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A combinatorial approach to the settheoretic solutions of the YangBaxter equation
 J.MATH.PHYS
, 2004
"... A bijective map r: X 2 − → X 2, where X = {x1, · · · , xn} is a finite set, is called a settheoretic solution of the YangBaxter equation (YBE) if the braid relation r12r23r12 = r23r12r23 holds in X 3. A nondegenerate involutive solution (X, r) satisfying r(xx) = xx, for all x ∈ X, is called s ..."
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A bijective map r: X 2 − → X 2, where X = {x1, · · · , xn} is a finite set, is called a settheoretic solution of the YangBaxter equation (YBE) if the braid relation r12r23r12 = r23r12r23 holds in X 3. A nondegenerate involutive solution (X, r) satisfying r(xx) = xx, for all x ∈ X, is called squarefree solution. There exist close relations between the squarefree settheoretic solutions of YBE, the semigroups of Itype, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of squarefree solutions (X, r) and the associated YangBaxter algebraic structures — the semigroup S(X, r), the group G(X, r) and the k algebra A(k, X, r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r. We study the properties of the associated YangBaxter structures and prove a conjecture of the present author that the three notions: a squarefree solution of (settheoretic) YBE, a semigroup of I type, and a semigroup of skewpolynomial type, are equivalent. This implies that the YangBaxter algebra A(k, X, r) is PoincaréBirkhoffWitt type algebra, with respect to some appropriate ordering of X. We conjecture that every squarefree solution of YBE is retractable, in the sense of EtingofSchedler.
Grouptheoretical properties of nilpotent modular categories, eprint arXiv:0704.0195v2 [math.QA
"... Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects o ..."
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Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects ofC have integral FrobeniusPerron dimensions then C is grouptheoretical in the sense of [ENO]. As a consequence, we obtain that semisimple quasiHopf algebras of prime power dimension are grouptheoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasiLie bialgebras in terms of Manin pairs given in [Dr]). 1. introduction In this paper we work over an algebraically closed field k of characteristic 0. By a fusion category we mean a klinear semisimple rigid tensor category C with finitely many isomorphism classes of simple objects, finite dimensional spaces of morphisms, and such that the unit object 1 of C is simple. We refer the reader to [ENO] for a general theory of such categories. A fusion category is pointed if all its simple objects are invertible. A pointed fusion category is equivalent to Vec ω G, i.e., the category of Ggraded vector spaces with the associativity constraint given by some cocycle ω ∈ Z 3 (G, k × ) (here G is a finite group). 1.1. Main results. Theorem 1.1. Any braided nilpotent fusion category has a unique decomposition into a tensor product of braided fusion categories whose FrobeniusPerron dimensions are powers of distinct primes. The notion of nilpotent fusion category was introduced in [GN]; we recall it in Subsection 2.2. Let us mention that the representation category Rep(G) of a finite group G is nilpotent if and only if G is nilpotent. It is also known that fusion categories of prime power FrobeniusPerron dimension are nilpotent [ENO]. On the other hand, Vec ω G is nilpotent for any G and ω. Therefore it is not true that any nilpotent fusion category is a tensor product of fusion categories of prime power dimensions.
The multiple Zeta value algebra and the stable derivation algebra
, 2003
"... Abstract. The MZV algebra is the graded algebra over Q generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the GrothendieckTeichmüller group. We shall show that there is a canonical surjective Qlinear map from the graded dual vector space of the ..."
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Abstract. The MZV algebra is the graded algebra over Q generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the GrothendieckTeichmüller group. We shall show that there is a canonical surjective Qlinear map from the graded dual vector space of the stable derivation algebra over Q to the newzeta space, the quotient space of the subvector space of the MZV algebra whose grade is greater than 2 by the square of the maximal ideal. As a corollary, we get an upperbound for the dimension of the graded piece of the MZV algebra at each weight in terms of the corresponding dimension of the graded piece of the stable derivation algebra. If some standard conjectures by Y. Ihara and P. Deligne concerning the structure of the stable derivation algebra hold, this will become a bound conjectured in Zagier’s talk at 1st European Congress of Mathematics. Via the stable derivation algebra, we can compare the newzeta space with the ladic Galois image Lie algebra which is associated with so the Galois representation on the prol fundamental group of P1 − {0,1, ∞}.
