We give an explicit description of the tracial state simplex C ⁎ -algebra ( G ) arbitrary connected, second countable, locally compact, solvable group . show that every lifts from a abelianized group, and intersection kernels all states is proper ideal unless abelian. As consequence, connected nonabelian Lie cannot embed into simple unital AF-algebra.

Let C(X) be the algebra generated by the curvature two-forms of standard holomorphic hermitian line bundles over the complex homogeneous manifold X = G/B. The cohomology ring of X is a quotient of C(X). We calculate the Hilbert polynomial of this algebra. In particular, we show that the dimension of C(X) is equal to the number of independent subsets of roots in the corresponding root system. We...

1 Background and Motivation We start with an example of affine Kac-Moody algebras and the Virasoro algebra. In this talk, F will be a field with characteristic 0, and all the vector spaces are assumed over F. Denote by Z the ring of integers and by N the set of nonnegative integers. Let 2 ≤ n ∈ N. Set sl(n,F) = {A ∈ Mn×n(F) | tr A = 0}, (1.1) 〈A,B〉 = tr AB for A,B ∈ sl(n,F), (1.2) where Mn×n(F)...

We will drop the compactness hypothesis on G in the results of §6, doing this in such a way that problems can be reduced to the compact case. This involves the notions of reductive Lie groups and algebras and Cartan involutions. Let © be a Lie algebra. A subalgebra S c © is called a reductive subaU gebra if the representation ad%\® of ίΐ on © is fully reducible. © is called reductive if it is a...

Let $${{\mathfrak {A}}}\, $$ and '$$ be two $$C^*$$ -algebras with identities $$I_{{{\mathfrak }$$ '}$$ , respectively, $$P_1$$ $$P_2 = I_{{{\mathfrak } - P_1$$ nontrivial projections in . In this paper, we study the characterization of multiplicative $$*$$ -Lie–Jordan-type maps, where notion these maps arise here. particular, if $${\mathcal {M}}_{{{\mathfrak is a von Neumann algebra relative $...

The Chen groups of a finitely-presented group G are the lower central series quotients of its maximal metabelian quotient, G/G′′. The direct sum of the Chen groups is a graded Lie algebra, with bracket induced by the group commutator. If G is the fundamental group of a formal space, we give an analog of a basic result of D. Sullivan, by showing that the rational Chen Lie algebra of G is isomorp...

A structure preserving Sort-Jacobi algorithm for computing eigenvalues or singular values is presented. The proposed method applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra an...

We establish several results regarding the algebra of derivations of tensor product of two algebras, and its connection to finite order automorphisms. These results generalize some well-know theorems in the literature. Dedicated to Professor Bruce Allison on the occasion of his sixtieth birthday 0. Introduction In 1969, R. E. Block [B] showed that the algebra of derivations of tensor product of...

Let g be a simple finite-dimensional Lie algebra and ĝκ, where κ is an invariant inner product on g, the corresponding affine Kac-Moody algebra. Consider the vacuum module Vκ over ĝκ (see Section 2 for the precise definitions). According to the results of [FF, Fr], the algebra of endomorphisms of Vκ is trivial, i.e., isomorphic to C, unless κ = κc, the critical value. The algebra Endĝκc Vκc is ...

The Yangian Y (g) associated to a finite-dimensional complex simple Lie algebra g is a Hopf algebra deformation of the universal enveloping algebra of the Lie algebra g[u] of polynomial maps C → g (with Lie bracket defined pointwise). It is important to understand the finite-dimensional representations of Y (g). For one thing, they are closely related to rational solutions of the so-called quan...