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arXiv2007

F_q[M_n], F_q[GL_n] and F_q[SL_n] as quantized hyperalgebras

The quantized universal enveloping algebra U_q(gl(n)) has two integral forms - over Z[q,q^{-1}] - the restricted (by Lusztig) and the unrestricted (by De Concini and Procesi) one. Dually, the quantum function algebra F_q[GL(n)] has two integral forms, namely those of all elements - of F_q[GL(n)] - which take values in Z[q,q^{-1}] when paired respectively with the restricted or the unrestricted form of U_q(gl(n)). The first one is the well-known form generated over Z[q,q^{-1}] by the entries of a q-matrix and the inverse of its quantum determinant. In this paper instead we study the second integral form, say F'_q[GL(n)], i.e. that of all elements which are Z[q,q^{-1}]-valued over the unrestricted form of U_q(gl(n)). In particular we yield a presentation of it by generators and relations, and a PBW-like theorem: in short, it is an algebra of "quantum divided powers" and "quantum binomial coefficients". Moreover, we give a direct proof that F'_q[GL(n)] is a Hopf subalgebra of F_q[GL(n)], and that its specialization at q=1 is the Z-hyperalgebra over gl(n)^*, the Lie bialgebra dual to gl(n). In addition, we describe explicitly the specializations of F'_q[GL(n)] at roots of 1, and the associated quantum Frobenius (epi)morphism. The same analysis is done for F'_q[SL(n)] and (as a key step) F'_q[Mat(n)]: in fact, for the latter the strongest results are obtained. This work extends to general n>2 the results for n=2, already treated in math.QA/0411440.

math.QA↗

arXiv2002

Braiding structures on formal Poisson groups and classical solutions of the QYBE

If g is a quasitriangular Lie bialgebra, one can asks what is the geometrical meaning of its r-matrix. A first answer was given in a paper by Weinstein and Xu, using purely geometrical means: roughly, one has that the formal Poisson group F[[g^*]] is endowed with a "braiding", i.e. a distinguished operator on its tensor square which satisfy quasitriangularity conditions (in particular, it is a solution of the QYBE). Independently, the authors also found, by means of quantum groups, that F[[g^*]] has a braiding. In this paper we compare these two approaches and their outcomes. First, we show that the two braidings obtained in the two processes do share several similar properties (in particular, the construction is functorial). Second, in the simplest case (G = SL_2) we prove that the two braidings do coincide. The question then rises of whether they are always the same: this problem is addressed and solved in math.QA/0207235, in a much broader context, in which unicity of braidings is proved.

math.QA↗

arXiv2009

Twisted Homology of Quantum SL(2) - Part II

We complete the calculation of the twisted cyclic homology of the quantised coordinate ring of SL(2) that we began in math.QA/0405249. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

math.QA↗

arXiv2015

Classification of quantum groups and Belavin--Drinfeld cohomologies for orthogonal and symplectic Lie algebras

In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric $r$-matrices from the Belavin-Drinfeld list for simple Lie algebras of type $B$, $C$, and $D$.

math.QA↗

arXiv2010

Non-cyclotomic fusion categories

Etingof, Nikshych and Ostrik ask in arXiv:math.QA/0203060 if every fusion category can be completely defined over a cyclotomic field. We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor arXiv:math.OA/9803044 and one coming from the newly constructed extended Haagerup subfactor arXiv:0909.4099 can not be completely defined over a cyclotomic field. On the other hand, we show that the double of the even part of the Haagerup subfactor is completely defined over a cyclotomic field. We identify the minimal field of definition for each of these fusion categories, compute the Galois groups, and identify their Galois conjugates.

math.QA↗

arXiv2016

On the Kontsevich $\star$-product associativity mechanism

The deformation quantization by Kontsevich [arXiv:q-alg/9709040] is a way to construct an associative noncommutative star-product $\star=\times+\hbar \{\ ,\ \}_{P}+\bar{o}(\hbar)$ in the algebra of formal power series in $\hbar$ on a given finite-dimensional affine Poisson manifold: here $\times$ is the usual multiplication, $\{\ ,\ \}_{P}\neq0$ is the Poisson bracket, and $\hbar$ is the deformation parameter. The product $\star$ is assembled at all powers $\hbar^{k\geq0}$ via summation over a certain set of weighted graphs with $k+2$ vertices; for each $k>0$, every such graph connects the two co-multiples of $\star$ using $k$ copies of $\{\ ,\ \}_{P}$. Cattaneo and Felder [ arXiv:math/9902090 [math.QA] ] interpreted these topological portraits as the genuine Feynman diagrams in the Ikeda-Izawa model [arXiv:hep-th/9312059] for quantum gravity. By expanding the star-product up to $\bar{o}(\hbar^3)$, i.e., with respect to graphs with at most five vertices but possibly containing loops, we illustrate the mechanism Assoc = Operator(Poisson) that converts the Jacobi identity for the bracket $\{\ ,\ \}_{P}$ into the associativity of $\star$. Key words: Deformation quantization, associative algebra, Poisson bracket, graph complex, star-product PACS: 02.40.Sf, 02.10.Ox, 02.40.Gh, also 04.60.-m

