Related papers: Mirabolic quantum $\mathfrak{sl}_2$
We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $\operatorname{GL}_n$, the two-parameter…
Considering anyonic oscillators in a two-dimensional lattice, we realize the quantum semi-group $sl_{(q,s)}(2)$ by means of a generalized Schwinger construction. We find that the parameter $q$ of the algebra is connected to the statistical…
We give a supersymmetric generalization of the sine algebra and the quantum algebra $U_{t}(sl(2))$. Making use of the $q$-pseudo-differential operators graded with a fermionic algebra, we obtain a supersymmetric extension of sine algebra.…
The quantum modularity conjecture, first introduced by Don Zagier, is a general statement about a relation between $\mathfrak{sl}_2$ quantum invariants of links and 3-manifolds at roots of unity related by a modular transformation. In this…
We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n,r) for 2 < r < n, using Elias-Khovanov and Khovanov-Lauda type diagrams. We also define the affine analogue of the Elias-Khovanov and the…
In this paper, we study the relationship between the representation theory of the quantum affine algebra $\mathcal{U}_q(\widehat{\mathfrak{sl}_\infty})$ of infinite rank, and that of the quantum toroidal algebra…
The affine quantum Schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine $\frak{gl}_n$. Finite dimensional irreducible modules for the affine quantum Schur…
In this paper we study weighted versions of Fourier algebras of compact quantum groups. We focus on the spectral aspects of these Banach algebras in two different ways. We first investigate their Gelfand spectrum, which shows a connection…
An enhanced algebraic group $\uG$ of $G=\GL(V)$ over $\bbc$ is a product variety $\GL(V)\times V$, endowed with an enhanced cross product. Associated with a natural tensor representation of $\uG$, there are naturally Levi and parabolic…
Let $k$ be an algebraically closed field of odd characteristic $p$, and let $D_n$ be the dihedral group of order $2n$ such that $p\mid 2n$. Let $D(kD_n)$ denote the quantum double of the group algebra $kD_n$. In this paper, we describe the…
In the present paper we construct all typical finite-dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ at generic deformation parameter $q$. As in the non-deformed case the finite-dimensional…
The finite dimensional irreducible representations of the quantum supergroup $U_q(gl(m|n))$ are constructed geometrically using techniques from the Bott - Borel - Weil theory and vector coherent states.
We use Drinfeld style generators and relations to define an algebra $\mathfrak{U}_n$ which is a ``$q=0$'' version of the affine quantum group of $\mathfrak{gl}_n.$ We then use the convolution product on the equivariant $K$-theory of…
We construct a polynomial family of semisimple left module categories over the representation category of the Drinfeld-Jimbo deformation, with the fusion rule of the representation category of each Levi subalgebra. In this construction we…
All finite dimensional irreducible representations of the simple Lie-Kac super algebra SU(2/1) are explicitly constructed in the Chevalley basis as complex matrices. For typical representations, the distinguished Dynkin label is not…
Inspired by a result in [Ga], we locate two $ k[q,q^{-1}] $-integer forms of $ F_q[SL(n+1)] $, along with a presentation by generators and relations, and prove that for $ q=1 $ they specialize to $ U({\mathfrak{h}}) $, where $…
We establish algebraically and geometrically a duality between the Iwahori-Hecke algebra of type D and two new quantum algebras arising from the geometry of N-step isotropic flag varieties of type D. This duality is a type D counterpart of…
The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible…
We show that the structure of an almost-commutative spectral triple emerges in a semi-classical limit from a geometric construction on a configuration space of gauge connections. The geometric construction resembles that of a spectral…
We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to…