Involutary G-algebras
Speaker: Dr. John Murray, Senior Lecturer, Maynooth University Department of Mathematics & Statistics.
Date: Monday, November 10th
Time: 4:00pm
Location: Seminar Room, Ag 1.01
Abstract:
An involutary G-algebra consists of an algebra A, an involutary algebra anti-automorphism * on A and a finite group G acting on A as algebra automorphisms, where the action of G commutes with *. We are interested in particular in the case that the underlying field has characteristic 2. The motivating example is the algebra of endomorphisms of a G-module M which affords a symmetric G-form B. Then the involution * is the adjoint of B. Orthogonal decompositions of M as G-module correspond to expressions for 1_M as a sum of pairwise orthogonal *-invariant idempotents. The theory of G- modules, including induction from and restriction to the subgroups of G, can be expressed in the language of G-algebras. We show that the theory can be strengthened to encompass the theory of G-modules with symmetric G-forms. This allows us to generalize classical notions such as `relative projectivity' to modules with forms. In particular we define a new object: the `symplectic vertex' of a non-trivial self-dual irreducible G-module.
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