math.QA↗

arXiv2004

Spectral Decomposition and Baxterisation of Exotic Bialgebras and Associated Noncommutative Geometries

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev-Reshetikhin-Takhtajan RTT prescription with non-triangular R-matrices which are denoted $R_{03}$ and $R_{14}$ in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1/1). We give the spectral decomposition which involves two, resp., three, projectors. These projectors are then used to provide the Baxterisation procedure with one, resp., two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess-Zumino prescription. In all these constructions there appear non-standard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators.

math.QA↗

arXiv2008

Opers with irregular singularity and spectra of the shift of argument subalgebra

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singulairity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, we obtain the structure of a Gorenstein ring on any such module. This fact may have geometric significance related to the intersection cohomology of Schubert varieties in the affine Grassmannian.

math.QA↗

arXiv2009

The braid group surjects onto $G_2$ tensor space

Let V be the 7-dimensional irreducible representation of the quantum group U_q(g_2). For each n, there is a map from the braid group B_n to the endomorphism algebra of the n-th tensor power of V, given by R-matrices. We can extend this linearly to a map on the braid group algebra. Lehrer and Zhang (MR2271576) prove this map is surjective, as a special case of a more general result. Using Kuperberg's spider for G_2 from arXiv:math.QA/9201302, we give an elementary diagrammatic proof of this result.

math.QA↗

arXiv2015

Populations of Solutions to Cyclotomic Bethe Equations

We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain "extended" master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an "extended" non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of [Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017], for diagram automorphisms of Kac-Moody Lie algebras. In the case of type A with the diagram automorphism, we associate to each population a vector space of quasi-polynomials with specified ramification conditions. This vector space is equipped with a ${\mathbb Z}_2$-gradation and a non-degenerate bilinear form which is (skew-)symmetric on the even (resp. odd) graded subspace. We show that the population of cyclotomic critical points is isomorphic to the variety of isotropic full flags in this space.

math.QA↗

arXiv2022

Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians

We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=\langle α_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $α_{i}\in{\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\check{S}^{\dagger}_{W}$ of the quotient difference operator $\check{S}_{W}$ satisfying $\widehat{S} =\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\dim W$ annihilating $W$, and $\widehat{S}$ is a linear difference operator with constant coefficients depending on $α_{i}$ and $°p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\operatorname{ord} \check{S}^{\dagger}_{W}$, which is annihilated by $\check{S}^{\dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\in\mathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.

math.QA↗

arXiv2023

Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras

We construct a separable Frobenius monoidal functor from $\mathcal{Z}\big(\mathsf{Vect}_H^{ω|_H}\big)$ to $\mathcal{Z}\big(\mathsf{Vect}_G^ω\big)$ for any subgroup $H$ of $G$ which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in $\mathcal{Z}\big(\mathsf{Vect}_G^ω\big)$, recovering the classification of étale algebras in these categories by Davydov-Simmons [J. Algebra 471 (2017), 149-175, arXiv:1603.04650] and generalizing their classification to algebraically closed fields of arbitrary characteristic. Categories of local modules over such algebras are modular tensor categories by results of Kirillov-Ostrik [Adv. Math. 171 (2002), 183-227, arXiv:math.QA/0101219] in the semisimple case and Laugwitz-Walton [Comm. Math. Phys., to appear, arXiv:2202.08644] in the general case.

