A bialgebra is a structure which is simultaneously an algebra and a coalgebra, such that the algebraic and coalgebraic parts are “compatible”. Bialgebras are normally studied over a field or commutative ring. In this paper, we show how to apply the defining diagrams of algebras, coalgebras, and bialgebras to categories of semimodules and semimodule homomorphisms over a commutative semiring. We ...

We consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole subtraction. We apply this form...

OF THE DISSERTATION Global Weyl Modules for Twisted and Untwisted Loop Algebras by Nathaniael Jared Manning Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, June 2012 Dr. Vyjayanthi Chari, Chairperson A family of modules called global Weyl modules were defined in [7] over algebras of the form g⊗A, where g is a simple finite–dimensional complex Lie algeb...

Let C be a unital AH-algebra and let A be a unital separable simple C-algebra with tracial rank no more than one. Suppose that φ, ψ : C → A are two unital monomorphisms. With some restriction on C, we show that φ and ψ are approximately unitarily equivalent if and only if [φ] = [ψ] in KL(C,A) τ ◦ φ = τ ◦ ψ for all tracial states of A and φ = ψ, where φ and ψ are homomorphisms from U0(C)/CU(C) →...

Abstract. A graphical expansion formula for non-commutative matrix integrals with values in a finite-dimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their non-orientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the inte...

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, implies the existence of the (non-homogeneous) cd-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian po...

We show that the E-theory of Connes and Higson can be formulated in terms of C∗extensions in a way quite similar to the way in which the KK-theory of Kasparov can. The essential difference is that the role played by split extensions should be taken by asymptotically split extensions. We call an extension of a C∗-algebra A by a stable C∗algebra B asymptotically split if there exists an asymptoti...

We prove a complexity dichotomy theorem for all non-negative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms [19, 8, 18, 12] and the celebrated dichotomy theorem for unweighted #CSP [6, 4, 21, 22]. Our dichotomy theorem gives a succinct criterion for tractability. If...

— We investigate the representation theory of the polynomial core T S of the quantum Teichmüller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of T S are classified, up to finitely many choices, by group hom...

We introduce a generalization of the Hermite normal form for matrices over a principal ideal ring R which may contain zero divisors. That normal form allows us to solve the following basic linear algebra problems. Equality decision, containment test and element test for submodules of R k , k 2 IN. The determination of images, kernels, and inverse images of homomorphisms ' : R l ! R k , l; k 2 I...