Séminaire d'algèbre et de géométrie du 22-11-2024

Exposé de Praveen Manju

Le 22-11-2024 à 13:20, en P108 et en ligne.

Twisted Derivations of Group Algebras with Coding Theory Applications

Résumé

In this talk, I will talk about twisted or (𝜎,𝜏)-derivations of group algebras. Let R be a commutative unital ring, G be any group with the presentation G = < X | Y > (X the set of generators for G and Y the set of relators) and (𝜎,𝜏) be a pair of R-algebra endomorphisms of the group algebra RG which are R-linear extensions of the group endomorphisms of G. We give a characterization under which a map f:X → RG can be extended uniquely to a (𝜎,𝜏)-derivation of RG. Using this characterization, we classify all 𝜎-derivations of commutative group algebras over fields of positive characteristic. We further see a classification of inner (𝜎,𝜏)-derivations of the group algebra RG of an arbitrary group G over an arbitrary commutative unital ring R. We obtain several of its significant consequences for finite and (𝜎,𝜏)-FC groups. We also see that all the (𝜎,𝜏)-derivations of RG are inner if G is a finite group whose order is invertible in R. We see the application of the above-obtained results in answering the 𝜎-derivation problem for dihedral group algebras over fields of arbitrary characteristic. Finally, we see some beautiful applications of our work in coding theory, where we define and construct a new code, namely, IDD code. This helps generate several examples of nice parameter codes.