Related papers: The finite dual coalgebra as a quantization of the…

In arXiv:math/0606241v2 M. Kontsevich and Y. Soibelman argue that the category of noncommutative (thin) schemes is equivalent to the category of coalgebras. We propose that under this correspondence the affine scheme of a k-algebra A is the…

Affine Hecke algebras arise naturally in the study of smooth representations of reductive $p$-adic groups. Finite dimensional complex representations of affine Hecke algebras (under some restriction on the isogeny class and the parameter…

We construct and study in detail various dual pairs acting on some Fock representations between a finite dimensional Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit…

Inspired by the perspective of Reyes' noncomutative spectral theory, we attempt to develop noncommutative algebraic geometry by introducing ringed coalgebras, which can be thought of as a noncommutative generalization of schemes over a…

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially ``sewn…

We describe the problem of Sweedler's duals for bialgebras as essentially characterizing the domain of the transpose of the multiplication. This domain is the set of what could be called ``representative linear forms'' which are the…

We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and…

In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations…

We show that provided $n\ne 3$, the involutive Hopf *-algebra $A_u(n)$ coacting universally on an $n$-dimensional Hilbert space has enough finite-dimensional representations in the sense that every non-zero element acts non-trivially in…

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change,…

In this paper, we develop the theory of bimodules over von Neumann algebras, with an emphasis on categorical aspects. We clarify the relationship between dualizability and finite index. We also show that, for von Neumann algebras with…

A notion of Drinfeld polynomials is introduced for modules of two-parameter quantum affine algebras. Finite dimensional representations are then characterized by sets of $l$-tuples of pairs of Drinfeld polynomials with certain conditions.

A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let $A$ be a noetherian complete basic semiperfect algebra over an algebraically closed field, and $C$ be its dual…

A new (in)finite dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite dimensional representations are constructed and…

We classify the category of finite-dimensional real division composition algebras having a non-abelian Lie algebra of derivations. Our complete and explicit classification is largely achieved by introducing the concept of a…

Given an affine hyperplane arrangement with some additional structure, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul…

We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call "cyclic double affine Lie algebra". We focus on type A : in the finite (resp. affine, double affine) case, we…

In this survey paper we study the relationships between the coarse moduli space which parameterizes the finite dimensional linear representations of an associative alegebra, the non commutative hilbert scheme and the affine scheme which is…

We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for…

We define a family of universal finite-dimensional highest weight modules for affine Lie algebras, we call these Weyl modules. We conjecture that these are the classical limits of the irreducible finite--dimensional representations of the…