Representations of semisimple Lie algebras in the BGG category í µí²ª, by James E.

We provide a systematic approach to obtain formulas for characters and Kostant u-homology groups of the oscillator modules of the finite dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highes...

Let g denote a Lie algebra, and let ĝ denote the tensor product of g with a ring of truncated polynomials. The Lie algebra ĝ is called a truncated current Lie algebra. The highest-weight theory of ĝ is investigated, and a reducibility criterion for the Verma modules is described. Let g be a Lie algebra over a field k of characteristic zero, and fix a positive integer N . The Lie algebra (1) ĝ =...

Recently I. E. Segal [2] has proposed a study of the unitary representations of a complex semisimple Lie group G by studying "analytic" holomorphic representations of G by normal operators. To this end he proved that every unitary representation U of G may be written U(g)=R(g)R(g~1)* (gEG) where R is an analytic holomorphic representation of G by normal operators such that if R(gi) and R(g2) ar...

In this article, using the definitions of central series and nilpotency in the Lie algebras, we give some results similar to the works of Hulse and Lennox in 1976 and Hekster in 1986. Finally we will prove that every non trivial ideal of a nilpotent Lie algebra nontrivially intersects with the centre of Lie algebra, which is similar to Philip Hall's result in the group theory.

We study the relation between Poisson algebras and representations of Lie conformal algebras. establish a Gröbner–Shirshov basis theory framework for modules over associative apply this technique to considered as appropriate envelopes current As result, we obtain setting calculation in algebra.

We construct a new efficient algorithm for finding Gröbner–Shirshov bases for noncommutative algebras and their representations. This algorithm uses the Macaulay matrix [F.S. Macaulay, On some formula in elimination, Proc. London Math. Soc. 33 (1) (1902) 3–27], and can be viewed as a representation theoretic analogue of the F4 algorithm developed by J.C. Faugère. We work out some examples of un...

This article is an exposition of the 1967 paper by Parthasarathy, Ranga Rao, and Varadarajan, on irreducible admissible Harish-Chandra modules over complex semisimple Lie groups and Lie algebras. It was written in Winter 2012 to be part of a special collection organized to mark 10 years and 25 volumes of the series Texts and Readings in Mathematics (TRIM). Each article in this collection is int...

In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra an $L_\infty$-algebra respectively. Then introduce representations cohomologies show that there is long exact sequence connecting various cohomologies. As applications, classify infinitesimal deformations central extensions using second cohomology gr...

In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_\infty$-algebra, whose Maurer-Cartan elements give rise to new algebras by twisting. particular, recover Lie $3$-algebra are O-operators (also called relative Rota-Baxter operators) on algebras. Then generalized matched pairs using representations algebras, which will The usual correspond special class ...