We prove that in a locally finite variety that has definable principal congruences (DPC), solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size.

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent ∆ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie br...

We provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group algebra of a nilpotent group.

Let G be an adjoint algebraic group of type B, C, or D over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of G. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.

We obtain inductive and enumerative formulas for the multiplicities of weights spin module Clifford algebra a Levi subalgebra in complex semisimple Lie algebra. Our involve only matrices tableaux, our techniques combine linear algebra, theory, combinatorics. Moreover, this suggests relationship with nilpotent orbits. The case special $\mathfrak{sl}(n,{\mathbb C})$ is emphasized.

We prove that if L = lim ←−Ln (n ∈ N), where each Ln is a finite dimensional semisimple Lie algebra, and A is a finite codimensional ideal of L, then L/A is also semisimple. We show also that every finite dimensional homomorphic image of the cartesian product of solvable (nilpotent) finite dimensional Lie algebras is solvable (nilpotent). Mathematics Subject Classification: 14L, 16W, 17B45

We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits of Q̌/(h + 1)Q̌ under the Weyl group (Q̌ being the coroot lattice and h the Coxeter number of g). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.

Let g be an affine Kac-Moody algebra with symmetric Cartan datum, n be the maximal nilpotent subalgebra of g. By the Hall algebra approach, we construct integral bases of the Z-form of the enveloping algebra U(n). In particular, the representation theory of tame quivers is essentially used in this paper.

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The classification is the same as in the characteristic zero case obtained by D.I. Panyushev in 1994, [32]: for e a nilpotent element in the Lie algebra of G, the ...

We interpret geometrically a variant of the Robinson-Schensted correspondence which links Brauer diagrams with updown tableaux, in the spirit of Steinberg’s result [32] on the original Robinson-Schensted correspondence. Our result uses the variety of all (N , ω, V) where V is a complete flag in C2n, ω is a nondegenerate alternating bilinear form on C2n, and N is a nilpotent element of the Lie a...