## Quantization and quantum geometric representation theory

Much of geometric representation theory is concerned with realizing representations of algebraic groups, their Lie algebras, Weyl groups, and associated Hecke algebras in geometric terms, i.e. in terms of various categories of sheaves on algebraic spaces constructed from the group. Quantum geometric representation theory is a relatively young field, in which one first $q$-deforms the classical geometric constructions, in a way compatible with the deformation to the quantum group, and studies the resulting categories.

### Collaborators

M. Balagovic, D. Ben-Zvi, A. Brochier, P. Etingof, I. Ganev, S. Gunningham, K. Kremnizer, X. Ma

### Finished Papers

*Fourier transform for quantum D-modules via the punctured torus mapping class group.*

w/ A. Brochier arXiv:1403.1841.We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to $H$. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in arXiv:0805.2766, and hence construct a homomorphism to the Heisenberg double $D_H$, which is an isomorphism if $H$ is factorizable.

The universal property of $E_H$ endows it with an action by algebra automorphisms of the mapping class group $\widetilde{SL_2(\mathbb{Z})}$ of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when $H=U_q(\mathfrak{g})$, the quantum Fourier transform degenerates to the classical Fourier transform on $D(\mathfrak{g})$ as $q\to 1$.*Quantized Multiplicative Quiver Varieties.*

Adv Math. Vol 250 (2014). arXiv:1010.4076Beginning with the data of a quiver $Q$, and its dimension vector $\mathbf{d}$, we construct an algebra $D_q=D_q(Mat_d(Q))$, which is a flat $q$-deformation of the algebra of differential operators on the affine space $Mat_d(Q)$. The algebra $D_q$ is equivariant for an action by a product of quantum general linear groups, acting by conjugation at each vertex. We construct a quantum moment map for this action, and subsequently define the Hamiltonian reduction $A^\lambda_\mathbf{d}(Q)$ of $D_q$ with moment parameter $\lambda$. We show that $A^\lambda_d(Q)$ is a flat formal deformation of Lusztig's quiver varieties, and their multiplicative counterparts, for all dimension vectors satisfying a flatness condition of Crawley-Boevey: indeed the product on $A^\lambda_d(Q)$ yields a Fedosov quantization the of symplectic structure on multiplicative quiver varieties. As an application, we give a description of the category of representations of the spherical double affine Hecke algebra of type $A_{n-1}$, and its generalization constructed by Etingof, Oblomkov, and Rains, in terms of a quotient of the category of equivariant $D_q$-modules by a Serre sub-category of aspherical modules.

**Quantum symmetric pairs and representations of double affine Hecke algebras of type $C^\vee C_n$.**

w/ X. Ma. Selecta Mathematica, Vol. 17, No. 1 (2011). arXiv:0908.3013.We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type $BC$ generalization of the results in arXiv:0805.2766.

*Quantum D-modules, elliptic braid groups, and double affine Hecke algebras.*

Int. Math. Res. Not. (2009). arXiv:0805.2766We build representations of the elliptic braid group from the data of a quantum $D$-module $M$ over a ribbon Hopf algebra $U$. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain geometric constructions of Calaque, Enriquez, and Etingof concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of $U=U_t(sl_N)$, as $t$ limits to $1$. In the latter case, we produce representations of the double affine Hecke algebra of type $A_{n-1}$, for each $n$.

### Works in Progress

- The quantum geometric Langlands TFT. w/ D. Ben-Zvi and A. Brochier. The aim of this project is to compute factorization homology of Rep $U_q\mathfrak{g}$ on surfaces, and relate this to a host of constructions new and old in the theory of quantum geometric representation theory. In particular, all of the structures of the papers I have written below can be readily understood in the context of the QGL TFT.
- The quantum geometric Langlands TFT for GL_2. w/ M. Balagovic. With Martina Balagovic, we aim to give a complete treatement of the geometric Langlands TFT for GL_2, including its relation to the double affine Hecke algebras for generic and root of unity parameters.
- The quantum geometric Langlands TFT in dimension 3. w/ A. Brochier and N. Snyder. The aim of this project is to extend the construction of the quantum geometric Langlands TFT rigorously to dimension 3.
- Quantum Springer theory. w/ D. Ben-Zvi and S. Gunningham. The aim of this project is to study natural domain walls between reductive $G$ and its Levi subgroups $L$ corresponding to parabolic subgroups $P$, and develop their implications to the respective QGL theories of $G$ and $L$.
- Quantization of multiplicative hypertoric varieties. w/ N.Cooney, I. Ganev, K. Kremnitzer. The aim of this project is to construct the aforementioned quantizations, and study their geometry when quantum parameters become a root of unity.
- Quantization of the stack $BG$. w/ P. Safronov. Exciting recent work of T. Pantev, B. Toen, M. Vaquie, G. Vezzosi constructs a shifted symplectic structure on $BG$. It is generally expected that there should correspond a $P_2$ structure on $BG$, and that the category $Rep U_q\mathfrak{g}$ gives rise to a quantization of this structure. We are working to make sense of this statement.

