Welcome to the Research page of my website. Below is an informal discussion of each of the works I have completed while in graduate school at MIT. I am using "MathJax", an open source LaTex compiler for Javascript in order to display the mathematics on this page. Please email me if you have trouble viewing anything on this page. Please see the linked papers for complete details, and for references mentioned in the summaries.

## Research Interests

I study various algebraic systems which lie between representation theory and physics. More precisely, I study tensor categories in their various flavors (fusion, braided, symmetric, ...), and also quantization problems involving quantum groups and their differential geometry, specifically via the algebra of quantum differential operators. I am currently trying to develop a braided generalization of quiver varieties.

## Quantum differential operators, braid groups, and double affine Hecke algebras

An early success in the theory of quantum groups was the construction of Reshetikhin-Turaev invariants of knots and braids. The braid group $$B_n$$ may be described as the first fundamental group of a configuration space, $B_n:= \pi_1(\mathrm{Conf}_n(\mathbb{C})).$ Here the nth configuration space is roughly the space of size-n subsets of X. Representation categories of quantum groups give many examples of braided tensor categories $$\mathcal{C}$$, from which one can assign, to each tuple $$V_1,\ldots,V_n$$ of objects of $$\mathcal{C}$$, a representation of the (pure) braid group to the space $$V_1\otimes \cdots \otimes V_n$$. In this line of research, we ask what further data is required in order to encode representations of more general braid groups associated to higher genus surfaces. That is, we let X denote an oriented, smooth, real surface, and set: $B_n(X) := \pi_1(\mathrm{Conf}_n(X)).$ For X a topologcal torus, these braid groups are called double affine braid groups, and have a distinguished quotient called the double affine Hecke algebra. Double affine Hecke algebras are connected to many important problems in representation theory. One may now ask what categorical information is required to produce representations of these more general braid groups? We pose answers to these questions in the following two articles:
• Quantum D-modules, elliptic braid groups, and double affine Hecke algebras. 2008 arXiv:0805.2766, published in International Math Research Notes.
• In this article, we construct representations of the double affine braid group and the double affine Hecke algebra following the ideas above. In this case, the answer to the question of what extra information is required comes in the way of a certain algebra in the category $$\mathcal{C}$$, known as the algebra $$D_q(G)$$ of quantum differential differential operators on the quantum group $$G$$. These algebras are $$q$$-deformations of the algebras of differential operators on an algebraic group $$G$$, and as such, their representation theory is well-understood. Given a representation $$M$$ of $$D_q(G)$$, and an object $$V$$ of $$\mathcal{C}$$, we construct an action of $$B_n(X)$$ on the space: $W:= \mathrm{Hom}_{\mathcal{C}}(\mathbf{1}, V^{\otimes n}\otimes M).$ I won't go into the details of the construction here, but let me say a few words about its motivations and connections to Lie theory. First, it is important to observe that the braid group of any surface $$X$$ receives a homomorphism from the braid group of $$\mathbb{C}$$, induced by the inclusion of $$\mathbb{C}\cong\mathbb{R}^2$$ into a small contractible neighborhood. Second, while the configurations of points on a plane organize into an $$E_2$$-operad, the configurations of points on a topological torus do not have a natural composition. However, it is possible to construct a module $$M$$ over the $$E_2$$-operad, which encodes the geometry of $$\mathrm{Conf}_n(X)$$. Thus, in order to extend the work of Reshetikhin and Turaev, it is natural to seek a module cateory over $$\mathcal{C}$$. It is well known that the category of modules over a $$\mathcal{C}$$-algebra has such a structure. Why it happens that $$D_q(G)$$ is precisely the correct algebra is discussed in more detail in the paper. There is an interesting parallel between the triples, which was explained to me by Pavel Etingof:

(Additive group $$\mathbb{G}_a$$, Multiplicative group $$\mathbb{G}_m$$, Elliptic curve $$\mathbb{E_\tau}$$)
(Lie algebras, Lie Groups, Quantum groups)
(Rat'l Cherednik algebra, Trig. Cherednik algebra, Double Affine Hecke algebra).

