MIT Infinite Dimensional Algebra Seminar (Spring 2018)

We study some tiling problems involving Axtec diamonds and non-convex lozenge tiling, leading to various related determinantal processes, and we indicate some of the combinatorial techniques involved. Large scale limits will be mentioned, but the emphasis shall be on the combinatorics.

Etingof and Kazhdan constructed a functorial quantisation U_h(b) of a Lie bialgebra b over an arbitrary field k of characteristic zero, which depends on the choice of an associator Phi. An important ingredient in, and byproduct of their construction is a Tannakian equivalence F_b between two braided tensor categories. The first is that of Drinfeld-Yetter modules over U_h(b) (equivalently, modules over the quantum double of U_h(b)). The second is that of Drinfeld-Yetter modules over b, with deformed associativity constraints given by Phi.

Motivated by the theory of quasi-Coxeter algebras and categories, Andrea Appel and I recently proved that the equivalence F_b is itself functorial in b.

I will discuss a combinatorial classification of finite dimensional irreducible modules over rational Cherednik algebras with focus on type B_n (and, more general, a computation of the support of an irreducible module). The answer is stated in terms of crystals coming from Kac-Moody and Heisenberg algebra actions on higher level Fock spaces (level 2 in case of B_n). The talk is based on arXiv:1509.00526.

The moduli space of flat connections on a surface naturally carries a Poisson structure, which can be seen as induced by the intersection of loops on the surface. Also taking self-intersections of loops into consideration one can define the Goldman-Turaev Lie bialgebra structure on the space of loops. On a genus zero surface with three boundary components the linearization problem of this structure is equivalent to the Kashiwara-Vergne problem. Motivated by this result a generalization of the Kashiwara-Vergne problem in higher genera is proposed and solutions are constructed in analogy with elliptic associators.

I will describe the construction of a Lax type operator L(z) with coefficients in the quantum finite W-algebra W(g,f). We show that for the classical linear Lie algebras gl_N, sl_N, so_N and sp_N, such operator L(z) satisfies a generalized Yangian identity. In the case of classical affine W-algebras, the analogue of this operator is a Lax operator L(\partial), which is used to construct an integrable Hamiltonian hierarchy of Lax type equations. All the result presented are joint work with V. Kac and D. Valeri.

There are two geometric models for representations of semisimple Lie algebras: one uses affine Grassmannians, the other uses quiver varieties. A long-standing open problem is to relate these two geometric models. I will explain how this has recently been accomplished using symplectic duality. In particular, I will discuss a recent result (joint with Tingley, Webster, Weekes, and Yacobi) which gives an equivalence of categories between modules for truncated shifted Yangians and modules for Khovanov-Lauda-Rouquier-Webster algebras.

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. By definition, the semisimplification of a tensor category is its quotient by the tensor ideal of negligible morphisms, i.e., morphisms $f$ such that $Tr(fg)=0$ for any morphism $g$ in the opposite direction. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic $p$ in terms of representations of the normalizer of its Sylow $p$-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group $S_{n+p}$ in characteristic $p$, where $n=0,...,p-1$, and of the abelian envelope of the Deligne category, $Rep^{ab} S_t$. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of $sl_2$. Finally, we study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). This is joint work with Victor Ostrik.

The universal centralizer Z of a semisimple algebraic group G of adjoint type is the family of centralizers in G of regular elements in Lie(G), parametrized by the regular conjugacy classes. It has a natural symplectic structure coming from a Hamiltonian reduction of the cotangent bundle T^*G. We construct a partial compactification of Z where each centralizer fiber is compactified inside the wonderful compactification of G. We show that the canonical symplectic structure on Z extends to a log-symplectic Poisson structure on this partial compactification. We identify the closures of the centralizer fibers with certain subvarieties of the flag variety, and we use this to describe the symplectic leaves of the Poisson structure.

A theorem of Drinfeld's from the 1980's succinctly explained how to rebuild the Yangian of a simple Lie algebra from an R-matrix associated to any of its finite-dimensional irreducible modules, yielding the so-called R-matrix presentation of the Yangian. In the first part of my talk, I will attempt to shed new light on this construction by presenting a generalization of Drinfeld's result and a more concrete description of the resulting presentation. A fundamental role in this theory is played by a ternary matrix relation called the RTT-relation. Replacing it with a quaternary relation called the reflection equation leads instead to the R-matrix presentation of twisted Yangians, which are certain co-ideal subalgebras of Yangians. In the second part of my talk, I will overview the theory of twisted Yangians of orthogonal and symplectic type and present a recent classification of their finite-dimensional irreducible modules using Drinfeld polynomials, which was obtained in joint work with N. Guay and V. Regelskis.

