We study of the category O for the queer Lie superalgebra q(n), and the corresponding block decomposition induced by infinitesimal central characters. In particular, we show that the so-called typical blocks correspond to standardly stratified algebras, in the sense of Cline, Parshall and Scott. By standard arguments for Lie algebras, modified to the superalgebra situation, we prove that these ...

In our recent papers the centralizer construction was applied to the series of classical Lie algebras to produce the quantum algebras called (twisted) Yangians. Here we extend this construction to the series of the symmetric groups S(n). We study the ‘stable’ properties of the centralizers of S(n − m) in the group algebra C[S(n)] as n → ∞ with m fixed. We construct a limit centralizer algebra A...

The theory of minimal types for representations of complex semisimple Lie groups [K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Ann. of Math. (2) 85 (1967), 383-429, Chapters 1, 2 and 3] is reformulated so that it can be generalized, at least partially, to real semisimple Lie groups. A rather complete extension of the complex theory is obtained for the semisimple Lie groups of real r...

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...

Olshanski' s centralizer construction provides a realization of the Yangian Y(m) for the Lie algebra gl(m) as a subalgebra in the projective limit algebra A m = lim proj A m (n) as n → ∞, where A m (n) is the centralizer of gl(n − m) in the enveloping algebra U(gl(n)). We give a modified version of this construction based on a quantum analog of Sylvester's theorem. We then use it to get an alge...

A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands by the use of t-roots. We present new G-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics. 2000 Mat...

We compute matrix units for Brauer’s centralizer algebras and Hecke algebras of type A. This can be used to construct a complete system of matrix units of the centralizers of tensor products of classical Lie groups (except S0(2n)) and their quantum deformations. The calculation is done by induction inspired by path models for special operator algebras. It is similar to the calculation of Young’...

Let g be the Lie algebra of a compact Lie group and let θ be any automorphism of g. Let k denote the fixed point subalgebra g . In this paper we present LiE programs that, for any finite dimensional complex representation π of g, give the explicit branching π|k of π on k. Cases of special interest include the cases where θ has order 2 (corresponding to compact Riemannian symmetric spaces G/K), ...

Let X be an F -rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F . Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F -split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or ...