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249
On Posets and Hopf Algebras
 Adv. Math
, 1996
"... this paper have a finite number of elements, a minimum element 0 # and a maximum element 1 . For two elements x and y in a poset P, such that x#y, define the interval [x, y]=[z#P:x#z#y]. We will consider [x, y] as a poset, which inherits its order relation from P. Observe that [x, y] is a poset with ..."
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Cited by 85 (14 self)
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this paper have a finite number of elements, a minimum element 0 # and a maximum element 1 . For two elements x and y in a poset P, such that x#y, define the interval [x, y]=[z#P:x#z#y]. We will consider [x, y] as a poset, which inherits its order relation from P. Observe that [x, y] is a poset with minimum element x and maximum element y. For x, y # P such that x#y, we may define the Mo# bius function +(x, y) recursively by +(x, y)= & : x#z<y +(x, z), if x<y, 1, if x=y
Coproducts and the CdIndex
, 1998
"... The linear span of isomorphism classes of posets, has a Newtonian coalgebra structure. We observe that the abindex is a Newtonian coalgebra map from the vector space to the algebra of polynomials in the noncommutative variables a and b. This enables us to obtain explicit formulas showing how the c ..."
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Cited by 53 (33 self)
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The linear span of isomorphism classes of posets, has a Newtonian coalgebra structure. We observe that the abindex is a Newtonian coalgebra map from the vector space to the algebra of polynomials in the noncommutative variables a and b. This enables us to obtain explicit formulas showing how the cdindex of the face lattice of a convex polytope changes when taking the pyramid and the prism of the polytope and the corresponding operations on posets. As a corollary, we have new recursion formulas for the cdindex of the Boolean algebra and the cubical lattice. Moreover, these operations also have interpretations for certain classes of permutations, including simsun and signed simsun permutations. We prove an identity for the shelling components of the simplex. Lastly, we show how to compute the abindex of the Cartesian product of two posets given the abindexes of each poset.
Noncommutative Riemannian and Spin Geometry of the Standard qSphere
 Comm. Math. Phys
"... Abstract. We study the quantum sphere Cq[S 2] as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2dimensional cotangent bundle as a direct sum Ω 0,1 ⊕ Ω 1,0 in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operato ..."
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Cited by 36 (8 self)
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Abstract. We study the quantum sphere Cq[S 2] as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2dimensional cotangent bundle as a direct sum Ω 0,1 ⊕ Ω 1,0 in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators and projective module structure. We show that the qmonopole as spin connection induces a natural LeviCivita type connection and find its Ricci curvature and qDirac operator ∇/. We find the possibility of an antisymmetric volume form quantum correction to the Ricci curvature and Lichnerowicztype formulae for ∇ / 2. We also remark on the geometric qBorelWeilBott construction. 1.
Invariants of 3–manifolds and projective representations of mapping class groups via quantum groups at roots of unity
 Comm. Math. Phys
, 1995
"... Abstract. An example of a finite dimensional factorizable ribbon Hopf Calgebra is given by a quotient H = uq(g) of the quantized universal enveloping algebra Uq(g) at a root of unity q of odd degree. The mapping class group Mg,1 of a surface of genus g with one hole projectively acts by automorphis ..."
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Cited by 36 (1 self)
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Abstract. An example of a finite dimensional factorizable ribbon Hopf Calgebra is given by a quotient H = uq(g) of the quantized universal enveloping algebra Uq(g) at a root of unity q of odd degree. The mapping class group Mg,1 of a surface of genus g with one hole projectively acts by automorphisms in the Hmodule H ∗⊗g, if H ∗ is endowed with the coadjoint Hmodule structure. There exists a projective representation of the mapping class group Mg,n of a surface of genus g with n holes labelled by finite dimensional Hmodules X1,..., Xn in the vector space HomH(X1 ⊗ · · · ⊗ Xn, H ∗⊗g). An invariant of closed oriented 3manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most of uq(g) at roots of unity q of even degree) are described. After works of Moore and Seiberg [44], Witten [62], Reshetikhin and Turaev [51], Walker [61], Kohno [22, 23] and Turaev [59] it became clear that any semisimple abelian ribbon category with finite number of simple objects satisfying some nondegeneracy condition gives rise to projective representations of mapping class groups
Classification of finitedimensional triangular Hopf algebras with the Chevalley property
 Mathematical Research Letters
"... Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose ..."
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Cited by 32 (10 self)
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Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne’s theorem on Tannakian categories and on Movshev’s theory in an essential way. We explain Movshev’s theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property. 1.
Integrals for braided Hopf algebras
 J. Pure Appl. Algebra
, 2000
"... Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided ve ..."
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Cited by 29 (3 self)
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Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided version of Radford’s formula for the fourth power of the antipode is obtained. Connections of integration with crossproduct and transmutation are studied. 1991 Mathematics Subject Classification. Primary 16W30, 18D15, 17B37; Secondary 18D35.
Hopf (Bi)Modules and Crossed Modules in Braided Monoidal Categories
 J. Pure Appl. Algebra
, 1995
"... Hopf (bi)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra ..."
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Cited by 27 (2 self)
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Hopf (bi)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. A generalization of the MajidRadford criterion for a braided Hopf algebra to be a cross product is obtained as an application of these results. Keywords: Braided category, Braided Hopf algebra, Crossed Module, Hopf (Bi)Module Mathematical Subject Classification (1991): 16W30, 17B37, 18D10, 81R50 1 Introduction For bialgebras over a field k the smash product and the smash coproduct are investigated extensively in the literature [Rad, Mol]. Let H be a bialgebra, B be an Hright module algebra and an Hright comodule coalgebra. If the smash product algebra structure and the smash coproduct coalgebra structure on H\Omega B form a bialgebra then Ra...
Invariants and coinvariants of the symmetric group in noncommuting variables
, 2005
"... We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan [12]. The bases for this algebra are indexed by set partitions. We show that there exist a natural inclusion of the Hopf algebra of noncom ..."
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Cited by 26 (6 self)
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We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan [12]. The bases for this algebra are indexed by set partitions. We show that there exist a natural inclusion of the Hopf algebra of noncommutative symmetric functions studied in [17] in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials.
Crossed modules and quantum groups in braided categories
, 1995
"... Let A be a Hopf algebra in a braided category C. Crossed modules over A are introduced and studied as objects with both module and comodule structures ..."
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Cited by 25 (1 self)
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Let A be a Hopf algebra in a braided category C. Crossed modules over A are introduced and studied as objects with both module and comodule structures