Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H 1 , the second term in the coradical filtration of H. Using these results, we are able to show that every Hopf algebra of dimension 14, 55 and 77 is semisimple and thus isomorphic to a group...

In this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to efficiently compute certain differential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivat...

Title: The Dixmier-Moeglin equivalence for D-groups Abstract: The Dixmier-Moeglin equivalence is a characterization of the primitive ideals of an algebra that holds for many classes of rings, including affine PI rings, enveloping algebras of finite-dimensional Lie algebras, and many quantum algebras. For rings satisfying this equivalence, it says that the primitive ideals are precisely those pr...

Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of H...

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...

In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This of course not only includes the case of a finite-dimensional Hopf algebra but also the case of any Hopf algebra with integrals (co-Frobenius Hopf algebras). The proof follows in a few lines from well-known formula...

In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This of course not only includes the case of a finite-dimensional Hopf algebra but also the case of any Hopf algebra with integrals (co-Frobenius Hopf algebras). The proof follows in a few lines from well-known formula...

Let CRF S denote the category of S-colored rooted forests, and HCRFS denote its Ringel-Hall algebra as introduced in [6]. We construct a homomorphism from a K+ 0 (CRF S)–graded version of the Hopf algebra of noncommutative symmetric functions to HCRFS . Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a K+ 0 (CRF S)–graded version of the algebra of quasisymmetric func...

We analyze the structure of the Malvenuto-Reutenauer Hopf algebraSSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. In addition, we describe the structure constants of the...

We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees, etc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like ...