We describe a practical method for constructing a nontrivial homomorphism between two Verma modules of an arbitrary semisimple Lie algebra. With some additions the method generalises to the affine case. A theorem of Verma, Bernstein-Gel’fand-Gel’fand gives a straightforward criterion for the existence of a nontrivial homomorphism between Verma modules. Moreover, the theorem states that such hom...

The Lie algebra of a nuclear group is a locally convex nuclear vector space. Mathematics Subject Classification (2000): 22B05, 22E65, 46Axx 1. The Lie Algebra of Abelian Topological Groups Let G be an abelian topological group. The set of all one–parameter–subgroups L(G) := {λ : R→ G : λ is a continuous homomorphism} is called the Lie algebra of G . We define addition pointwise and scalar multi...

We classify biharmonic and harmonic homomorphisms f:(G,g1)⟶(G,g2) where G is a connected simply three-dimensional unimodular Lie group g1 g2 are left invariant Riemannian metrics.

‎highest weight modules of the double affine lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎singular vectors of verma modules are‎ ‎determined using a similar condition with horizontal affine lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

Let B be a stable continuous trace C*-algebra with spectrum Y . We prove that the natural suspension map S, : [Co(X), B ] -+ [Co(X)@ Co(R), B @ Co(R)] is a bijection, provided that both X and Y are locally compact connected spaces whose one-point compactifications have the homotopy type of a finite CW-complex and X is noncompact. This is used to compute the second homotopy group of 9 in terms o...

A current Lie algebra is contructed from a tensor product of a Lie algebra and a commutative associative algebra of dimension greater than 2. In this work we are interested in deformations of such algebras and in the problem of rigidity. In particular we prove that a current Lie algebra is rigid if it is isomorphic to a direct product g× g × ...× g where g is a rigid Lie algebra. 1 Current Lie ...