We present an entirely new, elementary, explicit and combinatorial approach towards higher analogs of the simple Lie groups

Part I

Dynkin diagrams, roots, weights, quivers and their representations are built naturally from quantum subgroups of SU(2), by new constructions. We solve a conjecture of V.Kac on Lie exponents as eigenvalues of Dynkin diagrams. We show that the equation [3]+[5]=[9] encodes E_8, and we unify the exponents of exceptionals E_n as [n-5]+[n-3]=[n+1]. We build naturally from quantum subgroups of SU(N), which we classify for N=3,4: higher Dynkin diagrams, higher roots and weights as euclidean lattices, higher exponents, higher matrices and higher inner products. We construct higher Chebyshev polynomials. We extend all of these explicitly and elementarily to a much wider framework.

Part II

We study permutohedral cones and affine permutohedral cones. We write these in terms of trees, and define and study a new partial derivative of a tree with respect to another tree. We describe and study affine shards, the simplicial complex obtained by cutting root linear spaces with special hyperplanes. Their vertices are connected to determinants of Hadamard matrices. New combinatorial results are used, such as a permutation on a double chessboard, from which rooks are running away. They describe degenerations of generalized permutohedra. We show that in 4 coordinates the relations satisfied by affine permutohedral cones are precisely the ones of the Riemann curvature. We build explicitly higher hives and higher honeycombs and a curvature construction, the geode, which unites the two. We presented these topics in a graduate course at Harvard in physics, and in its continuation aimed at a book on our work.

Let B be a commutative F_p-algebra and let B_h be a quantization of B. A key favorable property of B_h is that it contains the Frobenius twist of B in its center. In that case we say that B_h is a Frobenius-constant quantization of B (see work of Bezrukavnikov et al.). One large and important class of examples of commutative algebras and their quantizations, known as (quantum) Coulomb branches, was recently defined by Braverman-Finkelberg-Nakajima (motivated by physics). We use Steenrod's construction to show that the quantum Coulomb branch is a Frobenius-constant quantization. This is an application of the general idea that one can approximate an action of the circle by the action of its subgroup C of order p. The key geometric input is a version of the Beilinson-Drinfeld Grassmannian in which the loop-rotation action of C on the affine Grassmannian Gr is deformed to the action of C on Gr^p by cyclic permutations.

Towards a functor between affine and finite Hecke categories in type A

In this thesis we construct a functor from a perfect subcategory of a coherent version of an affine Hecke category in type A to a finite constructible Hecke category, partly categorifying a certain natural homomorphism of the corresponding Hecke algebras. This homomorphism sends generators of the Bernstein's commutative subalgebra inside the affine Hecke algebra to Jucys-Murphy elements in the finite Hecke algebra. Construction employs a general strategy devised by Bezrukavnikov to prove an equivalence of coherent and constructible variants of the affine Hecke category. Namely, we identify an action of the category Rep(GL(n))$ on the finite Hecke category, and lift this action to a functor from a perfect derived category of a Steinberg variety, by equipping it with various additional data.

Different parts of this work are joint with Lusztig and Oblomkov respectively. Trigonometric and rational Cherednik algebras are degenerations of the double affine Hecke algebra. We will construct representations of the trigonometric algebra (with certain unequal parameters) from perverse sheaves on the nilpotent cone of Vinberg theta-groups. The geometry involved are those of Hessenberg varieties (and secretly affine Springer fibers). In certain cases, we can define a filtration on these representations using the geometry of the Hitchin system, and get representations of the rational Cherednik algebra by taking the associated graded.

The extended Toda hierarchy is an integrable system satisfied by the Gromov--Witten total descendant potential of $\mathbb{CP}^1$. It is an extension of the 1D Toda lattice hierarchy by a set of flows obtained by taking a logarithm of the Lax operator. We describe explicitly the action of its additional symmetries and Darboux transformations on the Lax operator, wave operators, and tau-function of the hierarchy. On the tau-function, the infinitesimal symmetries act as elements of the Virasoro algebra and the Darboux transformations act as a vertex operator. Joint work with William Wheeless and Anila Yadavalli.

Given two complex surfaces with boundary together with an analytic parametrization of a boundary component of each, they can be glued along the boundary parametrizations to form a new complex surface with boundary (in general, of higher genus). I will give an analogous picture on the algebro-geometric side: namely, I will give an algebraic notion of "smooth curve with parametrized boundary" (involving formal geometry and log geometry), and of glueing such curves. A consequence of this new theory is that the classifying spaces of "curves with boundary" form certain (modular) operad structures in the derived category of (cdh) motives, answering a question of Kontsevich. This has topological consequences: in particular, it endows the homology of the framed E2 operad (classifying configurations of little two-disks with choice of basepoint) with coefficients in any finite, or p-adic field, with action by the Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),$ and implies a new proof of the conjecture of Deligne on formality of the framed E2 operad, as well as a new higher-genus generalization.