We construct operators t(z) in the elliptic algebra Aq,p(ŝl(2)c). They close an exchange algebra when pm = qc+2 for m ∈ Z. In addition they commute when p = q2k for k integer non-zero, and they belong to the center of Aq,p(ŝl(2)c) when k is odd. The Poisson structures obtained for t(z) in these classical limits are identical to the q-deformed Virasoro Poisson algebra, characterizing the exchang...

For an endofunctor H an initial algebra I has, by Lambek’s Lemma, invertible structure map, thus, it can be considered as a coalgebra. We call an H-coalgebra wellfounded if it has a coalgebra homomorphism into I. This is equivalent to the wellfoundedness in Taylor’s “Practical Foundations”, defined by means of an induction principle. And it is also equivalent to the recursivity studied by Capre...

A nonzero 2-cocycle Γ ∈ Z(g,R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie-Poisson structure on the dual Lie algebra g∗, leading to a Poisson algebra C∞(g∗(Γ)). Similarly, a multiplier c ∈ Z (G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C∗algebra C∗(G, c). Further to some supe...

We show that a C *-algebra A is nuclear iff there is a number α < 3 and a constant K such that, for any bounded homomorphism u : A → B(H), there is an isomorphism ξ : H → H satisfying ξ −1 ξ ≤ Ku α and such that ξ −1 u(.)ξ is a *-homomorphism. In other words, an infinite dimensional A is nuclear iff its length (in the sense of our previous work on the Kadison similarity problem) is equal to 2. ...

It is shown that the elliptic algebra Aq,p(sl(2)c) at the critical level c = −2 has a multidimensional center containing some trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and containing the q-deformed Virasoro algebra is constructed on this center. We show also that t(z) close an exchange algebra when p m = q c+2 for m ∈ Z, they commute when in add...

We introduce the notion of a Poisson symmetric space and the associated infinitesimal object, a symmetric Lie bialgebra. They generalize corresponding notions for Lie groups due to V. G. Drinfel’d. We use them to give some geometric insight to certain Poisson brackets that have appeared before in the literature. 1 Motivation Let us recall briefly the best-known examples of Poisson manifolds. Th...

It is well known that a measured groupoid G defines a von Neumann algebra W ∗(G), and that a Lie groupoid G canonically defines both a C∗-algebra C∗(G) and a Poisson manifold A∗(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C∗-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of obj...

If X is a topological space we denote by C(X) ® Mn the algebra of continuous functions from X to the algebra Mn of n x n complex matrices. A complete characterization of those topological spaces Y is given (in terms of vector bundles on Y) such that each unital algebrahomomorphism Φ: C(X) Θ Mn —• C(Y) Mkn is of the form a o (Φ' ® idΛ) for some homomorphism Φ': C(X) -+ C(Y) ® Λ/* and some su...