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 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.     

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 planar algebra associated to a Kac algebra

We obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the ‘rotation on 2-boxes’.     

Gaps in the poset of projections in the Calkin algebra

We study the gap structure of the partial order of projections of the Calkin algebra of a complex, separable, infinite-dimensional Hilbert space. We prove the existence of an analytic Hausdorff gap in this partial order. As a consequence we obtain that under Todorcevic’s Axiom and MA the gap spectrum of P(C(H)) is strictly bigger than the gap spectrum of P(ω)/Fin.     

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...     

Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

We consider the combinatorial Dyson-Schwinger equation X=B⁺(P(X)) in the non-commutative Connes-Kreimer Hopf algebra of planar rooted trees HNCK, where B⁺ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra AN,P of HNCK. We describe all the formal series P such that AN,P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of HNCK, organized into three isomorphism classes: a first one, restricted to a polynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faà di Bruno Hopf algebra. By taking the quotient, the last class gives an infinite set of embeddings of the Faà di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Faà di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, together with a non-commutative version of this embedding.     

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.     

The planar algebra of group-type subfactors

If G is a countable, discrete group generated by two finite subgroups H and K and P is a II₁ factor with an outer G-action, one can construct the group-type subfactor PH⊂P?K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones' planar operad is determined explicitly.     

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.     

Cotorsion dimensions and Hopf algebra actions

Let H be a finite-dimensional Hopf algebra over a field k, and let A be an H-module algebra. In this paper, we discuss the cotorsion dimension of the smash product A # H. We prove that $$l.\cot .D\left( {A\# H} \right) \leqslant l.\cot .D\left( A \right) + r.D\left( H \right),$$ which generalizes the result of group rings. Moreover, we give some sufficient conditions for which $$l.\cot .D\left( {A\# H} \right) = l.\cot .D\left( A \right).$$ As applications, we study the invariants of IF properties and Gorenstein global dimensions.     

IntegrableN-dimensional systems on the Hopf algebra andq-deformations

We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative integration of linear differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 373–390, September, 2000     

Primitive elements in the matroid-minor Hopf algebra

We introduce the matroid-minor coalgebra C, which has labeled matroids as distinguished basis and coproduct given by splitting a matroid into a submatroid and complementary contraction in all possible ways. We introduce two new bases for C; the first of these is related to the distinguished basis by Möbius inversion over the rank-preserving weak order on matroids, the second by Möbius inversion over the suborder excluding matroids that are irreducible with respect to the free product operation. We show that the subset of each of these bases corresponding to the set of irreducible matroids is a basis for the subspace of primitive elements of C. Projecting C onto the matroid-minor Hopf algebra H, we obtain bases for the subspace of primitive elements of H.     

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.