## Tensor Categories: structure theory, classification and representation theory

I study tensor categories in all their various flavours: fusion, braided, symmetric, stable $\infty$-. I am especially interested in understanding tensor categories as categorifications of Frobenius algebras; this analogy is espeically strong when the category is moreover pivotal. Studying these structures typically involves higher categorical and homotopical techniques, but it has the attractive feature of being very complete, thus bringings these often abstract tools down to earth.

Fusion categories possess a host of finiteness properties, which make classification schemes (by global dimension, by number of simple objects, by dimensions of generators, etc...) feasible; their structure and representation theory mimics that of finite dimensional Frobenius algebras. Braided tensor categories play an important role in quantum computation, and their structure and representation theory mimics that of commutative Frobenius algebras (especially if when they are ribbon).

### Collaborators

Pinhas Grossman, Pavel Etingof, Eric Larson, Scott Morrison, Noah Snyder

### Finished Papers

*Cyclic extensions of fusion categories via the Brauer-Picard groupoid*

w/ P. Grossman and N. Snyder. to appear in Quantum Topology, arXiv:1211.6414.We construct a long exact sequence computing the obstruction space, $\pi_1(BrPic(\mathcal{C}_0))$, to $G$-graded extensions of a fusion category $\mathcal{C}_0$. The other terms in the sequence can be computed directly from the fusion ring of $\mathcal{C}_0$. We apply our result to several examples coming from small index subfactors, thereby constructing several new fusion categories as $G$-extensions. The most striking of these is a $\mathbb{Z}/2\mathbb{Z}$-extension of one of the Asaeda-Haagerup fusion categories, which is one of only two known 3-supertransitive fusion categories outside the ADE series.

In another direction, we show that our long exact sequence appears in exactly the way one expects: it is part of a long exact sequence of homotopy groups associated to a naturally occuring fibration. This motivates our constructions, and gives another example of the increasing interplay between fusion categories and algebraic topology.

*On the classification of certain fusion categories*

w/ Eric Larson. J. Noncommutative Geometry Vol 3, Issue 3 (2009). arXiv:0812.1603We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of $\mathbb{Z}/3\mathbb{Z}$-graded fusion categories, which are generalizations of the $\mathbb{Z}/2\mathbb{Z}$-graded Tambara-Yamagami categories. Our proofs are based on the recently developed theory of extensions of fusion categories.

## Non-commutative algebra and differential geometry

Many approaches to non-commutative algebraic geometry follow one of two tracks: either (1) to study non-commutative algebras which are "close to" commutative algebras, i.e. which depend on a parameter and such that at special values of the parameter they become commtuative -- this is the spirit of "quantum geometric representation theory" above, or (2) to develop fundamental results about representation theory of general non-commutative rings by analogy with Groethendieck-style algebraic geometry.

The line of research below was pioneered by Kapronov, and Feigin-Shoikhet, and instead studies those non-commutative algebras which are complete in their commutator filtration. Such "Lie-nilpotent" algebras can be imagined to lie on the normal cone to the space of commutative algebras within the space of all algebras, viewing the commutator ideal as the defining ideal of commutative algebras. The papers below each take this idea seriously, and relate basic algebraic invariants of the non-commutative algebra in terms of algebro-geometric constructions on its abelianization.

### Collaborators

N. Arbesfeld, A. Bapat, S. Bhupatiraju, Y. Chen, P. Etingof, W. Kuszmaul, J. Li, H. Orem, M. Zhang

### Papers

*An algebro-geometric construction of lower central series of associative algebras*

w/ H. Orem. To appear in Int. Math. Res. Not. arXiv:1302.2992The lower central series invariants $M_k$ of an associative algebra $A$ are the two-sided ideals generated by $k$-fold iterated commutators; the $M_k$ provide a filtration of $A$. We study the relationship between the geometry of $X = Spec A_{ab}$ and the associated graded components $N_k$ of this filtration. We show that the $N_k$ form coherent sheaves on a certain nilpotent thickening of $X$, and that Zariski localization on $X$ coincides with noncommutative localization of $A$. Under certain freeness assumptions on $A$, we give an alternative construction of $N_k$ purely in terms of the geometry of $X$ (and in particular, independent of $A$). Applying a construction of Kapranov, we exhibit the $N_k$ as natural vector bundles on the category of smooth schemes.