Modular categories and orbifold models
 Comm. Math. Phys
"... Abstract. In this paper, we try to answer the following question: given a modular tensor category A with an action of a compact group G, is it possible to describe in a suitable sense the “quotient ” category A/G? We give a full answer in the case when A = Vec is the category of vector spaces; in th ..."
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Abstract. In this paper, we try to answer the following question: given a modular tensor category A with an action of a compact group G, is it possible to describe in a suitable sense the “quotient ” category A/G? We give a full answer in the case when A = Vec is the category of vector spaces; in this case,
Braided Hopf Algebras
, 2005
"... The term “quantum group” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics a ..."
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The term “quantum group” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first stage is that integral and Maschke’s theorem are found. Maschke’s theorem gives a nice criterion of semisimplicity, which is due to M.E. Sweedler. The second stage is that the Lagrange’s theorem is proved, which is due to W.D. Nichols and M.B. Zoeller. The third stage is the research of the actions of Hopf algebras, which unifies actions of groups and Lie algebras. S. Montgomery’s “Hopf algebras and their actions on rings ” contains the main results in this area. S. Montgomery and R.J. Blattner show the duality theorem. S. Montegomery, M. Cohen, H.J. Schneider, W. Chin, J.R. Fisher and the author study the relation between algebra R and the crossed product R#σH of the semisimplicity, semiprimeness, Morita equivalence and radical. S. Montgomery and M. Cohen ask whether R#σH is semiprime when R is semiprime for a finitedimensional semiprime Hopf algebra H. This is a famous semiprime problem. The
Notes on the KazhdanLusztig theorem on equivalence of the Drinfeld category and the category of Uq(g)modules, Algebr
 Represent. Theory
"... Abstract. We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category D(g, ~) of gmodules and the category of finite dimensional Uqgmodules, q = epii~, for ~ ∈ C\Q∗. Aiming at operator algebraists the result is formulated as the existence for each ~ ∈ iR of a normal ..."
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Abstract. We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category D(g, ~) of gmodules and the category of finite dimensional Uqgmodules, q = epii~, for ~ ∈ C\Q∗. Aiming at operator algebraists the result is formulated as the existence for each ~ ∈ iR of a normalized unitary 2cochain F on the dual G ̂ of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by F is ∗isomorphic to the convolution algebra of the qdeformation Gq of G, while the coboundary of F−1 coincides with Drinfeld’s KZassociator defined via monodromy of the KnizhnikZamolodchikov equations.
Koszul algebras and the quantum MacMahon Master Theorem
 q” PRINCIPLE 9
"... ABSTRACT. We give a new proof of the quantum version of MacMahon’s Master Theorem due to Garoufalidis, Le and Zeilberger by deriving it from known facts about Koszul algebras. 1. ..."
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ABSTRACT. We give a new proof of the quantum version of MacMahon’s Master Theorem due to Garoufalidis, Le and Zeilberger by deriving it from known facts about Koszul algebras. 1.
THE DIRAC OPERATOR ON COMPACT QUANTUM GROUPS
, 2007
"... For the qdeformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element ..."
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For the qdeformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element of Ug⊗Cl(g). The commutator of Dq with a regular function on Gq consists of two parts. One is a twist of a classical commutator and so is automatically bounded. The second is expressed in terms of the commutator of the associator with an extension of D. We show that in the case of the Drinfeld associator the latter commutator is also bounded.
QUASICOXETER ALGEBRAS, DYNKIN DIAGRAM COHOMOLOGY AND QUANTUM WEYL GROUPS
, 2005
"... ... independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection ∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pai ..."
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... independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection ∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pairs (g, V) including vector and spin representations of classical Lie algebras and the adjoint representation of all complex, simple Lie algebras. The aim of this paper, and of its sequel [TL4] is to prove this conjecture for all g. Our strategy is inspired by Drinfeld’s proof of the equivalence of the monodromy of the Knizhnik–Zamolodchikov equations for g and the R–matrix representations coming from the quantum group U�g. It relies on the use of quasi–Coxeter algebras, which are to the generalised braid group of type g what Drinfeld’s quasitriangular quasibialgebras are to Artin’s braid groups Bn. Using this notion, and the associated deformation cohomology, which we call Dynkin diagram
From Quantum Groups to Unitary Modular Tensor Categories
 CONTEMPORARY MATHEMATICS
"... Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently propos ..."
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Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are “physically feasible.” We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.