Anti-Frobenius Algebras and Anti-Bialgebras
Gbêvèwou Damien Houndedji
Department of Mathematics, Gamal Abdel Nasser University of Conakry, Conakry, Republic of Guinea.
Cyrille Essossolim Haliya *
University of Abomey-Calavi, International Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, 072 BP 50, Cotonou, Republic of Benin.
Bakary Kourouma
Department of Mathematics, Gamal Abdel Nasser University of Conakry, Conakry, Republic of Guinea.
*Author to whom correspondence should be addressed.
Abstract
We introduce the notion of q-generalized associative algebras, which unifies associative (q = 1) and antiassociative (q = −1) structures, and investigate their bimodule and matched pair theories. Specializing to antiassociative algebras, we develop the double construction of quadratic antiassociative algebras — termed anti-Frobenius algebras — by equipping the direct sum A⊕A∗ with a compatible antiassociative product and a non-degenerate symmetric invariant bilinear form.
We prove that such double constructions are equivalent to matched pairs of antiassociative algebras and to antisymmetric infinitesimal anti-bialgebras, characterized by suitable co-derivation and antisymmetry conditions on the comultiplication. Furthermore, we establish a direct link to Mock-Lie structures: the anticommutator of an anti-Frobenius algebra yields a Manin triple of Mock-Lie algebras, and the corresponding antisymmetric infinitesimal anti-bialgebra induces a Mock-Lie bialgebra via symmetrization of the comultiplication.
A detailed low-dimensional example and a relation analysis highlight the analogies and distinctions with classical Frobenius and Mock-Lie bialgebra theories.
Keywords: Antiassociative algebras, anti-Frobenius algebras, antisymmetric infinitesimal anti-bialgebras, matched pairs, Mock-Lie bialgebras, q-generalized algebras