Some deformations of the fibred biset category

2020 Turkish Journal of Mathematics  
We prove the well-definedness of some deformations of the fibred biset category in characteristic zero. 4 The method is to realize the fibred biset category and the deformations as the invariant parts of some categories 5 whose compositions are given by simpler formulas. Those larger categories are constructed from a partial category of 6 subcharacters by linearizing and introducing a cocycle. 7 8 17 an inverse in R . The inversion condition, expressed differently, is that the field of rational
more » ... e field of rational numbers Q embeds 18 in R . We let K be an algebraically closed field of characteristic zero. We let A be a multiplicatively written 19 abelian group. 20 After reviewing some background in Section 2, we shall introduce the notion of an interior R -linear 21 category L with set of objects G . Each G acts on the endomorphism algebra End L (G) via an algebra map 22 from the group algebra RG. We shall construct a category L , called the invariant category of L . 23 Informally, borrowing a term from algebraic geometry, we call L a "polarization" of L . Let us retain 24 the scare-quotes, because we do not propose a general definition, and we wish only to use the term when the 25 composition for L is easier to describe than the composition for L . A "polarization" of the biset category was 26 introduced in [4], and that was extended to some deformations of the biset category in [2]. In Section 3, as 27 rather a toy illustration, we shall introduce a "polarization" of a K -linear category associated with K -character 28 rings. More substantially, in Section 4, we shall introduce a partial category called the A -subcharacter partial 29 * Correspondence: 2010 AMS Mathematics Subject Classification: 19A22; 16B50 This work is licensed under a Creative Commons Attribution 4.0 International License. category and, in Section 5, we shall show that a twisted R -linearization of the A-subcharacter partial category 1 serves as a "polarization" of the R -linear A-fibred biset category discussed in Boltje-Coşkun [3]. One direction 2 for further study may be towards reassessing the classification, in [3], of the simple A -fibred biset functors. We 3 shall comment further on that at the end of the paper. 4 Also in Section 5, we shall present some deformations of the R -linear A-fibred biset category. To prove 5 the associativity of the deformed composition, we shall make use of the fact that those deformations, too, admit 6 "polarizations" in the form of twisted R -linearizations of the A-subcharacter partial category. 7 Our hypothesis on R is not significantly more general than the case of an arbitrary field of characteristic 8 zero. Adaptations to other coefficient rings would require further techniques. 9 2. Interior linear categories 10 Categories and partial categories arise in our topic mainly as combinatorial structures (in the sense that some 11 familar "up to" qualifications are absent, to wit, all the equivalences of categories below are isomorphisms of 12 categories). Let us organize our notation and terminology accordingly. The main idea behind the less standard 13 among the following definitions goes back at least as far as Schelp [9]. For clarity, let us present the material 14 in a self-contained way. We define a partial magma to be a set P equipped with a relation ∼, called the 15 matching relation, together with a function P ϕψ → (ϕ, ψ) ∈ Γ(P) , called the multiplication, where 16 Γ(P) = {(ϕ, ψ) ∈ P × P : ϕ ∼ ψ} . 17 We call P a partial semigroup provided the following associativity condition holds: given θ, ϕ, ψ ∈ P 18 such that θ ∼ ϕ and ϕ ∼ ψ , then θ ∼ ϕψ if and only if θϕ ∼ ψ , in which case θ(ϕψ) = (θϕ)ψ . When θ ∼ ϕψ , 19
doi:10.3906/mat-2001-52 fatcat:ynojjy62knavvdelkx2ssmdh5a