math.QA↗

arXiv2008

Quantification pour les paires symetriques et diagrammes de Kontsevich

In this article we use the expansion for biquantization described in Cattaneo-Felder [math.QA/0309180] for the case of symmetric spaces. We introduce a function of two variables $E(X,Y)$ for any symmetric pairs. This function has an expansion in terms of Kontsevich's diagrams. We recover most of the known results though in a more systematic way by using some elementary properties of this $E$ function. We prove that Cattaneo and Felder's star product coincides with Rouviere's for any symmetric pairs. We generalize some of Lichnerowicz's results for the commutativity of the algebra of invariant differential operators and solve a long standing problem posed by M. Duflo for the expression of invariant differential operators on any symmetric spaces in exponential coordinates. We describe the Harish-Chandra homomorphism in the case of symmetric spaces by using all these constructions. We develop a new method to construct characters for algebras of invariant differential operators. We apply these methods in the case of $σ$-stable polarizations.

math.RT↗

arXiv2007

Depth three towers and Jacobson-Bourbaki correspondence

We introduce a notion of depth three tower of three rings C < B < A as a useful generalization of depth two ring extension. If A = End B_C and B | C is a Frobenius extension, this also captures the notion of depth three for a Frobenius extension in math.RA/0107064 and math.RA/0108067 such that if B | C is depth three, then A | C is depth two (cf. math.QA/0001020). If A, B and C correspond to a tower of subgroups G > H > K via the group algebra over a fixed base ring, the depth three condition is the condition that subgroup K has normal closure K^G contained in H. For a depth three tower of rings, there is a pre-Galois theory for the ring End {}_BA_C and coring (A \otimes_B A)^C involving Morita context bimodules and left coideal subrings. This is applied in the last two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.

math.RA↗

arXiv2011

Subfactors, Planar Algebras and Rotations

Growing out of the initial connections between subfactors and knot theory that gave rise to the Jones polynomial, Jones' axiomatization of the standard invariant of an extremal finite index $II_1$ subfactor as a spherical $C^*$-planar algebra, presented in arXiv:math.QA/9909027, is the most elegant and powerful description available. We make the natural extension of this axiomatization to the case of finite index subfactors of arbitrary type. We also provide the first steps toward a limited planar structure in the infinite index case. The central role of rotations, which provide the main non-trivial part of the planar structure, is a recurring theme throughout this work. In the finite index case the axioms of a $C^*$-planar algebra need to be weakened to disallow rotation of internal discs, giving rise to the notion of a rigid $C^*$-planar algebra. We show that the standard invariant of any finite index subfactor has a rigid $C^*$-planar algebra structure. We then show that rotations can be re-introduced with associated correction terms entirely controlled by the Radon-Nikodym derivative of the two canonical states on the first relative commutant, $N' \cap M$. By deforming a rigid $C^*$-planar algebra to obtain a spherical $C^*$-planar algebra and lifting the inverse construction to the subfactor level we show that any rigid $C^*$-planar algebra arises as the standard invariant of a finite index $II_1$ subfactor equipped with a conditional expectation, which in general is not trace preserving. Jones' results thus extend completely to the general finite index case. We conclude by applying our machinery to the $II_1$ case, shedding new light on the rotations studied by Huang [11] and touching briefly on the work of Popa [29]. (continued in article)

math.OA↗

arXiv2024

DAHAs of Type $C^\vee C_n$ and Character Varieties

This paper studies the spherical subalgebra of the double affine Hecke algebra of type $C^\vee C_n$ and relates it, at the classical level $q = 1$, to a certain character variety of the four-punctured Riemann sphere. This establishes a conjecture from math.QA/0504089. As a by-product, we find a completed phase space for the trigonometric van Diejen system, explicitly integrate its dynamics and explain how it can be obtained via Hamiltonian reduction.

math.RT↗

arXiv2002

Quantum group symmetry of integrable models on the half-line

This contribution to the Proceedings of the Workshop on Integrable Theories, Solitons and Duality in Sao Paulo in July 2002 summarizes results from the papers hep-th/0112023 and math.QA/0208043. We derive the non-local conserved charges in the sine-Gordon model and affine Toda field theories on the half-line. They generate new kinds of symmetry algebras that are coideals of the usual quantum groups. We show how intertwiners of tensor product representations of these algebras lead to solutions of the reflection equation. We describe how this method for finding solutions to the reflection equation parallels the previously known method of using intertwiners of quantum groups to find solutions to the Yang-Baxter equation.

hep-th↗

arXiv2006

The N to Infinity Limit of the Chiral Potts Model

We consider the N to infinity limits of the N-state chiral Potts model. We find new weights that satisfy the star-triangle relations with spin variables either taking all the integer values or having values from a continous interval. The models provide chiral generalizations of Zamolodchikov's Fishnet Model. (For the more complete version, see math.QA/9906029, where the misprints in eq. (12) are also corrected.)

hep-th↗