1,788 research outputs found
The classification ofseparable simple C*-algebras which are inductive limits of continuous-trace C*-algebraswith spectrum homeomorphic to the closed interval [0,1]
A classification is given of certain separable nuclear C*-algebras not
necessarily of real rank zero, namely, the class of separable simple
C*-algebras which are inductive limits of continuous-trace C*-algebras whose
building blocks have spectrum homeomorphic to the closed interval [0,1], or to
a disjoint union of copies of this space. Also, the range of the invariant is
calculated.Comment: 41 pages, 6figure
Poset topology and homological invariants of algebras arising in algebraic combinatorics
We present a beautiful interplay between combinatorial topology and
homological algebra for a class of monoids that arise naturally in algebraic
combinatorics. We explore several applications of this interplay. For instance,
we provide a new interpretation of the Leray number of a clique complex in
terms of non-commutative algebra.
R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie
combinatoire et l'alg\`ebre homologique d'une classe de mono\"ides qui figurent
naturellement dans la combinatoire alg\'ebrique. Nous explorons plusieurs
applications de cette interaction. Par exemple, nous introduisons une nouvelle
interpr\'etation du nombre de Leray d'un complexe de clique en termes de la
dimension globale d'une certaine alg\`ebre non commutative.Comment: This is an extended abstract surveying the results of arXiv:1205.1159
and an article in preparation. 12 pages, 3 Figure
Current algebras, highest weight categories and quivers
We study the category of graded finite-dimensional representations of the
polynomial current algebra associated to a simple Lie algebra. We prove that
the category has enough injectives and compute the graded character of the
injective envelopes of the simple objects as well as extensions between simple
objects. The simple objects in the category are parametized by the affine
weight lattice. We show that with respect to a suitable refinement of the
standard ordering on affine the weight lattice the category is highest weight.
We compute the Ext quiver of the algebra of endomorphisms of the injective
cogenerator of the subcategory associated to a interval closed finite subset of
the weight lattice. Finally, we prove that there is a large number of
interesting quivers of finite, affine and tame type that arise from our study.
We also prove that the path algebra of star shaped quivers are the Ext algebra
of a suitable subcategory.Comment: AMSLaTeX, 25 page
Projections in free product C*-algebras, II
Let (A,phi) be the reduced free product of infinitely many pairs (A_i,phi_i)
of C*-algebras with faithful states. Assume that the A_i are not too small, in
a specific sense. It is shown that if phi is a trace then K_0(A) is determined
entirely by K_0(phi). If, furthermore, the image of K_0(phi) is dense in the
reals then A has real rank zero. On the other hand, if phi is not a trace then
A is simple and purely infinite
Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras
Bell polynomials appear in several combinatorial constructions throughout
mathematics. Perhaps most naturally in the combinatorics of set partitions, but
also when studying compositions of diffeomorphisms on vector spaces and
manifolds, and in the study of cumulants and moments in probability theory. We
construct commutative and noncommutative Bell polynomials and explain how they
give rise to Fa\`a di Bruno Hopf algebras. We use the language of incidence
Hopf algebras, and along the way provide a new description of antipodes in
noncommutative incidence Hopf algebras, involving quasideterminants. We also
discuss M\"obius inversion in certain Hopf algebras built from Bell
polynomials.Comment: 37 pages, final version, to appear in IJA
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