Given a reductive Lie algebra g and choice of Cartan subalgebra h, the HarishChandra map gives an isomorphism between the center z of the universal enveloping algebra U(g) of g and the algebra of Weyl group invariant symmetric tensors of h, denoted S(h) . We define these objects and give a sketch of the proof, using sl(2,C) as a motivating example. The proofs and examples mirror those found in ...

Nigel Higson and John Roe Abstract: We connect the assembly map in C∗-algebra K-theory to rigidity properties for relative eta invariants that have been investigated by Mathai, Keswani, Weinberger and others. We give a new and conceptual proof of Keswani’s theorem that whenever the C∗-algebra assembly map is an isomorphism, the relative eta invariants associated to the signature operator are ho...

We study the isomorphism problem of two “natural” algebraic structures – F-algebras and cubic forms. We prove that the F-algebra isomorphism problem reduces in polynomial time to the cubic forms equivalence problem. This answers a question asked in [AS05]. For finite fields of the form 3 6 |(#F − 1), this result implies that the two problems are infact equivalent. This result also has the follo...

in this paper, we classify the indecomposable non-nilpotent solvable lie algebras with $n(r_n,m,r)$ nilradical,by using the derivation algebra and the automorphism group of $n(r_n,m,r)$.we also prove that these solvable lie algebras are complete and unique, up to isomorphism.

For von Neumann algebras M,N not isomorphic to C C and without type I2 summands, we show that for an order-isomorphism f : AbSub M! AbSub N between the posets of abelian von Neumann subalgebras of M and N , there is a unique Jordan ⇤-isomorphism g : M! N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of ...

An equivariant Thom isomorphism theorem in operator K-theory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a non-commutative C-algebra associated to a bundle E → M , equipped with a compatible connection ∇, which plays the role of the algebra of functions on the infinit...

We classify all essential extensions of the form $$ 0 \rightarrow \mathcal{B} \mathcal{D} C(X) where $\mathcal{B}$ is a nonunital, simple, separable, finite, real rank zero, $\mathcal{Z}$-stable $C^\*$-algebra with continuous scale, and $X$ finite CW complex. In fact, we prove that there group isomorphism \mathbf{Ext}(C(X), \mathcal{B}) KK(C(X), \mathcal{M}(\mathcal{B})/\mathcal{B}).

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C *-algebra of G is an isomorphism.

The PostLie algebra is an enriched structure of the Lie algebra that has recently arisen from operadic study. It is closely related to pre-Lie algebra, Rota-Baxter algebra, dendriform trialgebra, modified classical Yang-Baxter equations and integrable systems. This paper gives a complete classification of PostLie algebra structures on the Lie algebra sl(2,C) up to isomorphism. The classificatio...

Assume A is a Fréchet algebra equipped with a smooth isometric action of a vector group V , and consider Rieffel’s deformation AJ of A. We construct an explicit isomorphism between the smooth crossed products V n AJ and V n A. When combined with the Elliott-Natsume-Nest isomorphism, this immediately implies that the periodic cyclic cohomology is invariant under deformation. Specializing to the ...