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Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored,…
We give an axiomatic framework for studying the representation theory of towers of algebras. We introduce a new class of algebras, contour algebras, generalising (and interpolating between) blob algebras and cyclotomic Temperley-Lieb…
The results here presented are a continuation of the algebraic research line which attempts to find properties of multiple-valued systems based on a poset of two agents. The aim of this paper is to exhibit two relationships between some…
We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on…
Lie groups and quantum algebras are connected through their common universal enveloping algebra. The adjoint action of Lie group on its algebra is naturally extended to related q-algebra and q-coalgebra. In such a way, quantum structure can…
The objective of this paper is to determine the finite dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to…
We survey a number of results regarding the representation theory of $W$-algebras and their connection with the resent development of the four dimensional $N=2$ superconformal field theories in physics.
We introduce a graphical calculus for the representation theory of the quantized enveloping algebra of type $F_4$. We do this by giving a diagrammatic description of the category of invariant tensors on the 26-dimensional fundamental…
The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of…
A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, i.e. subvarieties of the varieties of representations. The study of this monoid leads to interesting…
We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.
We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the…
We introduce the notion of quantum Schur (or $q$-Schur) superalgebras. These algebras share certain nice properties with $q$-Schur algebras such as base change property, existence of canonical $\mathbb Z[v,v^{-1}]$-bases, and the duality…
A catalogue of explicit realizations of representations of (super) Lie algebras and quantum algebras in Fock space is presented.
We aim to explore if inside a quantum vertex algebras, we can find the right notion of a quantum conformal algebra.
The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring…
Quantum N-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper we construct a level-one vertex representation of the quantum N-toroidal algebra for type C. In particular, we also…
We give a brief survey of recent developments in the highest weight representation theory and the crystal basis theory of the quantum queer superalgebra $U_q(\mathfrak{q}(n))$.
A description of the quantum superalgebra U_q[sl(n+1|m)] via creation and annihilation generators (CAGs) is given. A statement that the Fock representations of the CAGs provide microscopic realizations of exclusion statistics is formulated.
We consider a twisted version of quantum groups corepresentations. This generalization amounts to include in the theory the case where quantum space coordinates and its endomorphism matrix entries belong to a non-commutative quadratic…