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Equivariant Cohomology, Koszul Duality, and the Localization Theorem
 Invent. Math
, 1998
"... This paper concerns three aspects of the action of a compact group K on a space ..."
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Cited by 147 (4 self)
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This paper concerns three aspects of the action of a compact group K on a space
Special Values of Multiple Polylogarithms
 Sém. Bourbaki, 53 e année, 2000–2001, n ◦ 885, Mars 2001; Astéisque 282 (2002
"... Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recen ..."
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Cited by 59 (18 self)
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Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier. 1.
The Structure of Sectors Associated with the LongoRehren Inclusions I. General Theory
 Commun. Math. Phys
, 1999
"... We investigate the structure of the LongoRehren inclusion for a finite closed system of endomorphisms of factors, whose categorical structure is known to be the same as the asymptotic inclusion of A. Ocneanu. In particular, we obtain a precise description of the sectors associated with the LongoRe ..."
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Cited by 37 (0 self)
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We investigate the structure of the LongoRehren inclusion for a finite closed system of endomorphisms of factors, whose categorical structure is known to be the same as the asymptotic inclusion of A. Ocneanu. In particular, we obtain a precise description of the sectors associated with the LongoRehren inclusions in terms of half braidings, which do not necessarily satisfy the usual condition of braidings. In doing so, we give new proofs to most of the known statements concerning asymptotic inclusions. We construct a complete system of matrix units of the tube algebra using the half braidings, which will be used in the second part to describe concrete examples of the LongoRehren inclusions arising from the Cuntz algebra endomorphisms. We also discuss the case where the original system has a braiding, and generalize Ocneanu and EvansKawahigashi's method for the analysis of the asymptotic inclusions of the Hecke algebra subfactors. 1 Introduction The notion of the asymptotic inclusio...
Quantum invariants of 3manifolds: integrality, splitting, and perturbative expansion
 In Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of ThreeManifolds
, 1999
"... Abstract. We consider quantum invariants of 3manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that t ..."
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Cited by 36 (9 self)
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Abstract. We consider quantum invariants of 3manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that the projective quantum invariant is always an algebraic integer, if the quantum parameter is a prime root of unity. We also show that the projective quantum invariant of rational homology 3spheres has a perturbative expansion a la Ohtsuki. The presentation of the theory of quantum 3manifold is selfcontained. 0.1. For a simple Lie algebra g over C with Cartan matrix (aij) let d = maxi̸=j aij. Thus d = 1 for the ADE series, d = 2 for BCF and d = 3 for G2. The quantum group associated with g is a Hopf algebra over Q(q 1/2), where q 1/2 is the quantum parameter. To fix the order let us point out that our q is q 2 in [Ka, Ki, Tu] or v 2 in the book [Lu2]. For example, the quantum
In bialgebras, preLie and dendriform algebras, Nov. 2002, preprint: ArXiv:math.QA/0211074, to appear
 DePaul University", Marcel Dekker
"... Abstract. We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but nonobvious way to construct a preLie algebra f ..."
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Cited by 22 (1 self)
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Abstract. We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but nonobvious way to construct a preLie algebra from an arbitrary infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bialgebra. As consequences, we obtain a preLie structure on the space of paths on an arbitrary quiver, and a striking dendriform structure on the space of endomorphisms of an arbitrary infinitesimal bialgebra, which combines the convolution and composition products. We extend the previous constructions to the categories of Hopf, preLie and dendriform bimodules. We construct a brace algebra structure from an arbitrary infinitesimal bialgebra; this refines the preLie algebra construction. In two appendices, we show that infinitesimal bialgebras are comonoid objects in a certain monoidal category and discuss a related construction for counital infinitesimal bialgebras. 1.
L.: Constructing free Boolean categories
, 2005
"... By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *autonomous category ..."
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Cited by 21 (6 self)
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By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *autonomous category and not in a weakly distributive one, which simplifies issues like the Mix rule. An important axiom, which is introduced later, is a “graphical ” condition, which is closely related to denotational semantics and the Geometry of Interaction. Then we show that a previously
Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 19 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
Unitary solutions to the YangBaxter equation in dimension four
 Quantum Information Processing, Vol 2 Ns 1,2
, 2003
"... In this paper, we determine all unitary solutions to the YangBaxter equation in dimension four. Quantum computation motivates this study. This set of solutions will assist in clarifying the relationship between quantum entanglement and topological entanglement. We present a variety of facts about ..."
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Cited by 17 (1 self)
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In this paper, we determine all unitary solutions to the YangBaxter equation in dimension four. Quantum computation motivates this study. This set of solutions will assist in clarifying the relationship between quantum entanglement and topological entanglement. We present a variety of facts about the YangBaxter equation for the reader unfamiliar with the equation. 1 Acknowledgement. This effort was sponsored by the Defense Advanced
The peak algebra and the descent algebras of types
 B and D, Trans. Amer. Math. Soc
"... Abstract. We show the existence of a unital subalgebra Pn of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that Pn is the image of the descent algebra of type B under the map to the descent algebra of type A which ..."
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Cited by 17 (9 self)
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Abstract. We show the existence of a unital subalgebra Pn of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that Pn is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra Pn contains a twosided ideal ◦ P n which is defined in terms of interior peaks. This object was introduced in previous work by Nyman [28]; we find that it is the image of certain ideals of the descent algebras of types B and D introduced in [4] and [9]. We derive an exact sequence of the form 0 → ◦ Pn → Pn → Pn−2 → 0. We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions n and ⌊ n 2 ⌋ + 1) by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces Pn and ◦ Pn over n ≥ 0 and explain the connection with previous work of Stembridge [31]; we also obtain new properties of his descentstopeaks map and construct a type B analog.
Perverse sheaves on affine flags and Langlands dual group
"... Abstract. This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a (split) simple group in terms the Langlands dual group. In the present paper we provide such a description for categories which are geometric counterparts of a maximal com ..."
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Cited by 17 (10 self)
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Abstract. This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a (split) simple group in terms the Langlands dual group. In the present paper we provide such a description for categories which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra H; of the antispherical module for H; and of the space of Iwahoriinvariant Whittaker functions. As a byproduct we obtain some new properties of central sheaves introduced in [G]. Acknowledgements. This project was conceived during the IAS special year in Representation Theory (1998/99) led by G. Lusztig, as a result of conversations with D. Gaitsgory, M. Finkelberg and I. Mirkovic. The outcome was strongly influenced by conversations with A. Beilinson and V. Drinfeld. The stimulating interest of A. Braverman, D. Kazhdan, G. Lusztig and V. Ostrik was crucial for keeping the project alive. We are very grateful to all these people. We thank I. Mirkovic and D. Gaitsgory for the permission to use their unpublished results; and M. Finkelberg and D. Gaitsgory for taking the trouble to read the text and point out various lapses in the exposition. The second author was supported by NSF and Clay Institute. 1.