In their paper, "Duality between $${\frak{sl}}_N(\mathbb{C})$$ and the degenerate affine Hecke algebra", Arakawa and Suzuki initiated a sort of Schur-Weyl dualtiy, giving a functor from $$\mathfrak{sl}_N$$-modules to representations of the degenerate affine Hecke algebra. In their paper, "Universal KZB equations I: the elliptic case", Calaque, Enriquez, and Etingof extended these constructions to Rational and Trigonometric Cherednik algebras. The results of this paper may be considered an elliptic version of those constructions. It gives a concrete connection between the final component in each row above.

Finally, I should add that each of the above constructions is not so much about the categories of $$D(\frak{g})$$-, $$D(G),$$- or $$D_q(G)$$-modules, but really about the categories of $$D(\mathfrak{g}//G)$$-, $$D(G//G)$$-, or $$D_q(G//G)$$-modules, of $$D$$-modules on the adjoint quotients. This is encoded in the appearance of the invariants $$\mathrm{Hom}(\mathbf{1},-)$$ in each construction. What is really going on is that we are giving a sort of twisted Hamiltonian reduction, on which construction the Schur-Weyl-dual algebras act.

• Quantum symmetric pairs and representations of double affine Hecke algebras of type $$C^\vee C_n$$, with Xiaoguang Ma. arXiv:0908.3013, to appear in Selecta Mathematica.
• This paper extends the results above to certain curves $$\mathbf{E}_\tau / \Gamma$$, where $$\Gamma=\mathbb{Z}/2$$ is acting by $$x\mapsto -x$$. The fact that $$\mathbb{Z}/2\mathbb{Z}$$-quotients of $$E_\tau$$ might be related to the type $$C$$ (and thus, type $$B$$) root systems is perhaps not too surprising, given the construction of the latter in terms of "folding" root systems of type $$A$$. In particular, we consider a $$q$$-deformed category of $$D$$-modules on the quotient $$K\backslash G/K$$, where $$G=GL_N$$, and $$K=GL_p\times GL_r$$, with $$p+r=N$$. We construct an action of the double affine Hecke algebra of type $$C^\vee C_n$$, which is the correct "double affine" version of the type $$C$$ Hecke algebra.

The proofs here are considerably more involved than in the type $$A$$ constructions above, and involve a great deal of work by many authors working in the area of "quantum symmetric pairs". One side bonus of our constructions is that they provide some extra organization to this rather technical corner of the theory of quantum groups, in terms of elementary topology of surfaces.

## Lower central series filtrations

The lower central series of an associative algebra $$A$$ is the filtration by Lie ideals with $$L_1=A$$, and $$L_{k+1}=[A,L_k]$$. The successive quotients $$B_i=L_i/L_{i+1}$$ were first studied by Feigin and Shoikhet. When $$A=A_n$$ is the free algebra on $$n$$ generators, Feigin and Shoikhet demonstrated that each $$B_i$$ has a surprising amount of structure. More precisely, they exhibited an action of the Lie algebra $$W_n$$ of polynomial vector fields on each $$B_i$$, for $$i\geq2$$. This gives a drastic generalization of the ordinary action of $$W_n$$ on $$\mathbb{C}[x_1,\ldots,x_n]=A_n/A_nL_2A_n$$ by partial differentiation. Moreover, it follows from their work, and classical work of Rudakov on cohomology of Lie algebras that each B_i admits a further filtration by so-called tensor field modules $$F_\lambda$$, whose origins lie in differential geometry - as their name suggests, they are tensor fields on $$\mathbb{C}^n$$ which transform under local changes of coordinates like the $$\mathfrak{gl}_n$$-module $$V_\lambda$$.

Moreover, Feigin and Shoikhet identified which $$F_\lambda$$ appear in the first two quotients; as it turns out they are simply differential forms on $$\mathbb{C}^n$$. In particular, only finitely many $$F_\lambda$$ appear. They conjectured that this finiteness property held for all the quotients $$B_k$$. This conjecture was proven by Dobrovolska and Etingof, who gave an explicit bound on which $$F_\lambda$$ could occur.

What is intriguing about these results is that each $$F_\lambda$$, and therefore each $$B_i$$, for $$i\geq 2$$, has a rational Hilbert series, which means in particular that it grows polynomially in dimension as the monomial degree increases. This is in contrast to the free algebra $$A_n$$ itself, which exhibits exponential growth. One may interpret these relations as saying that the study of lower central series quotients is more firmly in the realm of elementary algebraic geometry and combinatorics than of non-commutative algebra. Indeed, since differential forms occur already in the lower central series, one can hope to find more connections between geometry of algebraic varieties, and lower central series of non-commutative algebras related to them.

