Details

Autor(en) / Beteiligte

• Marko, Frantisek

Titel

Quasi-hereditary algebras and their Borel subalgebras

Ort / Verlag

ProQuest Dissertations & Theses

Erscheinungsjahr

1996

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• Quasi-hereditary algebras were introduced by Cline-Parshall-Scott (see (CPS1) or (PS)) to describe highest weight categories which occur in the study of semi-simple complex Lie algebras and algebraic groups. Since then, quasi-hereditary algebras were found to play an important role in various other applications. Exact Borel subalgebras were found to play an important role in various other applications. Exact Borel subalgebras (for short, Borel subalgebras) were defined by Konig (see (K1)) to capture another distinctive feature of quasi-hereditary algebras appearing in the applications which is related to Cartan decomposition of Lie algebras. The objective of this thesis is to study properties of the quasi-hereditary algebras and their Borel subalgebras. We concentrate on homological properties of Borel subalgebras and their relationship to homological properties of quasi-hereditary algebras. For two important classes of quasi-hereditary algebras, namely, endomorphism algebras of a semilocal module over a selfinjective local algebra and endomorphism algebras of Auslander type, we establish the existence of Borel subalgebras and the relationship between their global dimension and the global dimensions of their Borel subalgebras. Later, we investigate the extension algebra of standard modules for quasi-hereditary algebras with Borel subalgebras and directed standard modules and describe the bimodule structure of this algebra over its two subalgebras, the subalgebra of morphisms between standard modules and the subalgebra isomorphic to a Yoneda algebra of a Borel subalgebra of the orginal algebra. This result is analogous to Cartan decomposition of Lie algebras. We also relate the species of the extension algebra of standard modules and the species of its above mentioned subalgebras. Finally, we characterize Borel subalgebras of separable quasi-hereditary algebra which are images of monomial subalgebras of the corresponding tensor algebra. These results are then used in a canonical construction of quasi-hereditary algebras from split standard system.