Finite dimensional algebras and algebraic groups. (English) Zbl 0682.20029

Parshall, Brian J.

Classical groups and related topics, Proc. Conf., Beijing/China 1987, Contemp. Math. 82, 97-114 (1989).

[For the entire collection see Zbl 0658.00005.]
The author uses the well-known correspondences between some categories of representations of algebraic groups and modules over a finite dimensional algebra A and develops methods to study the structure of such categories. The main results of the paper imply the following recollement of a derived category \[ D^ b(A/I)\quad D^ b(A)\quad D^ b(eAe) \] where \(e\in A\) is an idempotent, \(I=AeA\) and the symbol \(D^ b\) denotes the derived category of bounded complexes of A-modules. In particular, Theorem 2.1 presents a strong criterion equivalent to recollement for general idempotent ideals I. Finally, these results are interpreted as generalization of some classical theorems, for example, Green’s theorems about representations of \(GL_ n\).

Reviewer: V.Yanchevskij

MSC:

20G05	Representation theory for linear algebraic groups
20G15	Linear algebraic groups over arbitrary fields
16Gxx	Representation theory of associative rings and algebras
16P10	Finite rings and finite-dimensional associative algebras

Keywords:

categories of representations; algebraic groups; finite dimensional algebra; derived category of bounded complexes; recollement