The antipode of a finite-dimensional Hopf algebra over a field has finite order

Finite dimensional comodules over the Hopf algebra of rooted trees

In [1, 3, 4, 5], a Hopf algebra of rooted trees HR was introduced. It was shown that the antipode of this algebra was the key of a problem of renormalization ([8]). HR is related to the Hopf algebra HCM introduced in [2]. Moreover, the dual algebra of HR is the enveloping algebra of the Lie algebra of rooted trees L. An important problem was to give an explicit construction of the primitive ele...

Cosemisimple Hopf Algebras with Antipode of Arbitrary Finite Order

Let m ≥ 1 be a positive integer. We show that, over an algebraically closed field of characteristic zero, there exists cosemisimple Hopf algebras having an antipode of order 2m. We also discuss the Schur indicator for such Hopf algebras.

Finite Dimensional Pointed Hopf Algebras over S

Finite Semigroups whose Semigroup Algebra over a Field Has a Trivial Right Annihilator

An element a of a semigroup algebra F[S] over a field F is called a right annihilating element of F[S] if xa = 0 for every x ∈ F[S], where 0 denotes the zero of F[S]. The set of all right annihilating elements of F[S] is called the right annihilator of F[S]. In this paper we show that, for an arbitrary field F, if a finite semigroup S is a direct product or semilattice or right zero semigroup o...

On the Plesken Lie Algebra Defined over a Finite Field

Let G be a finite group and p a prime number. The Plesken Lie algebra is a subalgebra of the complex group algebra C[G] and admits a directsum decomposition into simple Lie algebras. We describe finite-field versions of the Plesken Lie algebra via traditional and computational methods. The computations motivate our conjectures on the general structure of the modular Plesken Lie algebra.