Decomposition of some pointed Hopf algebras given by the canonical Nakayama automorphism

MATÍAS GRAÑA, JORGE A. GUCCIONE Y JUAN JOSÉ GUCCIONE

JOURNAL OF PURE AND APPLIED ALGEBRA

Elsevier

Lugar: Amsterdam; Año: 2006

0022-4049

Let k be a field, A a finite dimensional k-algebra and DA the dual space Homk(A,k), endowed with the usual A-bimodule structure. Recall that A is said to be a Frobenius algebra if there exists a linear form φ: A→k, such that the map A→DA, defined by x→xφ, is a left A-module isomorphism. This linear form φ:A→k is called a Frobenius homomorphism. It is well known that this is equivalent to say that the map x→φx, from A to DA, is an isomorphism of right A-modules. From this it follows easily that there exists an automorphism ρ of A, called the Nakayama automorphism of A with respect to φ, such that xφ =φρ(x), for all x ε A. It is easy to check that a linear form wt{φ}:A→k is another Frobenius homomorphism if and only if there exists an invertible element x in A, such that wt{φ}=xφ. It is also easy to check that the Nakayama automorphism of A with respect to wt{φ} is the map given by a→ρ(x)1ρ(a)ρ(x).