Combinatorial Hopf algebra of superclass functions of type D

We provide a Hopf algebra structure on the space of superclass functions on the unipotent upper triangular group of type D over a finite field based on a supercharacter theory constructed by André and Neto in J. Algebra 305, 394–429 (2006) and 322 (2009). Also, we make further comments with respect to types B and C. Type A was explored by M. Aguiar et al. in Adv. Math. 229 (2012); thus this paper is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types.     

A note on the algebra of operations for Hopf cohomology at odd primes

Let p be any prime, and let \({\mathcal B}(p)\) be the algebra of operations on the cohomology ring of any cocommutative \(\mathbb {F}_p\)-Hopf algebra. In this paper we show that when p is odd (and unlike the \(p=2\) case), \({\mathcal B}(p)\) cannot become an object in the Singer category of \(\mathbb {F}_p\)-algebras with coproducts, if we require that coproducts act on the generators of \({\mathcal B}(p)\) coherently with their nature of cohomology operations.

     

The Hopf algebra of diagonal rectangulations

We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter permutations, which previously had only been described extrinsically as a Hopf subalgebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We describe the natural lattice structure on diagonal rectangulations, analogous to the Tamari lattice on triangulations, and observe that diagonal rectangulations index the vertices of a polytope analogous to the associahedron. We give an explicit bijection between twisted Baxter permutations and the better-known Baxter permutations, and describe the resulting Hopf algebra structure on Baxter permutations.     

Hopf algebra actions

Finite groups acting on rings by automorphisms, and group-graded rings are instances of Hopf algebras H acting on H-module algebras A. We study such actions. Let A^H = {a ε A¦h · A = (h) a, all h ε H}, the ring of H-invariants, and form the smash product A # H. We study the ring extensions A^H A A # H. We prove a Maschke-type theorem for A # H-modules. We form an associated Morita context [A^H, A, A, A # H] and use these to get connections between the various rings.     

The planar algebra of a semisimple and cosemisimple Hopf algebra

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.     

The Hopf algebra of uniform block permutations

We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Gebhard, Rosas, and Sagan. These two embeddings correspond to the factorization of a uniform block permutation as a product of an invertible element and an idempotent one. Aguiar supported in part by NSF grant DMS-0302423. Orellana supported in part by the Wilson Foundation.     

The combinatorial Hopf algebra of motivic dissection polylogarithms

We introduce a family of periods of mixed Tate motives called dissection polylogarithms, that are indexed by combinatorial objects called dissection diagrams. The motivic coproduct on the former is encoded by a combinatorial Hopf algebra structure on the latter. This generalizes Goncharov's formula for the motivic coproduct on (generic) iterated integrals. Our main tool is the study of the relative cohomology group corresponding to a bi-arrangement of hyperplanes.     

The Hopf graph algebra and renormalization group equations

We study the renormalization group equations implied by the Hopf graph algebra. The vertex functions are regarded as vectors in the dual space of the Hopf algebra. The renormalization group equations for these vertex functions are equivalent to those for individual Feynman integrals. The solution of the renormalization group equations can be represented in the form of an exponential of the beta function. We clearly show that the exponential of the one-loop beta function allows finding the coefficients of the leading logarithms for individual Feynman integrals. The calculation results agree with those obtained in the parquet approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 22–32, April, 2005.     

Supercharacters,symmetric functions in noncommuting variables,and related Hopf algebras

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.     

The Brauer group of Sweedler's Hopf algebra

We calculate the Brauer group of the four dimensional Hopf algebra introduced by M. E. Sweedler. This Brauer group is defined with respect to a (quasi-) triangular structure on , given by an element . In this paper is a field . The additive group of is embedded in the Brauer group and it fits in the exact and split sequence of groups: where is the well-known Brauer-Wall group of . The techniques involved are close to the Clifford algebra theory for quaternion or generalized quaternion algebras.

     

The quantum double of a factorizable weak Hopf algebra

Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k. We first construct a weak Hopf algebra [Δ(1)(H?H)Δ(1)]^R, which is based on the subalgebra of the tensor product algebra H?H. Next we verify that if H is factorizable, then the Drinfeld’s quantum double of H is isomorphic to [Δ(1)(H?H)Δ(1)]^R.     

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.     

Extension of a quantized enveloping algebra by a Hopf algebra

Suppose that H is a Hopf algebra,and g is a generalized Kac-Moody algebra with Cartan matrix A =(aij)I×I,where I is an index set and is equal to either {1,2,...,n} or the natural number set N.Let f,g be two mappings from I to G(H),the set of group-like elements of H,such that the multiplication of elements in the set {f(i),g(i)|i ∈I} is commutative.Then we define a Hopf algebra Hgf Uq(g),where Uq(g) is the quantized enveloping algebra of g.     

On the Hopf algebra of multi-complexes

Journal of Algebraic Combinatorics - We introduce a general class of combinatorial objects, which we call multi-complexes, which simultaneously generalizes graphs, multigraphs, hypergraphs and...     

Representations of the odd affine Temperley-Lieb algebra

We define the odd affine Temperley–Lieb algebra (OATLA)to be the category defined by planar diagrams in an annuluswith an odd number of marked points on each boundary connectedin pairs by disjoint strings and a modulus . This algebra isthe odd part of the annularization of the Temperley–Liebplanar algebra. A positivity result is proved, which allowsus to completely characterize the Hilbert space representationsof OATLA when the parameter is of the form . The results finish the project of describing theirreducible Hilbert representations of the affine Temperley–Liebalgebra, which naturally arises in the study of subfactors.