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The AdS5 × S 5 superstring worldsheet Smatrix and crossing symmetry
, 2008
"... An Smatrix satisying the YangBaxter equation with symmetries relevant to the AdS5 × S 5 superstring has recently been determined up to an unknown scalar factor. Such scalar factors are typically fixed using crossing relations, however due to the lack of conventional relativistic invariance, in thi ..."
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Cited by 226 (6 self)
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An Smatrix satisying the YangBaxter equation with symmetries relevant to the AdS5 × S 5 superstring has recently been determined up to an unknown scalar factor. Such scalar factors are typically fixed using crossing relations, however due to the lack of conventional relativistic invariance, in this case its determination remained an open problem. In this paper we propose an algebraic way to implement crossing relations for the AdS5 ×S 5 superstring worldsheet Smatrix. We base our construction on a Hopfalgebraic formulation of crossing in terms of the antipode and introduce generalized rapidities living on the universal cover of the parameter space which is constructed through an auxillary, coupling constant dependent, elliptic curve. We determine the crossing transformation and write functional equations for the scalar factor of the Smatrix in the generalized rapidity plane.
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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The q–characters of representations of quantum affine agebras and deformations
 of W–algebras, Preprint math.QA/9810055; in Contemporary Math 248, 163–205, AMS
, 2000
"... Abstract. We propose the notion of q–characters for finitedimensional representations of quantum affine algebras. It is motivated by our theory of deformed W– algebras. 1. ..."
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Cited by 113 (7 self)
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Abstract. We propose the notion of q–characters for finitedimensional representations of quantum affine algebras. It is motivated by our theory of deformed W– algebras. 1.
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 108 (9 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
On The Decomposition Matrices Of The Quantized Schur Algebra
 Duke Math. J
, 1999
"... We prove the decomposition conjecture for the Schur algebra stated in [LT]. We also give a new approach to the Lusztig conjecture via canonical bases of the Hall algebra. 0. ..."
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Cited by 92 (6 self)
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We prove the decomposition conjecture for the Schur algebra stated in [LT]. We also give a new approach to the Lusztig conjecture via canonical bases of the Hall algebra. 0.
Weyl Modules for Classical and Quantum Affine Algebras
 Represent. Theory
, 2001
"... The study of the irreducible finite–dimensional representations of quantum affine algebras has been the subject of a number of papers, [AK], [CP3], [CP5], [FR], [FM], [GV], [KS] to name a few. However, the structure of these representations is still unknown except in certain special cases. In this p ..."
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Cited by 87 (24 self)
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The study of the irreducible finite–dimensional representations of quantum affine algebras has been the subject of a number of papers, [AK], [CP3], [CP5], [FR], [FM], [GV], [KS] to name a few. However, the structure of these representations is still unknown except in certain special cases. In this paper, we approach the problem
FiniteDimensional Representations Of Quantum Affine Algebras
 Publ. RIMS, Kyoto Univ
, 1997
"... We present a conjecture on the irreducibility of the tensor products of fundamental representations of quantized affine algebras. This conjecture implies in particular that the irreducibility of the tensor products of fundamental representations is completely described by the poles of Rmatrices ..."
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Cited by 80 (5 self)
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We present a conjecture on the irreducibility of the tensor products of fundamental representations of quantized affine algebras. This conjecture implies in particular that the irreducibility of the tensor products of fundamental representations is completely described by the poles of Rmatrices. The conjecture is proved in the cases of type A n .
Noncommutative geometry and gravity
"... We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a starproduct. The class of noncommutative spaces studied is very rich. Nonanti ..."
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Cited by 77 (18 self)
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We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a starproduct. The class of noncommutative spaces studied is very rich. Nonanticommutative superspaces are also briefly considered. The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein’s equations for gravity on noncommutative manifolds.
Combinatorics of qcharacters of finitedimensional representations of quantum affine algebras
, 1999
"... We study finitedimensional representations of quantum affine algebras using q–characters. We prove the conjectures from [FR2] and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the corresp ..."
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Cited by 76 (10 self)
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We study finitedimensional representations of quantum affine algebras using q–characters. We prove the conjectures from [FR2] and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the corresponding normalized R–matrices has a pole.