*Lower central series of a free associative algebra over the integers and finite fields*

w/ S. Bhupatiraju, P. Etingof, D. Jordan, W. Kuszmaul, J. Li. J. Algebra 2012. arXiv:1203.1893.Consider the free algebra $A_n$ generated over $\mathbb{Q}$ by $n$ generators $x_1, \ldots, x_n$. Interesting objects attached to $A = A_n$ are members of its lower central series, $L_i = L_i(A)$, defined inductively by $L_1 = A, L_{i+1} = [A,L_{i}]$, and their associated graded components $B_i = B_i(A)$ defined as $B_i=L_i/L_{i+1}$. These quotients B_i, for i at least 2, as well as the reduced quotient $\bar{B}_1=A/(L_2+A L_3)$, exhibit a rich geometric structure, as shown by Feigin and Shoikhet and later authors, (Dobrovolska-Kim-Ma,Dobrovolska-Etingof,Arbesfeld-Jordan,Bapat-Jordan). We study the same problem over the integers $\mathbb{Z}$ and finite fields $\mathbb{F}_p$. New phenomena arise, namely, torsion in $B_i$ over $\mathbb{Z}$, and jumps in dimension over $\mathbb{F}_p$. We describe the torsion in the reduced quotient $\bar{B}_1$ and $B_2$ geometrically in terms of the De Rham cohomology of $\mathbb{Z}^n$. As a corollary we obtain a complete description of $\bar{B}_1(A_n(\mathbb{Z}))$ and $\bar{B}_1(A_n(\mathbb{F}_p))$, as well as of $B_2(A_n(\mathbb{Z}[\frac{1}{2}]))$ and $B_2(A_n(\mathbb{F}_p))$, $p>2$. We also give theoretical and experimental results for $B_i$ with $i>2$, formulating a number of conjectures and questions based on them. Finally, we discuss the supercase, when some of the generators are odd (fermionic) and some are even (bosonic), and provide some theoretical results and experimental data in this case.

*The lower central series of the symplectic quotient of a free associative algebra*

w/ B. Bond. J. Pure & Appl. Alg. arXiv:1111.2316.We study the lower central series filtration $L_k$ for a symplectic quotient $A=A_{2n}/\lt w \gt$; of the free algebra $A_{2n}$ on $2n$ generators, where $w=\sum [x_i,x_{i+n}]$. We construct an action of the Lie algebra $H_{2n}$ of Hamiltonian vector fields on the associated graded components of the filtration, and use this action to give a complete description of the reduced first component $\bar{B}_1(A)= A/(L_2 + AL_3)$ and the second component $B_2=L_2/L_3$, and we conjecture a description for the third component $B_3=L_3/L_4$.

*Lower central series of free algebras in symmetric tensor categories*

w/ A. Bapat. J. Alg, 2012. arXiv:1001.1375.We continue the study of the lower central series of a free associative algebra, initiated by B. Feigin and B. Shoikhet (arXiv:math/0610410). We generalize via Schur functors the constructions of the lower central series to any symmetric tensor category; specifically we compute the modified first quotient $\bar{B}_1$, and second and third quotients $B_2$, and $B_3$ of the series for a free algebra $T(V)$ in any symmetric tensor category, generalizing the main results of arXiv:math/0610410 and arXiv:0902.4899. In the case $A_{m|n}:=T(\mathbb{C}^{m|n})$, we use these results to compute the explicit Hilbert series. Finally, we prove a result relating the lower central series to the corresponding filtration by two-sided associative ideals, confirming a conjecture from arXiv:0805.1909, and another one from arXiv:0902.4899, as corollaries.

*New results on the lower central series of an free associative algebra*

w/ N. Arbesfeld. J. Alg, 2010. arXiv:0902.4899We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated by B. Feigin and B. Shoikhet. We establish a linear bound on the degree of tensor field modules appearing in the Jordan-Hoelder series of each graded component, which is conjecturally tight. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.

*Poisson traces in positive characteristic*

w/ Y. Chen, P. Etingof, M. Zhang. arXiv:1112.6385.We study Poisson traces of the structure algebra $A$ of an affine Poisson variety $X$ defined over a field of characteristic p. According to arXiv:0908.3868v4, the dual space $HP_0(A)$ to the space of Poisson traces arises as the space of coinvariants associated to a certain D-module $M(X)$ on $X$. If $X$ has finitely many symplectic leaves and the ground field has characteristic zero, then $M(X)$ is holonomic, and thus $HP_0(A)$ is finite dimensional. However, in characteristic $p$, the dimension of $HP_0(A)$ is typically infinite. Our main results are complete computations of $HP_0(A)$ for sufficiently large $p$ when $X$ is either 1) a quasi-homogeneous isolated surface singularity in the three-dimensional space, 2) a quotient singularity $V/G$, for a symplectic vector space V by a finite subgroup $G \subset Sp(V)$, and 3) a symmetric power of a symplectic vector space or a Kleinian singularity. In each case, there is a finite nonnegative grading, and we compute explicitly the Hilbert series. The proofs are based on the theory of D-modules in positive characteristic.