br Conclusion Membrane transporters remain relatively unders

Conclusion Membrane transporters remain relatively understudied in human disease, especially TBI (Cesar-Razquin et al., 2015), and considerably less is known in terms of expression, function and substrates of membrane transporters in pediatric patients. However, their importance in maintaining l lysine reviews homeostasis and modulating the brain milieu of numerous injury mediators and therapeutic drugs is emerging. Provocative genetic studies, exciting—albeit early, clinical studies including pharmacometabolomic evidence of drug transporter inhibitor target engagement, and recent studies in experimental TBI support further investigation to best define the scope of the effects of transporter manipulation and therapeutic potential.
Introduction Let G be a finite group and be a finite G-poset (we shall regard a G-set as a G-poset with trivial relations). The transporter category over is a Grothendieck construction, a specifically designed finite category. It may be thought as a semi-direct product between G and , and is considered as a generalized group. This construction has its roots in group theory, representation theory and algebraic topology. Our initiative comes from the observation that there exists a category equivalence for each subgroup . Representation theory of can be divided into two parts, usually through the category algebra, where R is a commutative ring with identity. One part examines how category representations contribute to the representations of G. It is known that the canonical functor induces functors from -mod to RG-mod, so that every -module can be induced up to an RG-module [12], [13]. Assuming to be connected (as a graph), if (the number of connected components of ) is invertible in R, then every RG-module is a direct summand of an induced module from some -module [13]. The other part contains the main goal of the present paper, where we examine transporter categories (as generalized groups) from an intrinsic point of view. Our treatment allows some classical settings in local representation theory (of groups), and the results that follow, to survive in this generality. To this end, let H be a subgroup of G and be a H-subposet of . The category is called a transporter subcategory of . We will discuss the structure theory of transporter categories, based on which we shall develop a local representation theory. It means that we will establish connections between the representations of and those of its transporter subcategories. The idea of using to understand may be traced back to the Quillen stratification of the equivariant cohomology ring , in which is indeed homotopy equivalent to the classifying space . Our ultimate aim is to investigate representations of various local categories, arisen in group representations and homotopy theory, and their applications, see for instance [2]. We shall carry on the above mentioned tasks with the help of transporter category algebras , where k is an (algebraically closed) field of characteristic p that divides the order of G. If H happens to be a p-subgroup, we shall call a p-transporter subcategory. We have the following comparison chart. The bulk of this paper contains a theory of vertices and sources, as well as a theory of blocks, for transporter category algebras.
To see a concrete example, we may choose , the poset of non-trivial p-subgroups of G, with conjugation action. Then (connected) is the usual p-transporter category , containing all p-local subgroups of G, as automorphism groups of objects. By definition, a representation of is a covariant functor from to , the category of finite-dimensional k-vector spaces. It can be thought as a diagram of representations of local subgroups , for a collection of . Since is coprime to p, every kG-module is a direct summand of an induced module from some -module [13]. It extends the well-known result that every kG-module is H-projective, as long as H contains a Sylow p-subgroup (it is equivalent to considering the G-poset ). As another application of local categories, one may find a new way to reformulate the Alvis–Curtis duality when G is a finite group with a split BN-pair in defining characteristic p[13].