Lie theory seminar

Abstract. Among finite dimensional modules of a quantum affine algebra, there is a distinguished family called KR modules. It is known that, by taking the restricted classical limit, a KR module becomes isomorphic to a certain Demazure module. In this talk, I will generalize this result to a tensor product of KR modules. In this case its restricted classical limit becomes isomorphic to a certain generalized Demazure module defined using Joseph functor. If time permitted, I will introduce some application of this result.

Abstract. In my talk (based on a joint work with David Kazhdan) I will introduce a class of quantum Hankel algebras which are flat deformations of polynomial algebras and admit a number of automorphisms and same number of derivations. The simplest example is the quadratic algebra $H_1$ generated by $\{X_n\}$, where $n$ runs over integers, with a single relation $X_1X_0=qX_0X_1$, where $q$ is not a root of unity and the remaining relations coming from an automorphism and a derivation of $H_1$ both sending $X_n$ to $X_{n+1}$. Quite surprisingly, $H_1$ is a flat deformation of polynomials in infinitely many variables and:

• admits a canonical basis,
• has a quantum cluster structure,
• contains a $q$-deformation of the so called $Q$-system of type $A$ (the latter one is the set of characters of Kirillov-Reshetikhin modules over affine quantum groups of type $A$),
• each member of this $q$-deformed $Q$-system is a quantum Hankel determinant in $\{X_n\}$,
• each subalgebra of $H_1$ generated by $X_1,\dots,X_n$ is a flat deformation of polynomials in $n$ variables.

I will also define the "$k$-dimensional" quantum Hankel algebra $H_k$ whose generators are labeled by the $k$-dimesnional lattice $\mathbb Z^k$ and whose relations are determined by some basic ones and by $k$ automorphisms and $k$ derivations; and will demonstrate that these algebras share many properties of $H_1$. Ultimately, I will explain that the flatness of $H_k$ and its generalizations follows from the (no less surprising) observation that Hecke algebras "look like" Hopf algebras, which allows to produce many new solutions of the quantum Yang-Baxter equation (QYBE) out of a given initial one.

Abstract. For any finite dimensional basic associative algebra, we study the presentation spaces and their relation to the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigid presentation and study the number of nonisomorphic direct summands and different complements. Based on that, we construct a simplicial complex governing the canonical decompositions of rigid presentations and provide some examples.

Abstract. I will present some systems of short exact sequences in the categories of finite-dimensional representations of quantum affine algebras of types $A$ and $B$. These systems contain the $T$-system of relations among Kirillov-Reshetikhin modules, and extend it to include, for example, all minimal affinizations. I will outline the proofs, which use the theory of $q$-characters, and comment on what can be expected in other types. This is joint work with E. Mukhin.

Abstract. We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux, which are periodic continuations of ordinary skew tableaux, employing a statistical lattice model and non-intersecting paths. We show that the cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra, which can be interpreted as a one-parameter deformation of the $\mathfrak{sl}(n)$ Verlinde algebra, i.e. the structure constants of the Frobenius algebra are polynomials in a variable t whose constant terms are the Wess-Zumino-Novikov-Witten fusion coefficients. The latter are known to coincide with dimensions of moduli spaces of generalized theta-functions and multiplicities of tilting modules of quantum groups at roots of unity. Alternatively, the deformed Verlinde algebra can be realised as a commutative subalgebra in the endomorphisms over a Kirillov-Reshetikhin module of the quantum affine $\mathfrak{sl}(n)$ algebra. Acting with special elements of this subalgebra on a highest weight vector, one obtains Lusztig's canonical basis.

Abstract. I will give a brief introduction to (infinitesimal) symplectic reflection algebras. Hochschild cohomology provides information about the deformation theory of an associative algebra. Etingof and Ginzburg have computed the Hochschild cohomology of symplectic reflection algebras. In the hope of obtaining an analogous result for the infinitesimal case, I will make a first step of computing the Hochschild cohomology for the undeformed algebra $SV \rtimes U(\mathfrak{sl}_2)$.

