Quantum Double for Quasi-Hopf Algebras

We introduce a quantum double quasi-triangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross-product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D(G) associated to a finite group G and group 3-cocycle .     

Comparison between algebraic and topological K-theory of locally convex algebras

This paper is concerned with the algebraic K-theory of locally convex C-algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological K-theory agree on the completed projective tensor product , and that the obstruction for the comparison map to be an isomorphism is (absolute) algebraic cyclic homology. We prove the existence of an exact sequence (Theorem 6.2.1)We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Fréchet algebra whose topology is generated by a countable family of sub-multiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family (Theorem 8.3.3). This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper [M. Wodzicki, Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485-496].We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. Every Fréchet ideal is sub-harmonic, but not conversely; for example the Schatten ideal Lp is sub-harmonic for all p>0 but is Fréchet only if p?1. We prove a variant of the exact sequence above which essentially says that if A is a C-algebra and J is sub-harmonic, then the obstruction for the periodicity of K_∗(AC_⊗J) is again cyclic homology (Theorem 7.1.1). This generalizes to all algebras a result of Wodzicki for H-unital algebras announced in [M. Wodzicki, Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485-496].The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize in Theorem 6.1.6), and the excision theorem for infinitesimal K-theory, due to the first author.     

Kernel estimates and -spectral independence of generators of -semigroups

After the appearance of W. Arendt's result that “Gaussian estimate of a semigroup implies the L^p-spectral independence of the generator,” various generalizations have been obtained. This paper shows that a certain kernel estimate of a semigroup implies the L^p-spectral independence of the generator, generalizing the case of upper Gaussian estimate and “Gaussian estimate of order α(0,1] [S. Miyajima, H. Shindoh, Gaussian estimates of order α and L^p-spectral independence of generators of C₀-semigroups, Positivity 11 (1) (2007) 15–39], Definition 3.1.” The proof uses S. Karrmann's result about the L^p-spectral independence and B.A. Barnes' theorem about the spectrum of integral operators. As an application, the L^p-spectral independence of −[(−Δ)^α+V] (α(0,1]) for a suitable V is proved with the help of a recent result by V. Liskevich, H. Vogt and J. Voigt [V. Liskevich, H. Vogt, J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006) 245–277].     

Optimal embeddings in quaternion algebras

Let A be a quaternion algebra over a number field k and assume that A satisfies the Eichler condition so that some infinite place of k is unramified in A. Let L be a quadratic extension of k which embeds in A. Let R_k denote the ring of integers of k and let B be an R_k-order in L. Suppose that is an Eichler order of A of square-free level S. In this paper, we determine when there exists an embedding σ:L→A over k which gives an optimal embedding of B into in the sense that . This generalises previous work of Eichler [M. Eichler, Zur Zahlentheorie der Quaternionenalgebren, J. Reine Angew. Math. 195 (1955) 127–155] and Chinburg and Friedman [T. Chinburg, E. Friedman, An embedding theorem for quaternion algebras, J. London Math. Soc. 60 (1999) 33–44].     

Varieties of dialgebras and conformal algebras

We introduce and study the concept of a variety of dialgebras which is closely related to the concept of a variety of conformal algebras: Each dialgebra of a given variety embeds into an appropriate conformal algebra of the same variety. In particular, the Leibniz algebras are exactly Lie dialgebras, and each Leibniz algebra embeds into a conformal Lie algebra.     

Conformal representations of Leibniz algebras

We study the embedding construction of Lie dialgebras (Leibniz algebras) into conformal algebras. This construction leads to the concept of a conformal representation of Leibniz algebras. We prove that each (finite-dimensional) Leibniz algebra possesses a faithful linear representation (of finite type). As a corollary we give a new proof of the Poincaré-Birkhoff-Witt theorem for Leibniz algebras.     

Dual <Emphasis Type="Italic">R</Emphasis>-matrix integrability

Using the R-operator on a Lie algebra satisfying the modified classical Yang-Baxter equation, we define two sets of functions that mutually commute with respect to the initial Lie-Poisson bracket on . We consider examples of the Lie algebras with the Kostant-Adler-Symes and triangular decompositions, their R-operators, and the corresponding two sets of mutually commuting functions in detail. We answer the question for which R-operators the constructed sets of functions also commute with respect to the R-bracket. We briefly discuss the Euler-Arnold-type integrable equations for which the constructed commutative functions constitute the algebra of first integrals. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 147–160, April, 2008.     

A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

In this paper, we construct explicitly a noncommutative symmetric ( CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed CS system with the order polynomials of rooted trees is also given and proved.     

Lie powers and Witt vectors

In the study of Lie powers of a module V in prime characteristic p, a basic role is played by certain modules B _n introduced by Bryant and Schocker. The isomorphism types of the B _n are not fully understood, but these modules fall into infinite families , one family B(k) for each positive integer k not divisible by p, and there is a recursive formula for the modules within B(k). Here we use combinatorial methods and Witt vectors to show that each module in B(k) is isomorphic to a direct sum of tensor products of direct summands of the kth tensor power V ^{⊗ k} . To the memory of Manfred Schocker.