It is safe to say that the deeper significance of the combinatorial observations in these papers is yet to become completely clear.

• New results on the lower central series quotients of a free associative algebra, with Noah Arbesfeld. arXiv:0908.3013, published in Journal of Algebra, 2009.
• In this paper, we improve the bound of Dobrovolska and Etingof on which $$F_\lambda$$ can appear in the Jordan-Hoelder series from a quadratic to a linear bound. This bound was suggested by computer experiment. Perhaps more interesting than the numerical significance of the new bound is the identities in the free algebra which we find in order to prove the bound. The key technical step is to show, for $$m\geq 3$$, that: $B_m = [L,B_{m-1}] + [Q,B_{m-1}] + [C,B_{m-1}],$ where L, Q, and C are the span of linear, quadratic and cubic expressions in the variables, respectively. We conjectured in this paper, and proved in the work below with Asilata Bapat, that one may eliminate the cubic terms in the sum above. The identities we find are universal, since they lie in the free algebra, and they may be interpreted as generalizations of the Jacobi identity. The techniques in this paper are completely elementary.
• Lower central series of free algebras in a symmetric tensor category, with Asilata Bapat. arXiv:1001.1375.

In this paper, we generalize many constructions and results in this area to algebras in symmetric tensor categories. In particular, we obtain explicit results concerning the structure of the lower central series quotients for super-algebras $$A_{m|n}$$. We also establish a technique whereby one may study the quotients $$B_m(A_n)$$ for $$n\to\infty$$, and make conclusions about arbitrary tensor categories. Somewhat tangentially, we find a crucial identity allowing to remove the cubic terms appearing in the previous work.

## Classification of Fusion categories in small dimensions

Fusion categories are certain mild generalizations of finite groups. Maschke's theorem tells us that the category Rep(G) of representations of a finite group over $$\mathbb{C}$$, is a semi-simple tensor category with finitely many isomorphism classes of simple objects. A fusion category is an abstract tensor category with these properties: it should be semi-simple and have finitely many isomorphism classes of simple objects. A very general program is to classify fusion categories with various restrictions on their dimensions, braidings/symmetries, number of simple objects, etc. One doesn't seek a full classification (which would encompass the classification of finite groups, as well as compact Lie groups), except in certain "small" cases of interest. Pioneering work by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik enables us to combine methods from the theory of finite groups, such as arithmetic of algebraic integers, Galois actions, Burnside's theorem, nilpotency, solvability, etc. with modern techniques in category theory, in particular homotopy theory and higher groupoids. Thus the study of fusion categories has become a sort of playground, where many sophisticated notions from higher category theory obtain concrete and computable formulations.
• On the classification of certain fusion categories, with Eric Larson. arXiv:0812.1603, published in Journal of Noncommutative Geometry. 2008

In this paper, we consider the classification problem for fusion categories of dimension $$pq^2$$. Before this work, fusion categories of dimensions $$p^k, pq,$$ and $$pqr$$ (p,q,r distinct) were classified. They were shown to be "group-theoretical", a definition introduced by Victor Ostrik, which essentially means they are built from constructions in elementary group theory, rather than being genuinely novel categories. We give a complete and concrete description of all categories of dimension $$pq^2$$, which relies heavily on the impressive machinery developed in "Fusion categories and homotopy theory", by Etingof Nikshych, and Ostrik. A corollary of our results is that there exist non-group-theoretical categories in dimensions $$pq^2$$, for many p and q.

This paper is perhaps best seen as a companion paper to "Fusion categories and homotopy theory", in that the powerful techniques of that paper are brought to bear on a specific tractible problem, and allow a complete classification.

In this paper, we also apply our techniques to classify certain "$$\mathbb{Z}/3\mathbb{Z}$$" versions of Tambara-Yamagami categories, which arose in problems of conformal field theory.

Please note that there are two arXiv articles and several print articles published by another David A. Jordan. I did not, in fact, publish mathematics before I was born, nor am I the head of Pure Mathematics at Sheffield University. I am also, unfortunately, not the British pop sensation David Jordan. If you know of any other David Jordan's who I am not, please let me know!