A Cartan subalgebra of a finite-dimensional Lie algebra L is a nilpotent subalgebra H of L that coincides with its normalizer NL H . Such subalgebras occupy an important place in the structure theory of finite-dimensional Lie algebras and their properties have been explored in many articles (see, e.g., Barnes, 1967; Benkart, 1986; Wilson, 1977; Winter, 1969). In general (more precisely, when th...

We show that the characters of all highest weight modules over an affine Lie algebra with the highest weight away from the critical hyperplane are meromorphic functions in the positive half of the Cartan subalgebra, their singularities being at most simple poles at zeros of real roots. We obtain some information about these singularities. 0. Introduction 0.0.1. Let g be a simple finite-dimensio...

This paper illustrates the notion of a Cartan subalgebra in a C*algebra through a number of examples and counterexamples. Some of these examples have a geometrical flavour and are related to orbifolds and nonHausdorff manifolds.

Not every quasihereditary algebra $(A,\Phi,\unlhd)$ has an exact Borel subalgebra. A theorem by Koenig, K\"ulshammer and Ovsienko asserts that there always exists a Morita equivalent to $A$ regular subalgebra, but characterisation of such representative is not directly obtainable from their work. This paper gives criterion decide whether contains subalgebra provides method compute all the repre...

Let g be a reductive Lie algebra over C. We say that a g-module M is a generalized Harish-Chandra module if, for some subalgebra k ⊂ g, M is locally k-finite and has finite k-multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when k is a Cartan subalgebra. We ...

Thus our generators are not quite canonically normalized, but are all normalized equally, and β is positive definite. This is related to the fact, which we have already seen, that the group is compact. In writing down our generators, we have chosen, arbitrarily, one direction to make diagonal. Any rotation can be, by similarity transformation with at rotation, rotated into the z direction, so a...

Let g be a complex semisimple Lie algebra, with a Bore1 subalgebra b c g and Cartan subalgebra h c b. In classifying the finite dimensional representations of g, Cartan showed that any simple finite dimensional g-module has a generating element u, annihilated by n = [b, b], on which h acts by a linear form I E h*. Such an element is called a primitive vector (for the module). Harish-Chandra [9]...