-known class of purely massless theories. More generally, it is proved that the complete theory has to have a form of the flatness condition for a connection of a Lie algebra, which, provided there is a non-degenerate invariant bilinear form, can be derived from the Chern-Simons action. We also point out the existence of higher spin theories without the dynamical graviton in the spectrum. As an application of a more general statement that the frame-like formulation can be systematically constructed starting from the metric one by employing a combination of the local BRST cohomology technique and the parent formulation approach, we also obtain an explicit uplift of any given metric-like vertex to its frame-like counterpart. This procedure is valid for general gauge theories while in the case of higher spin fields in d-dimensional Minkowski space one can even use as a starting point metric-like vertices in the transverse-traceless gauge. In particular, this gives the fully off-shell lift for transverse-traceless vertices.

A screw theory approach to compute instantaneous rotation axes of indeterminate spherical linkages

This paper presents a screw theory approach for the computation of the instantaneous rotation axes of indeterminate spherical linkages. Since the last part of the XIX century, the determination of the instantaneous rotation, or velocity, centers of planar mechanisms has been studied using the Aronhold-Kennedy theorem, extended to the spatial case until the second part of last century. In the beginning of the XX century, it was found that there were planar mechanisms for which the application of the Aronhold-Kennedy theorem was unable to find all the instantaneous rotation centers; these mechanisms where denominated complex or indeterminate. The beginning of this century saw a renewed interest on the complex or indeterminate mechanisms, both planar and spherical; and very soon, taken advantage of the results obtained for planar mechanisms, there were several methods for the determination of the indeterminate instantaneous rotation axes. The new screw theory approach, presented here, provides a more straightforward method for setting up the equations; furthermore, the algebraic equations to be solved are simpler than the ones published up to date. In addition, the method is part of a comprehensive method for the determination of the instantaneous screw axes of spatial mechanisms or their many special cases. The method is based on a systematic application of screw theory, which is isomorphic to the Lie algebra, se(3), of the Euclidean group, SE(3), and the invariant symmetric bilinear forms defined on se(3).

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.

Screening operators and parabolic inductions for affine [formula omitted]-algebras

(Affine) W-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra g over C, a nilpotent element f in [g,g], a good grading Γ and a symmetric invariant bilinear form κ on g. We introduce free field realizations of W-algebras by using Wakimoto representations of affine Lie algebras, where W-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free fields realizations of W-algebras. As applications, under certain conditions that are valid in all cases of type A, we construct parabolic inductions for W-algebras, which we expect to induce the parabolic inductions of finite W-algebras defined by Premet and Losev. In type A, we show that our parabolic inductions are a chiralization of the coproducts for finite W-algebras defined by Brundan-Kleshchev. In type BCD, we are able to obtain some generalizations of the coproducts in some special cases. This paper also contains an appendix by Shigenori Nakatsuka on the compatibility of screening operators with Miura maps.

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.