r-Bell polynomials in combinatorial Hopf algebras

Crossed products of restricted enveloping algebras

Let K be a eld of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials. Crossed products of gro...

Hopf Algebras of Formal Diffeomorphisms and Numerical Integration on Manifolds

B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge–Kutta methods. Connections to renormalization have been established in recent years. The algebraic structure of classical Runge–Kutta methods is described by the Connes–Kreimer Hopf algebra. Lie–Butcher theory is a generalization of B-series ai...

On the Coderivations of Some Hopf Algebras

Nonstandard Braid Relations and Chebyshev Polynomials

A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two Sr-irreducibles into irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical basis theory, by first constructing a nonstandard Hecke algebra Br, which, though not a Hopf al...

A generic Hopf algebra for quantum statistical mechan-

In this note we present a Hopf algebra description of a bosonic quantum model, using the elementary combinatorial elements of Bell and Stirling numbers. Our objective in doing this is the following. Recent studies have revealed that perturbative Quantum Field Theory (pQFT) displays an astonishing interplay between analysis (Riemann Zeta functions), topology (Knot theory), combinatorial graph th...