Related papers: The quantum $\mathfrak{sl}(n,\mathbb{C})$ represen…

For any Levi subalgebra of the form $\mathfrak{l}=\mathfrak{gl}_{l_{1}}\oplus\dots\oplus\mathfrak{gl}_{l_{d}}\subseteq\mathfrak{gl}_{n}$ we construct a quotient of the category of annular quantum $\mathfrak{gl}_{n}$ webs that is equivalent…

We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the given link.

This note is a write-up of a talk given by the author at the Meeting of the Sociedade Portuguesa de Matematica in July 2012. We describe Jaeger's HOMFLY-PT expansion of the Kauffman polynomial and how to generalize it to other quantum…

We analyze the hamiltonian quantization of Chern-Simons theory associated to the universal covering of the Lorentz group SO(3,1). The algebra of observables is generated by finite dimensional spin networks drawn on a punctured topological…

We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our…

For a positive braid link, a link represented as a closed positive braids, we determine the first few coefficients of its HOMFLY polynomial in terms of geometric invariants such as, the maximum euler characteristics, the number of split…

Let $H^\bullet(\mbox{OG})$ be the quantum cohomology (specialized at $q=1$) of the $2n-1$ dimensional quadric $\mbox{OG}$. We will calculate the characteristic polynomial of the linear operators induced by quantum multiplication in…

Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of…

We define a one-parameter family of two-sided coideals in U_q(gl(n)) and study the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group U_q(n). The Plancherel decomposition of these algebras with…

We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_q(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of diagonalization of these operators…

Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E/F$ be a quadratic extension and $\rm{SL}_{n}^{*}(E) = {\rm{GL}}_{n}(E) \cap \rm{SL}_{n}(D)$. We study distinguished…

We define a ribbon category $\mathsf{Sp}(\beta)$, depending on a parameter $\beta$, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta=m-n$ the monoidal category of representations of…

The recently suggested bipartite analysis extends the Kauffman planar decomposition to arbitrary $N$, i.e. extends it from the Jones polynomial to the HOMFLY polynomial. This provides a generic and straightforward non-perturbative calculus…

F. Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive…

We give a purely combinatorial construction of colored $\mathfrak{sl}_n$ link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between $\mathfrak{sl}_n$ webs; applying a…

We introduce a new approach to the study of finite-dimensional representations of the quantum group of the affine Lie superalgebra $\mathrm{L}\mathfrak{gl}_{M|N}=\mathbb{C}[t,t^{-1}]\otimes\mathfrak{gl}_{M|N}$ ($M\neq N$). We explain how…

A family of infinite-dimensional irreducible $*$-representations on $\mathcal{H}\simeq L^2(\mathbb{R})\otimes\mathbb{C}^N$ is defined for a quantum-deformed Lorentz algebra $\mathscr{U}_{\bf q}(sl_2)\otimes \mathscr{U}_{\widetilde{\bf…

We study a quantum (non-commutative) representation of the affine Weyl group mainly of type $E_8^{(1)}$, where the representation is given by birational actions on two variables $x$, $y$ with $q$-commutation relations. Using the tau…

We study the category $\textrm{Rep}(Q,\mathbb{F}_1)$ of representations of a quiver $Q$ over "the field with one element", denoted by $\mathbb{F}_1$, and the Hall algebra of $\textrm{Rep}(Q,\mathbb{F}_1)$. Representations of $Q$ over…

This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group $\mathrm{SL}(2,\mathbb{C})$ at an even level $k\in\mathbb{Z}_+$. Our approach follows the procedure of combinatorial…