arXiv subjects

David Jordan

Publications and source records attributed to David Jordan.

At least 37 records · Page 2Linked to original sources

arXiv2020

The quantum Frobenius for character varieties and multiplicative quiver varieties

We prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and we prove that the Azumaya locus of the Kauffman bracket skein algebras contains the smooth locus, proving a strong form of the Unicity Conjecture of Bonahon and Wong. The proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as it arises in each setting. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of Frobenius Poisson orders. We use these tools to construct canonical central subalgebras of quantum algebras, and explicitly compute the resulting Azumaya loci we encounter, using a natural nondegeneracy assumption.

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arXiv2020

Invertible braided tensor categories

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the semisimple case. More generally, we characterize invertibility for E_1- and E_2-algebras in an arbitrary symmetric monoidal oo-category, and we conjecture a similar characterization of invertible E_n-algebras for any n. Finally, we propose the Picard group of BrTens as a generalization of the Witt group of non-degenerate braided fusion categories, and pose a number of open questions about it.

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arXiv2019

Quantum Weyl algebras and reflection equation algebras at a root of unity

We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group $\mathrm{GL}_2$. These examples illustrate in elementary terms much more general phenomena explored further in [Ganev-Jordan-Safronov 2019].

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arXiv2018

On dualizability of braided tensor categories

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produce a framed 3-dimensional local TFT attached to the category of representations of a quantum group at any value of $q$.

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arXiv2018

Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants $\int_S\mathcal A$ of a surface $S$, determined by the choice of a braided tensor category $\mathcal A$, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a {\em braided module category} for $\mathcal A$, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called {\em quantum moment maps}. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided $\mathcal A$-modules are objects of the torus category $\int_{T^2}\mathcal A$. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of $\mathcal A=\operatorname{Rep_q} G$ with the category $\mathcal D_q(G/G)-\operatorname{mod}$ of equivariant quantum $\mathcal D$-modules. When $G=GL_n$, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra (DAHA) $\mathbb{SH}_{q,t}$.

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arXiv2018

Integrating quantum groups over surfaces

We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the $(0,1,2)$-dimensional part of Crane-Yetter-Kauffman 4D TFTs associated to modular categories. Starting from modules for the Drinfeld-Jimbo quantum group $U_q(\mathfrak g)$ we obtain in this way an aspect of topologically twisted 4-dimensional ${\mathcal N}=4$ super Yang-Mills theory, the setting introduced by Kapustin-Witten for the geometric Langlands program. For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of $G$-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to $U_q(\mathfrak g)$, and from the punctured torus we recover the algebra of quantum differential operators associated to $U_q(\mathfrak g)$. From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum $\mathcal D$-modules.

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arXiv2017

The rectangular representation of the double affine Hecke algebra via elliptic Schur-Weyl duality

Given a module $M$ for the algebra $\mathcal{D}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper we take $M$ to be the basic $\mathcal{D}_{\mathtt{q}}(G)$-module, i.e. the quantized coordinate algebra $M= \mathcal{O}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ combinatorially in terms of walks in the type $A$ weight lattice, and standard periodic tableaux, and subsequently identify $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ with the irreducible "rectangular representation" of height $N$ of the double affine Hecke algebra.

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arXiv2017

The center of the reflection equation algebra via quantum minors

We give simple formulas for the elements $c_k$ appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (REA) associated to the quantum group $U_q(\mathfrak{gl}_N)$, answering a question of Kolb and Stokman. The $c_k$'s are certain canonical generators of the center of the REA, and hence of $U_q(\mathfrak{gl}_N)$ itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known. As byproducts, we prove a quantum Girard-Newton identity relating the $c_k$'s to the so-called quantum power traces, and we give a new presentation for the quantum group $U_q(\mathfrak{gl}_N)$, as a localization of the REA along certain principal minors.

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arXiv2017

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

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 $q\to 1$.

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arXiv2016

The Harish-Chandra isomorphism for quantum GL_2

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra D_q(GL_2) of quantum differential operators on GL_2, to the spherical double affine Hecke algebra associated to GL2. The isomorphism holds for all deformation parameters non-zero q, t, such that t does not equal to +/-i, and q is not a non-trivial root of unity. We also discuss its extension to the root of unity case.

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arXiv2013

Cyclic extensions of fusion categories via the Brauer-Picard groupoid

We construct a long exact sequence computing the obstruction space, pi_1(BrPic(C_0)), to G-graded extensions of a fusion category C_0. The other terms in the sequence can be computed directly from the fusion ring of 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 Z/2Z-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.

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arXiv2013

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

The 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.

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arXiv2012

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

We study the lower central series filtration L_k for a symplectic quotient A=A_{2n}/ 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.

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arXiv2012

Lower central series of free algebras in symmetric tensor categories

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(\CC^{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.

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arXiv2012

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

Consider the free algebra A_n generated over Q by n generators x_1, ..., 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 Z and finite fields F_p. New phenomena arise, namely, torsion in B_i over Z, and jumps in dimension over F_p. We describe the torsion in the reduced quotient RB_1 and B_2 geometrically in terms of the De Rham cohomology of Z^n. As a corollary we obtain a complete description of \bar{B}_1(A_n(Z)) and \bar{B}_1(A_n(F_p)), as well as of B_2(A_n(Z[1/2])) and B_2(A_n(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.

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arXiv2011

Poisson traces in positive characteristic

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 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 in 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.

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arXiv2011

Quantized multiplicative quiver varieties

Beginning with the data of a quiver Q, and its dimension vector 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_d(Q) of D_q with moment parameter λ. We show that A^λ_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^λ_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.

math.QA↗

arXiv2010

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

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.

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