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