Abstract. We consider a twisted loop superalgebra built from a Lie superalgebra of type $Q$. After presenting some of its properties, we will introduce a quantization of a certain bisuperalgebra structure and we will explain how this new quantized enveloping algebra is related to affine Hecke-Clifford algebras. This is a $q$-version of previous work of M. Nazarov about the Yangian attached to Lie superalgebras of type $Q$.

Abstract. Complex semisimple Lie algebras, as well as a variety of other Lie algebras including the Virasoro algebra, possess a triangular decomposition: $\mathfrak g=\mathfrak n^- \oplus \mathfrak h \oplus \mathfrak n^+$ where $\mathfrak h$ is a Cartan subalgebra and $\mathfrak n^{\pm}$ are maximal nilpotent subalgebras. Whittaker modules are defined in terms of this decomposition and sit naturally inside a larger category of modules that I refer to as a Whittaker category. I'll discuss some of the historical development of these ideas, as well as my own work on the Whittaker category for the Virasoro algebra.

Abstract. Vertex algebroids can be regarded as distant relatives of rings of twisted differential operators (TDO) on smooth varieties. The latter are employed in the classical Beilinson-Bernstein equivalence result that relates $\mathfrak g$-modules to twisted $D$-modules on the flag variety of $\mathfrak g$. I will make a quick introduction to vertex algebroids and show how they can be used to construct a version of Beilinson-Bernstein localization for a class of modules over affine Lie algebras at the critical level.

Abstract. Global Weyl modules, for generalized loop algebras $\mathfrak g\otimes A$, where $\mathfrak g$ is a simple finite dimensional Lie algebra and $A$ is an associative commutative algebra, have been defined and studied for any dominant integral weight $\lambda$. We show that the space of morphisms between global Weyl modules shares some properties with the space of morphisms between Verma modules.

Abstract. Generalized Weyl algebras (GWAs) include well-known examples such as the Weyl algebra and classical and quantum ${\mathfrak{sl}}(2)$. At the same time, they contain "non-Noetherian examples" such as continuous Hecke algebras (defined by Etingof, Gan, and Ginzburg). We study blocks of the BGG category $\mathscr O$ over a GWA, with finitely many simple objects. We compute the Ext-quiver (with relations) of the endomorphism algebra of the projective generator. We also show that this algebra is Koszul and satisfies the Strong Kazhdan-Lusztig condition.

Abstract. Given a finite dimensional simple Lie algebra $\mathfrak g$ over $\mathbb C$ and a commutative associative $\mathbb C$-algebra with unity $A$, we exhibit a $\mathbb Z$-form for the universal enveloping algebra of $\mathfrak g\otimes A$ and an explicit $\mathbb Z$-basis for this $\mathbb Z$-form. We also produce explicit commutation formulas in the universal enveloping algebra of $\mathfrak{sl}_2\otimes A$ that allow us to write certain elements in Poincaré-Birkhoff-Witt order.

Abstract. Examples of almost commutative algebra are abundant in representation theory. In positive characteristic, these algebras tend to be finite over their centers. In this talk I will discuss Kac-Weisfeiler type estimates for dimensions of irreducible modules of an almost commutative algebra in terms of dimensions of symplectic leaves of the corresponding Poisson variety. Applications to symplectic reflection algebras will be discussed.

Abstract. In this talk I will present combinatorial definitions of Kostka numbers and Kostka polynomials, their connection to the algebra of invariant polynomials and some applications to Lie Theory.

Abstract. One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category $\rm{Vect}$. In the past this problem has been resolved by working with a weaker structure called a "twisted" bialgebra. In this talk we will present a different solution by first switching to a new underlying category ${\rm Vect}^K$ of vector spaces graded by a group $K$ called the Grothendieck group. We equip this category with a nontrivial braiding which depends on the $K$-grading. With this braiding, we find that the Hall algebra does satisfy the bialgebra condition exactly for the standard multiplication and comultiplication in this category, and can also be equipped with an antipode, making it a Hopf algebra object in ${\rm Vect}^K$.