We define a notion of ghost centre of a Lie superalgebra g = g0 ⊕ g1 which is a sum of invariants with respect to the usual adjoint action (centre) and invariants with respect to a twisted adjoint action (“anticentre”). We calculate the anticentre in the case when the top external degree of g1 is a trivial g0-module. We describe the Harish-Chandra projection of the ghost centre for basic classi...

The notion of Frobenius morphisms for finite dimensional associative algebras is introduced in [2] and applied there to the study of representations of finite dimensional algebras over finite fields. In this paper, we develop a parallel theory for Lie algebras. Thus, a connection between representations over finite fields Fq and over their algebraic closures k = F̄q is established analogously. T...

This paper generalize [7]: We construct new links in-variants from g, a type I basic classical Lie superalgebra. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module. Using this, we get a multivariable link invariant associated to any one parameter family of irreducible g-modules.

Quantum super 2-shpheres and the corresponding quantum super transformation group are introduced in analogy to the well-known quantum 2-shpheres and quantum SL(2), connection between little t-Jacobi polynomials and the finite dimensional representations of the quantum super group is formulated, and the Peter-Weyl theorem is obtained. The quantum group SUq(2) introduced by Woronowicz and the qua...

Chevalley’s theorem states that for any simple finite dimensional Lie algebra g: (1) the restriction homomorphism of the algebra of polynomials S(g∗) −→ S(h∗) onto the Cartan subalgebra h induces an isomorphism S(g∗)g ∼= S(h∗)W , where W is the Weyl group of g; (2) each g-invariant polynomial is a linear combination of the polynomials trρ(x)k , where ρ is a finite dimensional representation of ...

Simple associative superalgebra with 2 independent supertraces is presented. Its commutant is a simple Lie superalgebra which has at least 2 independent invariant bilinear forms.

‎let $l$ be a lie algebra‎, ‎$mathrm{der}(l)$ be the set of all derivations of $l$ and $mathrm{der}_c(l)$ denote the set of all derivations $alphainmathrm{der}(l)$ for which $alpha(x)in [x,l]:={[x,y]vert yin l}$ for all $xin l$‎. ‎we obtain an upper bound for dimension of $mathrm{der}_c(l)$ of the finite dimensional nilpotent lie algebra $l$ over algebraically closed fields‎. ‎also‎, ‎we classi...

The two–parametric quantum superalgebra U p,q [gl(2/2)] and its representations are considered. All finite–dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and non–typical ones according to a proposition proved in the present paper. This proposition is a nontrivial deformation from the one for the classical superal-gebra gl(2/2)...

We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating the problem to the classical KazhdanLusztig theory. We further verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category O of the general linear supera...

assume that $(n,l)$, is a pair of finite dimensional nilpotent lie algebras, in which $l$ is non-abelian and $n$ is an ideal in $l$ and also $mathcal{m}(n,l)$ is the schur multiplier of the pair $(n,l)$. motivated by characterization of the pairs $(n,l)$ of finite dimensional nilpotent lie algebras by their schur multipliers (arabyani, et al. 2014) we prove some properties of a pair of nilpoten...