Related papers: The Lie superalgebra of transpositions
For any $n \geq 3$ we obtain the decomposition in simple factors of the Lie subalgebra of the group algebra of the symmetric group on $n$ letters generated by the transpositions. This enables us to determine the algebraic hull of the braid…
In this paper, all symmetric super-biderivations of some Lie superalgebras are determined. As an application, commutative post-Lie superalgebra structures on these Lie superalgebras are also obtained.
We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of…
We describe explicitly Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on supermanifolds, showing in particular that any such isomorphism induces a diffeomorphism of the supermanifolds.…
We introduce families of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra of the symmetric group. Bethe subalgebras are deformations of the Gelfand-Zetlin subalgebra. We describe various properties of Bethe…
We consider the finite Weyl groups of classical type -- $W(A_{r})$ for $r \geq 1$, $W(B_{r}) = W(C_{r})$ for $r \geq 2$, and $W(D_{r})$ for $r \geq 4$ -- as supergroups in which the reflections are of odd superdegree. Viewing the…
We give a unified description of morphisms and comorphisms of Lie pseudoalgebras, showing that the both types of morphisms can be regarded as subalgebras of a Lie pseudoalgebra, called the $\psi$-sum. We also provide similar descriptions…
We describe a construction of an algebra over the field of order 2 starting from a conjugacy class of 3-transpositions in a group. In particular, we determine which simple Lie algebras arise by this construction. Among other things, this…
In this paper we describe the the category of Lie algebras of group algebras and the category of Plesken Lie algebras and explore the categorical relations between them. Further we provide the examples of the Lie algebra of the group…
We describe certain almost-simple algebraic supergroups over an algebraically closed field of odd or zero characteristic. In addition to supergroups with simple Lie superalgebras from Kac's theorem, we construct new supergroups whose Lie…
Let A_k denote the twisted group algebra of the symmetric group S_k, whose representations correspond to the nonlinear projective representations of S_k. We establish a duality relation between A_k and a Lie superalgebra q(n), sometimes…
A study is made of real Lie algebras admitting a hypersymplectic structure, and we provide a method to construct such hypersymplectic Lie algebras. We use this method in order to obtain the classification of all hypersymplectic structures…
Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of enveloping algebra of a Lie antialgebra and study its properties. We show that every Lie antialgebra is canonically…
We construct a subalgebra of the Hecke algebra of type A. This is a generalization of the group algebra of the alternating groups. All the equivalent classes of irreducible representations of the subalgebra and the q-analogue of the…
The main definitions and properties of Lie superalgebras are proposed a la facon de a short dictionary, the different items following the alphabetical order. The main topics deal with the structure of simple Lie superalgebras and their…
The survey contains a brief description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. Representations of hypergroups are considered, being treated as continuous…
We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a…
Lie superautomorphisms of prime associative superalgebras are considered. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in…
A class of representations of a Lie superalgebra (over a commutative superring) in its symmetric algebra is studied. As an application we get a direct and natural proof of a strong form of the Poincare'-Birkhoff-Witt theorem, extending this…
After briefly reviewing the methods that allow us to derive consistently new Lie (super)algebras from given ones, we consider enlarged superspaces and superalgebras, their relevance and some possible applications.