Invariant Symmetric Bilinear Form Research Articles

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.

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).

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.

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.

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).

Non-degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac--Moody (super)algebras are the most known examples.

Double extensions of Lie superalgebras in characteristic 2 with nondegenerate invariant supersymmetric bilinear form

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form will be called a NIS-Lie (super)algebra. The double extension of a NIS-Lie (super)algebra is the result of simultaneously adding to it a central element and an outer derivation so that the larger algebra has also a NIS. Affine loop algebras, Lie (super)algebras with symmetrizable Cartan matrix over any field, Manin triples, symplectic reflection (super)algebras are among the Lie (super)algebras suitable to be doubly extended.We consider double extensions of Lie superalgebras in characteristic 2, and concentrate on peculiarities of these notions related with the possibility for the bilinear form, the center, and the derivation to be odd. Two Lie superalgebras we discovered by this method are indigenous to the characteristic 2.

Invariant Symmetric Bilinear Forms and Derivation Algebras of Unitary Lie Algebras

Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized: ℬ(L) ≅ (R+/([R,R] ∩ R+))* and Der (L) = Inn (L)+ Der (L)0 = Inn (L)+ SDer (R), which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field 𝔽.

Symmetric bilinear form on a Lie algebra

Let \(\mathfrak {g}\) be the finite dimensional simple Lie algebra associated to an indecomposable and symmetrizable generalized Cartan matrix \(C=(a_{ij})_{n\times n}\) of finite type and let \(\mathfrak {d}\) be a finite dimensional Lie algebra related to a quantum group \(D_{q,p^{-1}}(\mathfrak {g})\) obtained by Hodges et al. (Adv Math 126:52–92, 1997) by deforming the quantum group \(U_q(\mathfrak {g})\). Here we see that \(\mathfrak {d}\) is a generalization of \(\mathfrak {g}\) and give a \(\mathfrak {d}\)-invariant symmetric bilinear form on \(\mathfrak {d}\).

Corrigendum to “Novikov algebras carrying an invariant Lorentzian symmetric bilinear form” [J. Geom. Phys. 82 (2014) 132–144

The purpose of this note is to correct the classification list of Novikov algebras admitting an invariant Lorentzian symmetric bilinear form in our paper mentioned above.

Polynomial invariants of Weyl groups for Kac–Moody groups

In this paper, we prove that the ring of polynomial invariants of the Weyl group for an indecomposable and indefinite Kac-Moody Lie algebra is generated by invariant symmetric bilinear form or is trivial depending on $A$ is symmetrizable or not. The result was conjectured by Moody and assumed by Kac. As applications we discuss the rational homotopy types of Kac-Moody groups and their flag manifolds.

Affine connections on three-dimensional pseudo-Riemannian homogeneous spaces. II

We describe all invariant affine connections on three-dimensional pseudo-Riemannian homogeneous spaces. We present the complete local classification of pseudo-Riemannian homogeneous spaces. It is equivalent to the description of effective pairs of Lie algebras supplied with an invariant nondegenerate symmetric bilinear form on the isotropy module. We describe all invariant affine connections on pseudo-Riemannian homogeneous spaces together with their curvature and torsion tensors and single out pseudo-Riemannian connections.

Novikov algebras carrying an invariant Lorentzian symmetric bilinear form

In this note, we shall classify Novikov algebras that admit an invariant Lorentzian symmetric bilinear form.

On low-dimensional manifolds with isometric SO0(p, q)-actions

Let G be a non-compact simple Lie group with Lie algebra \( \mathfrak{g} \). Denote with m(\( \mathfrak{g} \)) the dimension of the smallest non-trivial \( \mathfrak{g} \)-module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold M it is observed that dim(M) ≥ dim(G) + m(\( \mathfrak{g} \)) when M admits an isometric G-action with a dense orbit. The Main Theorem considers the case \( G = {\widetilde {\text{SO}}_0}\left( {p,q} \right) \), providing an explicit description of M when the bound is achieved. In such a case, M is (up to a finite covering) the quotient by a lattice of either \( {\widetilde {\text{SO}}_0}\left( {p + 1,q} \right) \) or \( {\widetilde {\text{SO}}_0}\left( {p,q + 1} \right) \).

The Socle of a Metric n-Lie Algebra

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.

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 \(\mathcal{R}^ \bot \subseteq \mathcal{R}\).

Nonexistence of Invariant Symmetric Forms on Generalized Jacobson–Witt Algebras Revisited

We provide a short alternative homological argument showing that any invariant symmetric bilinear form on simple modular generalized Jacobson–Witt algebras vanishes, and outline another, deformation-theoretic one, allowing to describe such forms on simple modular Lie algebras of contact type.