Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the global dimension of quasi--hereditary algebras with triangular decomposition


Author: Steffen König
Journal: Proc. Amer. Math. Soc. 124 (1996), 1993-1999
MSC (1991): Primary 16E10, 18G20; Secondary 16G10, 17B10, 17B35, 18G05, 20G05
DOI: https://doi.org/10.1090/S0002-9939-96-03549-6
MathSciNet review: 1346979
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $A$ be a quasi--hereditary algebra with triangular decomposition ${_C}A_{C^{op}} \simeq C \otimes _S C^{op}$ such that all Verma modules are semisimple over $C^{op}$. Then we show: $gldim(A) = 2 \cdot gldim(C)$. Applying this formula to the more special class of twisted double incidence algebras of finite partially ordered sets, we get a proof of a conjecture of Deng and Xi. Another application is to the so-called dual extensions of algebras.


References [Enhancements On Off] (What's this?)

  • 1. E.Cline, B.Parshall and L.Scott, Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85--99 (1988). MR 90d:18005
  • 2. B.M.Deng and C.C.Xi, Quasi--hereditary algebras which are dual extensions of algebras. Comm.Alg. 22, 4717--4736 (1994). CMP 94:15
  • 3. B.M.Deng and C.C.Xi, Quasi--hereditary algebras which are twisted double incidence algebras of posets. Contrib. Algebra and Geom. 36 (1995), 37--72.
  • 4. V.Dlab and C.M.Ringel, Quasi--hereditary algebras. Illinois J. Math. 33, 280--291 (1989). MR 90e:16023
  • 5. V.Dlab and C.M.Ringel, The module theoretical approach to quasi--hereditary algebras. In: H.Tachikawa and S.Brenner (Eds.), Representations of algebras and related topics. London Math.Soc.LN Series 168, 200--224 (1992). MR 94f:16026
  • 6. M.Dyer, Kazhdan--Lusztig--Stanley polynomials and quadratic algebras I. Preprint (1992).
  • 7. S.König, Exact Borel subalgebras of quasi--hereditary algebras, I. With an appendix by L.Scott. Math. Z. 220 (1995), 399--426.
  • 8. S.König, Exact Borel subalgebras of quasi--hereditary algebras, II. Comm. Alg. 23 (1995), 2331--2344. CMP 95:11
  • 9. S.König, Strong exact Borel subalgebras of quasi--hereditary algebras and abstract Kazhdan--Lusztig theory. To appear in Adv.in Math.
  • 10. S.König, Cartan decompositions and BGG--resolutions. Manuscr.Math. 86, 103--111 (1995). CMP 95:07
  • 11. B.Parshall and L.L.Scott, Derived categories, quasi--hereditary algebras and algebraic groups. Proc. of the Ottawa-Moosonee Workshop in Algebra 1987, Math. Lect. Note Series, Carleton University and Université d'Ottawa (1988).
  • 12. L.L.Scott, Simulating algebraic geometry with algebra I: The algebraic theory of derived categories. AMS Proc. Symp. Pure Math. 47, 271--281 (1987). MR 89c:20062a
  • 13. C.C.Xi, Quasi--hereditary algebras with a duality. J.reine angew.Math. 449, 201--215 (1993). MR 95f:16010
  • 14. C.C.Xi, Global dimensions of dual extension algebras. Manuscripta Math. 88 (1995), 25--31.
  • 15. K.Yamagata, A construction of algebras with large global dimension. J.Alg. 163, 57--67 (1994). MR 95a:16012

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 16E10, 18G20, 16G10, 17B10, 17B35, 18G05, 20G05

Retrieve articles in all journals with MSC (1991): 16E10, 18G20, 16G10, 17B10, 17B35, 18G05, 20G05


Additional Information

Steffen König
Affiliation: Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, D–70 569 Stuttgart, Federal Republic of Germany
Address at time of publication: Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D–33 501 Bielefeld, FR Germany
Email: koenigs@mathematik.uni-bielefeld.de

DOI: https://doi.org/10.1090/S0002-9939-96-03549-6
Received by editor(s): March 25, 1994
Received by editor(s) in revised form: July 1, 1994, and February 21, 1995
Communicated by: Ken Goodearl
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society