Lie theory seminar

Abstract. Complexity is an established invariant of modules in non-semisimple settings. For a given module $M$ it is defined as the rate of growth of the minimal projective resolution of $M$. Hence, in a sense, it is a measure of how far the module is from being projective. I'll introduce the complexity and its geometric interpretation in the setting of finite groups as an example. I'll also give the results of the recent calculation of the complexity of the simple modules for the complex Lie superalgebra $\mathfrak{gl}(m|n)$ (along with an intriguing geometric interpretation). I'll explain all the necessary background as we go, so all are welcome. Our work is joint with Brian Boe and Dan Nakano.

Abstract. We will discuss a realization of the crystal $B(\infty)$ adapted for the action of an admissible diagram automorphism, with the aim of describing a natural subcrystal for folded Cartan datum.

Abstract. We study finite-dimensional representations of twisted current algebras, especially Demazure and (twisted) Weyl modules. First we identify these Weyl modules with corresponding affine Demazure modules, then we give an explicit construction from untwisted Weyl modules which generalize the fusion

Abstract. I will talk about recent joint work with V. Futorny in which we prove that the quantized enveloping algebra $U_q(\mathfrak{gl}_n)$ has the structure of a Galois order, a certain subring of invariants in a skew group algebra. As an application we describe explicitly the structure of the division ring of fractions of $U_q(\mathfrak{gl}_n)$, in particular obtaining a new proof of the quantum Gelfand-Kirillov conjecture for $\mathfrak{gl}_n$. Secondly we prove that the Gelfand-Tsetlin subalgebra of $U_q(\mathfrak{gl}_n)$ is maximal commutative, and obtain a parametrization of irreducible Gelfand-Tsetlin modules over $U_q(\mathfrak{gl}_n)$.

Abstract. We will define and study deformations along several parameters of the enveloping algebra of a semisimple finite dimensional Lie algebra, called the Generalized Quantum Enveloping (GQE) algebras. We will see how GQE algebras can be used to define Langlands Interpolating Quantum (LIQ) groups and solve conjectures motivated by the geometric Langlands program. These conjectures have been suggested by E. Frenkel and D. Hernandez in an original work on different LIQ groups.

Abstract. The Kostant partition function can be used to determine the weight multiplicities associated to irreducible representations of Kac-Moody algebras. Its $t$-analog was used by Lusztig to define a $t$-analog of weight multiplicity. We study Lusztig's $t$-weight multiplicities associated to the level one representation of twisted affine Kac-Moody algebras. We will derive a closed form expression for the corresponding $t$-string function using constant term identities of Macdonald and Cherednik. We describe how generalized exponents of certain representations of the underlying finite dimensional simple Lie algebra enter the picture.

Abstract. The study of the structure of irreducible representations of a quantum affine algebra can be reduced to the so called prime representations, those which cannot be written as a tensor product of two non-trivial simple representations. In their recent paper, Prime Representations from a Homological perspective, V. Chari, A. Moura and C. Young work to understand these prime representations via self extensions. Namely, they conjecture that an irreducible finite dimensional representation $V$ is prime if and only if the space of self extensions has dimension 1. I will be presenting some of the results of this paper that prove the conjecture for the $\mathfrak{sl}_2$ case, and give partial evidence in the case for general $\mathfrak g$.

Abstract. B. Kostant introduced a class of modules for finite dimensional complex semisimple Lie algebras. He called them Whittaker modules because of their connection with the Whittaker equations that arise in the study of the associated Lie group. Since then, a number of others have further developed the idea of Whittaker modules for Lie algebras. Recently, in a joint work with K. Christodoulopoulou and E. Wiesner, we have adapted some of these ideas to the setting of Lie superalgebras.

Abstract. I will give an overview of the 2007 paper by S. Witherspoon, "Twisted Graded Hecke Algebras." Given a finite group $G$ acting on a finite dimensional complex vector space $V$ and a 2-cocycle $\alpha$, we may form the twisted crossed product algebra $TV \#_\alpha G$. The main theorem gives conditions for a PBW basis for certain quotients of this algebra. A twisted graded Hecke algebra is one that satisfies these conditions. One can think of these algebras as certain deformations of $SV \#_\alpha G$. I will describe several examples, including symplectic reflection algebras and the case $G \cong (\mathbb{Z}/m \mathbb{Z})^n$.

Abstract. Given an abelian category, one of the natural questions to be addressed is that of understanding the space of extensions between its simple objects. For the category of finite-dimensional representations of an affine Kac-Moody algebra, this question has been answered in the last few years. The quantum version of this category is far more complicated and the answer to this question remains open. We shall discuss some ideas towards the answer with the help of the concept of $q$-characters and show, via examples, that the quantum answer is different from the classical one in an essential way. An interesting feature of the category of finite-dimensional representations of a quantum affine is that it has simple objects which are not prime, i.e., which are isomorphic to a tensor product of two nontrivial simple objects. It is then natural to try to classify the prime ones. Although this classification is also unknown, the amount of known examples of prime modules has been growing. In the main part of this talk we shall present results from a joint paper with V. Chari and C. Young relating the study of prime representations to that of the space of extensions between simple modules. In particular, we show that, if the underlying simple Lie algebra is $\mathfrak{sl}(2)$, then a simple representation is prime if and only if the space of its self extensions is one-dimensional. It is tempting to conjecture that this is true in general and we construct a large class of prime representations satisfying this homological property.

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. 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. The Lie algebra $\mathfrak g[t] = \mathfrak g\otimes \mathbb C[t] $ associated to a simple Lie algebra $\mathfrak g$ is called the current algebra, and has connections to the representation theory of the loop and affine Lie algebras associated to $\mathfrak g$. Of particular interest is the graded representation theory of $\mathfrak g[t]$. In a joint work with V. Chari and N. Manning we proved a BGG-type reciprocity formula for the graded representations of $\mathfrak{sl}_2[t]$ which suggested the presence of a highest weight category theory and of tilting modules. In this talk I will outline a construction of tilting modules for $\mathfrak {sl_2}[t]$ and explain a sufficient condition for the construction to work in general.

Abstract. A Loewy series of a module of finite length is by definition a semisimple filtration whose length is smallest, and it is a fundamental problem to determine the Loewy series of a module. In this talk, we study this problem for Weyl modules of a current algebra for ADE type. In this case, we can show the Loewy series is unique and coincides with the natural filtration given by its grading. As an application of this result, we can show that the Weyl module is isomorphic as graded modules to a standard modules, which are defined as the homology groups of quiver varieties. Hence we can study the Poincare polynomials of quiver varieties using Weyl modules. This talk is based on the joint work with Ryosuke Kodera.

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.