[en] In this paper, we introduce an new combinatorial model, which we call generalized Young walls for classical Lie algebras, and we give two realizations of the crystal B(∞) over classical Lie algebras using generalized Young walls. Also, we construct natural crystal isomorphisms between generalized Young wall realizations and other realizations, for example, monomial realization, polyhedral realization and tableau realization. Moreover, as applications, we obtain a crystal isomorphism between two different polyhedral realizations of B(∞).

[en] Let U_ζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra $\mathfrak{g}$ and a root of unity ζ. When L, L^′ are irreducible U_ζ-modules having regular highest weights, the dimension of ${\text{Ext}}_{U_{\mathrm{\zeta <!--\; ??\; -->}}}^n(L,L^{\mathrm{\prime <!--\; ???\; -->}})$ can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of U_ζ. This paper shows for L, L^′ irreducible modules in a singular block that $\dim <!--\; ???\; -->{\text{Ext}}_{U_{\mathrm{\zeta <!--\; ??\; -->}}}^n(L,L^{\mathrm{\prime <!--\; ???\; -->}})$ is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. This also computes the corresponding cohomology for q-Schur algebras and many generalized q-Schur algebras. The result depends on a certain parity vanishing property which we obtain from the Kazhdan-Lusztig correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the corresponding affine Lie algebra.

[en] We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices ${\mathfrak{g}\mathfrak{l}}^M(V,V_{\ast <!--\; ???\; -->})$. Here ${\mathfrak{g}\mathfrak{l}}^M(V,V_{\ast <!--\; ???\; -->})$ is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces $V_{\ast <!--\; ???\; -->}\otimes <!--\; ???\; -->V\to <!--\; ???\; -->\mathbb{K}$, where $\mathbb{K}$ is the base field. Tensor representations of ${\mathfrak{g}\mathfrak{l}}^M(V,V_{\ast <!--\; ???\; -->})$ are defined as arbitrary subquotients of finite direct sums of tensor products (V^∗)^⊗m ⊗ (V_∗)^⊗n ⊗ V^⊗p where V^∗ denotes the algebraic dual of V. The category ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M(V,V_{\ast <!--\; ???\; -->})}^3$ which they comprise, extends a category ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M(V,V_{\ast <!--\; ???\; -->})}$ previously studied in Dan-Cohen et al. Adv. Math. 289, 205–278, (2016), Penkov and Serganova (2014) and Sam and Snowden Forum Math. Sigma 3(e11):108, (2015) . Our main result is that ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M(V,V_{\ast <!--\; ???\; -->})}^3$ is a finite-length, Koszul self-dual, tensor category with a certain universal property that makes it into a “categorified algebra” defined by means of a handful of generators and relations. This result uses essentially the general properties of ordered Grothendieck categories, which yield also simpler proofs of some facts about the category ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M(V,V_{\ast <!--\; ???\; -->})}$ established in Penkov and Serganova (2014). Finally, we discuss the extension of ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M(V,V_{\ast <!--\; ???\; -->})}^3$ obtained by adjoining the algebraic dual (V_∗)^∗ of V_∗.

[en] In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincaré-Birkhoff-Witt type bases under the “reverse” lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors.

[en] Given an elementary simple-minded collection in the derived category of a non-positive dg algebra with finite-dimensional total cohomology, we construct a silting object via Koszul duality.

[en] We define a quantum analog of a class of generalized cluster algebras which can be viewed as a generalization of quantum cluster algebras defined in Berenstein and Zelevinsky (Adv. Math. 195(2), 405–455 2005). In the case of rank two, we extend some structural results from the classical theory of generalized cluster algebras obtained in Chekhov and Shapiro (Int. Math. Res. Notices 10, 2746–2772 2014) and Rupel (2013) to the quantum case.

[en] In this paper, working over an algebraically closed field k of prime characteristic p, we introduce a concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional Hopf algebras which are Hopf deformations of restricted universal enveloping algebras. We illustrate this technique for the case when the restricted Lie algebra has dimension 3. Together with our previous classification results, we provide a complete classification of p³-dimensional connected Hopf algebras over k of characteristic p > 2.

[en] In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds.

[en] In this paper we study the subcategory of finite-length objects of the category of positive level integrable representations of a toroidal Lie algebra. The main goal is to characterize the blocks of the category. In the cases when the underlying finite type Lie algebra associated with the toroidal Lie algebra is simply-laced, we are able to give a parametrization for the blocks.

[en] Let n ≥ 1. The pro-unipotent completion of the pure braid group of n points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models (Bezrukavnikov), (b) the choice of a complex structure on the genus 1 surface, making it into an elliptic curve E, and an appropriate flat connection on the configuration space of n points in E (joint work of the authors with D. Calaque). Following a suggestion by P. Deligne, we give an interpretation of this isomorphism in the framework of the Riemann-Hilbert correspondence, using the total space E^{#Number Sign#} of an affine line bundle over E, which identifies with the moduli space of line bundles over E equipped with a flat connection.