This article presents a natural extension of the tensor algebra. This extended algebra is based on a vector space as the ordinary tensor algebra is. In addition to “left multiplications” by vectors, we can consider “derivations” by covectors as fundamental operators on this algebra. These two types of operators satisfy an analogue of the canonical commutation relations, and we can regard the al...

We study a finite-dimensional quotient of the Hecke algebra of type H n for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring. To appear in " M...

For a finite-dimensional Lie algebra $\mathfrak{L}$ over $\mathbb{C}$ with fixed Levi decomposition $\mathfrak{L} = \mathfrak{g} \oplus \mathfrak{r}$ where $\mathfrak{g}$ is semi-simple, we investigate $\mathfrak{L}$-modules which decompose, as $\mathfrak{g}$-modules, into direct sum of simple $\mathfrak{g}$-modules finite multiplicities. We call such modules $\mathfrak{g}$-Harish-Chandra modul...

1. The norm || • || in a Banach algebra A is said to be minimal [l ] if, given any other norm || •||1 in A (with respect to which A need not be complete), the condition ||a||iá||a|| for each oG^4 implies that ||a[|i = ||a||. We shall say that || •|| is absolutely minimal if, given any other norm ||-||i whatever in A, then ||a||iè||a|| for each aEA. An absolutely minimal norm is of course minima...

This paper is a natural extension of the previous note [Ar2]. Semiinfinite cohomology of Tate Lie algebra was defined in that note in terms of some duality resembling Koszul duality. The language of differential graded Lie algebroids was the main technical tool of the note. The present note is devoted to globalization of the main construction from [Ar2] in the following sense. The setup in [Ar2...

The Heisenberg Lie Group is the most frequently used model for studying representation theory of groups. This group modular-noncompact and its algebra nilpotent. elements can be expressed in form matrices size 3×3. Another specialty also inherited by three-dimensional called algebra. whose Algebra extended to dimension 2n+1 generalized it denoted H h_n. In this study, surjectiveness exponential...

Let L = L(x1, . . . , xm) be a graded Lie algebra generated by {x1, . . . , xm}. In this paper, we show that for any element P in L and any order k, exp(P ) may be approximated at the order k by a finite product of elementary factors exp(λixi). We give an explicit construction that avoids any calculation in the Lie algebra.

Let d \subset d' be finite-dimensional Lie algebras, H = U(d), H'=U(d') the corresponding universal enveloping algebras endowed with cocommutative Hopf algebra structure. We show that if L is a primitive pseudoalgebra over then all finite irreducible L' Cur_H^{H'} L-modules are of form V, where V an L-module, single class exceptions. Indeed, when H(d, \chi, \omega), we introduce non current L'-...

All finite-dimensional indecomposable solvable Lie algebras L(n, f), having the triangular algebra T (n) as their nilradical, are constructed. The number of nonnilpotent elements f in L(n, f) satisfies 1 ≤ f ≤ n− 1 and the dimension of the Lie algebra is dim L(n, f) = f + 1 2 n(n − 1).

It was proved by Jacobson [2] that every finite-dimensional Lie algebra of characteristic p has a finite-dimensional representation space which is not semisimple. In this note we prove a similar result for Jacobson's restricted Lie algebras [l]. The structure of a restricted Lie algebra L over a field F of characteristic p comprises, in addition to the ordinary Lie algebra structure, a map x—*-...