Results 11  20
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283
Grouptheoretical properties of nilpotent modular categories
"... 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 ..."
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Cited by 32 (6 self)
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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]).
Categorifying fractional Euler characteristics, JonesWenzl projector and 3 jsymbols with applications to Exts of HarishChandra bimodules
"... Abstract. We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3jsymbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3jsymbols. All th ..."
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Cited by 28 (3 self)
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Abstract. We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3jsymbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3jsymbols. All these formulas are realized as graded Euler characteristics. The 3jsymbols appear as new generalizations of KazhdanLusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the JonesWenzl projector, Θnetworks and tetrahedron networks. Networks and their evaluations play an important role in the TuraevViro construction of 3manifold invariants. We categorify these evaluations by Extalgebras of certain simple HarishChandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of 3manifolds will be studied in detail in subsequent papers.
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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Cited by 24 (10 self)
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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, ∞}.
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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Cited by 22 (8 self)
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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.
Chord Diagram Invariants of Tangles and Graphs
 Duke Math. J
, 1998
"... this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms are spanned by tangles and chord diagrams, respectively. More precisely, we fix a commutative ring R and consider categories T (R) and A(R) whose morphisms are f ..."
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Cited by 20 (2 self)
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this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms are spanned by tangles and chord diagrams, respectively. More precisely, we fix a commutative ring R and consider categories T (R) and A(R) whose morphisms are formal linear combinations of framed oriented tangles and chord diagrams with coefficients in R, cf. Section 2. The set of morphisms in T (R) has a canonical filtration given by the powers of an ideal I which we call the augmentation ideal. Functions on morphisms in T (R) vanishing on I
HomHopf algebras
"... Abstract. Homstructures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Homstructures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Homalgebras coincide with algebra ..."
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Cited by 20 (3 self)
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Abstract. Homstructures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Homstructures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Homalgebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras and Lie algebras.
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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Cited by 19 (7 self)
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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
Nahm sums, stability and the colored jones polynomial
 RESEARCH IN MATHEMATICAL SCIENCES
, 2012
"... ..."
Oystaeyen, On iterated twisted tensor products of algebras, arXiv:math.QA/0511280
"... ABSTRACT. We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We find conditions for constructing an iterated prod ..."
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Cited by 18 (8 self)
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ABSTRACT. We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We find conditions for constructing an iterated product of three factors, and prove that they are enough for building an iterated product of any number of factors. As an example of the geometrical aspects of our construction, we show how to construct differential forms and involutions on iterated products starting from the corresponding structures on the factors, and give some examples of algebras that can be described within our theory. We prove a certain result (called “invariance under twisting”) for a twisted tensor product of two algebras, stating that the twisted tensor product does not change when we apply certain kind of deformation. Under certain conditions, this invariance can be iterated, containing as particular cases a number of independent and previously unrelated results from Hopf algebra theory.