Non-degenerate 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.

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.

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).

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.

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

Quasifields of symplectic translation planes

We show that a translation plane is symplectic if and only at least one of its associated quasifields admits a non-degenerate invariant symmetric bilinear form. As an application we prove that a proper desarguesian, Moufang or nearfield plane can never be symplectic. Moreover, we give a purely algebraic characterization of the quasifields which coordinatize symplectic translation planes.

Pseudo-Riemannian Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.

Deformed Kac–Moody algebras and their representations

A class of Lie algebras G ( A ) associated to generalized Cartan matrices A is studied. The Lie algebras G ( A ) have much simpler structure than Kac–Moody algebras, but have the same root spaces with g ( A ) . In particular, G ( A ) has an abelian subalgebra of “half size.” We show that, G ( A ) has a non-degenerate invariant symmetric bilinear form if and only if A is symmetrizable; G ( X 1 ) ≅ G ( X 2 ) if and only if the GCMs X 1 and X 2 are the same up to a permutation of rows and columns. We study the lowest (respectively highest) weight Verma module V ¯ ( λ ) (respectively V ˜ ( λ ) ) over G ( A ) , and obtain the necessary and sufficient conditions for V ¯ ( λ ) to be irreducible, and also find its maximal proper submodule when V ¯ ( λ ) is reducible. Then using graded dual module of V ¯ ( λ ) we deduce the necessary and sufficient conditions for V ˜ ( λ ) to be irreducible.

Bilinear forms on fermionic Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian super-operator in a super-variable. In this paper, we show that there is a remarkable geometry on fermionic Novikov algebras with non-degenerate invariant symmetric bilinear forms, which we call pseudo-Riemannian fermionic Novikov algebras. They are related to pseudo-Riemannian Lie algebras. Furthermore, we obtain a procedure to classify pseudo-Riemannian fermionic Novikov algebras. As an application, we give the classification in dimension ⩽4. Motivated by the one in dimension 4, we construct some examples in high dimensions.

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H2 Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.

INVARIANT SYMMETRIC BILINEAR FORMS ON LIE TRIPLE SYSTEMS

ABSTRACT We first show that an invariant symmetric bilinear form on a Lie triple system can be uniquely extended to its standard imbedding Lie algebra, and then prove that any Lie triple system over a field of characteristic zero admitting a unique up to a scalar multiple, nondegenerate invariant symmetric bilinear form is necessarily simple. If the field is algebraically closed, that condition is also sufficent.

On the variation of the Poisson structures of certain moduli spaces

Let \(\pi\) be the fundamental group of a compact connected orientable topological surface with finitely many boundary circles. Our aim is to construct Poisson algebras of continuous functions on certain open subspaces \({\cal R}(\pi,G)_{\vec w}\) of the space of representations of \(\pi\) in a Lie group G whose Lie algebra is supposed to be endowed with a nondegenerate invariant symmetric bilinear form. When G is compact and connected, the spaces \({\cal R}(\pi,G)_{\vec w}\) may be taken to be dense in the space of all representations. The spaces \({\cal R}(\pi,G)_{\vec w}\) contain spaces of representations \({\rm Rep}(\pi,G)_{\vec C}\) where the values of those generators of the fundamental group which correspond to the boundary circles are constrained to lie in fixed conjugacy classes and, the representation spaces \({\rm Rep}(\pi,G)_{\vec C}\) being endowed with universal Poisson algebras constructed elsewhere earlier, the restrictions from the spaces \({\cal R}(\pi,G)_{\vec w}\) to any of the spaces \({\rm Rep}(\pi,G)_{\vec C}\) are Poisson maps. Thus the Poisson algebra on each \({\cal R}(\pi,G)_{\vec w}\) gives a description of the variation of the universal Poisson structures on the spaces \({\rm Rep}(\pi,G)_{\vec C}\) as the chosen conjugacy classes move across G. We also give twisted versions of these results.

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.