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

The linear span of isomorphism classes of posets, has a Newtonian coalgebra structure. We observe that the ab-index is a Newtonian coalgebra map from the vector space to the algebra of polynomials in the non-commutative variables a and b. This enables us to obtain explicit formulas showing how the cd-index 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 cd-index 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 ab-index of the Cartesian product of two posets given the ab-indexes of each poset.

this paper we will consider oriented matroids. The lattice of regions of an oriented matroid is an Eulerian poset, thus it is natural to ask how to compute its cd-index. We provide here an answer to this question

Abstract. An example of a finite dimensional factorizable ribbon Hopf C-algebra 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 H-module H ∗⊗g, if H ∗ is endowed with the coadjoint H-module 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 H-modules X1,..., Xn in the vector space HomH(X1 ⊗ · · · ⊗ Xn, H ∗⊗g). An invariant of closed oriented 3-manifolds 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

Algebraic techniques are used to find several new combinatorial interpretations for valuations of the Martin polynomial, M(G; s), for unoriented graphs. The Martin polynomial of a graph, introduced by Martin in his 1977 thesis, encodes information about the families of closed paths in Eulerian graphs. The new results here are found by showing that the Martin polynomial is a translation of a universal skein-type graph polynomial P(G) which is a Hopf map, and then using the recursion and induction which naturally arise from the Hopf algebra structure to extend known properties. Specifically, when P(G) is evaluated by substituting s for all cycles and 0 for all tails, then P(G) equals sM(G; s+2) for all Eulerian graphs G. The Hopf-algebraic properties of P(G) are then used to extract new properties of the Martin polynomial, including an immediate proof for the formula for M(G; s) on disjoint unions of graphs, combinatorial interpretations for M(G; 2+2 k) and M(G; 2&2 k) with k # Z 0, and a new formula for the number of Eulerian orientations of a graph in terms of the vertex degrees of its Eulerian subgraphs.

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Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum invariant of triangulated 4-manifolds using Crane-Yetter cocycles as Boltzmann weights. Our invariant generalizes the 3-dimensional invariants defined by Dijkgraaf and Witten and the invariants that are defined via Hopf algebras. We present diagrammatic methods for the study of such invariants that illustrate connections between Hopf categories and moves to triangulations. 1 Contents 1 Introduction 3 2 Quantum 2- and 3- manifold invariants 4 Topological lattice field theories in dimension 2 . . . . . . . . . . . . . . . . . . . 4 Pachner moves in dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Turaev-Viro inv...

Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional 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.

Abstract. We generalize the fundamental structure Theorem on Hopf (bi)modules by Larson and Sweedler to quasi-Hopf algebras H. For dim H < ∞ this proves the existence and uniqueness (up to scalar multiples) of integrals in H. Among other applications we prove a Maschke-type Theorem for diagonal crossed products as constructed by the authors in [HN, HN99].

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 Majid-Radford 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 H-right module algebra and an H-right comodule coalgebra. If the smash product algebra structure and the smash coproduct coalgebra structure on H\Omega B form a bialgebra then Ra...