Related papers: Extended graphical calculus for categorified quant…

This is an expanded version of the notes of my three lectures at a NATO Advanced Study Institute ``Symmetric functions 2001: surveys of developments and perspectives" (Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; June…

We consider a two-fold problem: on the one hand, the classification of a family of solution-generating techniques in (modified) supergravity and, on the other hand, the classification of a family of canonical transformations of…

We classify semisimple module categories over the tensor category of representations of quantum SL(2) extending previous results to the roots of unity and positive characteristic cases.

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of…

We show that parabolic Kazhdan-Lusztig polynomials of type $A$ compute the decomposition numbers in certain Harish-Chandra series of unipotent characters of finite groups of Lie types $B$, $C$ and $D$ over a field of non-defining…

We categorify one half of the small quantum sl(2) at a prime root of unity. An extension of this construction to an arbitrary simply-laced case is proposed.

We show that the reduced $\mathrm{SL}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of…

We provide a counter-example to Hutchinson's original proof of $C^{1,\alpha}$ representation of curvature $m$-varifolds with $L^q$-integrable second fundamental form and $q>m$ in [6]. We also provide an alternative proof of the same result…

We provide a definition for an extended system of $\gamma$-factors for products of generic representations $\tau$ and $\pi$ of split classical groups or general linear groups over a non-archimedean local field of characteristic $p$. We…

This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids ("quantum groupoids of separable type"), based on the purely algebraic notion of weak multiplier Hopf algebras. The…

In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztig's perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of…

We provide a holomorphic description of the Hilbert space H(j_1,..,j_n) of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for the decomposition of identity in H(j_1,..,j_n). Interestingly, the…

We introduce and describe the $2$-category $\mathsf{Grt}_{\flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $\boxtimes$ restricts nicely to…

The category $\cal{C}$ (studied by Andersen-Jantzen-Soergel) of representations of a Lusztig's small quantum group at a root of unity, together with its modular structure, is defined geometrically, using configuration spaces.

We develop the 2-representation theory of the odd one-dimensional super Lie algebra $gl(1|1)^+$ and show it controls the Heegaard-Floer theory of surfaces of Lipshitz, Ozsv\'ath and Thurston. Our main tool is the construction of a tensor…

We demonstrate that the matrix quantum group $SL_q(2)$ gives rise to nontrivial matrix product operator representations of the Lie group $SL(2)$, providing an explicit characterization of the nontrivial global $SU(2)$ symmetry of the XXZ…

Based on work done by Bonechi, Cattaneo, Felder and Zabzine on Poisson sigma models, we formally show that Kontsevich's star product can be obtained from the twisted convolution algebra of the geometric quantization of a Lie 2-groupoid, one…

We develop a calculus for $S_n$-equivariant Euler characteristics of moduli spaces of stable curves and stable maps. Our approach involves an enrichment of P\'olya's cycle index polynomial of a graph to a certain algebra $\Lambda^{[2]}$ of…

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra D_q(GL_2) of quantum differential operators on GL_2, to the spherical double affine Hecke algebra associated to GL2. The isomorphism…

Let $\mathcal{B}_\mathfrak{q}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q}$. We consider the graded dual $\mathcal{L}_{\mathfrak{q}}$ of the distinguished pre-Nichols algebra…