Individual ergodic theorem for unitary maps of random matrices

Using simple techniques of finite von Neumann algebras, we prove a limit theorem for random matrices.

     

Beyond Borcherds Lie algebras and inside

We give a definition for a new class of Lie algebras by generators and relations which simultaneously generalize the Borcherds Lie algebras and the Slodowy G.I.M. Lie algebras. After proving these algebras are always subalgebras of Borcherds Lie algebras, as well as some other basic properties, we give a vertex operator representation for a factor of them. We need to develop a highly non-trivial generalization of the square length two cut off theorem of Goddard and Olive to do this.

     

Affine Yangians and deformed double current algebras in type A

We study the structure of Yangians of affine type and deformed double current algebras, which are deformations of the enveloping algebras of matrix W1+∞-algebras. We prove that they admit a PBW-type basis, establish a connection (limit construction) between these two types of algebras and toroidal quantum algebras, and we give three equivalent definitions of deformed double current algebras. We construct a Schur-Weyl functor between these algebras and rational Cherednik algebras.     

Baxter algebras and Hopf algebras

By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.

     

Equivalences for blocks of the Weyl groups

We describe the source algebras of the blocks of the Weyl groups of type B and type D in terms of the source algebras of the blocks of the symmetric groups. As a consequence, we show that Puig's conjecture on the finiteness of the number of isomorphism classes of source algebras for blocks of finite groups with a fixed defect group holds for these classes of groups. We also show how certain isomorphisms between subalgebras of block algebras of the symmetric groups can be lifted to block algebras of the Weyl groups of type B.

     

Cluster-tilted algebras

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

     

Quasitilted algebras admit a preprojective component

Quasitilted algebras are generalizations of tilted algebras. As a main result we show here that the Auslander-Reiten quiver of such an algebra has a preprojective component

     

Mauldin-Williams graphs, Morita equivalence and isomorphisms

We describe a method for associating some non-self-adjoint algebras to Mauldin-Williams graphs and we study the Morita equivalence and isomorphism of these algebras.

We also investigate the relationship between the Morita equivalence and isomorphism class of the -correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.

     

A new characterization of semisimple Lie algebras

Using Casimir elements, we characterize the semisimple Lie algebras among the quadratic Lie algebras. This characterization gives, in particular, a generalization of a consequence of Cartan's second criterion.

     

Isomorphisms between sub-strongly reducible maximal triangular algebras

In this paper, we introduce the concept of sub-strongly maximal triangular algebras which form a large class of maximal triangular algebras, and prove that every algebraic isomorphism of sub-strongly maximal triangular algebras is spatially implemented, which generalizes the result by Ringrose in two respects.

     

Distributivity and stationary reflections

In this paper, we present some relations between generalized distributivity of quotient algebras and Mahlo operations, and show that the distributivity implies some variants of stationary relections.

     

Young wall realization of crystal bases for classical Lie algebras

In this paper, we give a new realization of crystal bases for finite-dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.

     

Bimodules over nest algebras and Deddens' theorem

We generalize Deddens' theorem for nest algebras in the case of w*-closed nest algebras bimodules. For each such bimodule, we introduce a norm closed sub-bimodule of it, which corresponds to the radical of a nest algebra and describe it in a number of ways, generalizing known facts about nest algebras.

     

Homological integral of Hopf algebras

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

     

A characteristic free approach to Brauer algebras

Brauer algebras arise in representation theory of orthogonal or symplectic groups. These algebras are shown to be iterated inflations of group algebras of symmetric groups. In particular, they are cellular (as had been shown before by Graham and Lehrer). This gives some information about block decomposition of Brauer algebras.

     

Logmodularity and isometries of operator algebras

We generalize some facts about function algebras to operator algebras, using the ``noncommutative Shilov boundary' or ``-envelope' first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are provided.

     

Deformation theory of abelian categories

In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations, and we construct several equivalences between deformation problems.

     

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

     

Jordan isomorphisms of triangular rings

We investigate Jordan isomorphisms of triangular rings and give a sufficient condition under which they are necessarily isomorphisms or anti-isomorphisms. As corollaries we obtain generalizations of two recent results: the one concerning Jordan isomorphisms of triangular matrix algebras by Beidar, Bresar and Chebotar, and the one concerning Jordan isomorphisms of nest algebras by Lu.