Alcahestic subalgebras of the alchemic algebra and a correspondence of simple modules

Abstract

The unified treatment of the five module-theoretic notions, transfer, inflation, transport of structure by an isomorphism, deflation and restriction, is given by the theory of biset functors, introduced by Bouc. In this paper, we construct the algebra realizing biset functors as representations. The algebra has a presentation similar to the well-known Mackey algebra. We adopt some natural constructions from the theory of Mackey functors and give two new constructions of simple biset functors. We also obtain a criterion for semisimplicity in terms of the biset functor version of the mark homomorphism. The criterion has an elementary generalization to arbitrary finite-dimensional algebras over a field. © 2008 Elsevier Inc. All rights reserved

Projective resolutions of globally defined Mackey functors in characteristic zero

“…In [4], a more general type of functor is introduced where it is also allowed to forget some of the induction and the restriction bisets. One can introduce two more families of groups to control the non-existence of these bisets but for the purposes of this paper, it is sufficient to consider the two extreme cases.…”

“…Thus we write 0, Y to mean the subalgebra of X , Y generated by all deflation, restriction and isogation maps in X , Y and 0, 0 for the subalgebra generated only by all isogations. See [4] for further details.…”

“…The simple modules of the algebra X , Y are determined in [4]. In the case that X and Y are non-zero, the simples are also determined by Bouc, [1].…”

“…In the case that X and Y are non-zero, the simples are also determined by Bouc, [1]. We recall the description from [4] since it covers all cases. Note that the results from [4] are more general then the versions that we include here.…”

“…We recall the description from [4] since it covers all cases. Note that the results from [4] are more general then the versions that we include here.…”