Based algebras and standard bases

for quasi-hereditary algebras

Authors:
Jie Du and Hebing Rui

Journal:
Trans. Amer. Math. Soc. **350** (1998), 3207-3235

MSC (1991):
Primary 16G99, 17B10; Secondary 20C20, 20C30

DOI:
https://doi.org/10.1090/S0002-9947-98-02305-8

MathSciNet review:
1603902

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Abstract | References | Similar Articles | Additional Information

Abstract: Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly full-based. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.

**[B]**C. Berge,*Principles of Combinatorics*, Academic Press, New York 1971. MR**42:5805****[BV]**D. Barbasch and D. Vogan,*Primitive ideals and orbital integrals in complex classical groups*, Math. Ann.**259**(1982), 153-199. MR**83m:22026****[BLM]**A.A. Beilinson, G.Lusztig, and R. Macpherson,*A geometric setting for the quantum deformation of GL*, Duke Math. J. (1990), 655-677. MR**91m:17012****[CPS1]**E. Cline, B. Parshall and L. Scott,*Finite dimensional algebras and highest weight categories*, J. Reine Angew. Math.**391**(1988), 85-99. MR**90d:18005****[CPS2]**E. Cline, B. Parshall and L. Scott,*Duality in highest weight categories*, Contemp. Math.**82**(1989), 7-22. MR**90g:17014****[CPS3]**E. Cline, B. Parshall and L. Scott,*Integral and graded quasi-hereditary algebras, I*, J. Algebra.**131**(1990), 126-160. MR**91c:16009****[DJ1]**R. Dipper and G. James,*Representations of Hecke algebras of general linear groups*, Proc. London Math. Soc.**52**(1986), 20-52. MR**88b:20065****[DJ2]**R. Dipper and G. James,*The q-Schur algebra*, Proc. London Math. Soc.**59**(1989), 23-50. MR**90g:16026****[DJ3]**R. Dipper and G. James,*-tensor spaces and -Weyl modules*, Trans. Amer. Math. Soc.**727**(1991), 251-282. MR**91m:20061****[DR1]**V. Dlab and C. M. Ringel,*Quasi-hereditary algebras*, Illinois J. Math.**33**(1989), 280-291. MR**90e:16023****[DR2]**V. Dlab and C. M. Ringel,*The module theoretical approach to quasi-hereditary algebras*, Representations of Algebras and Related Topics (H. Tachikawa and S. Brenner, eds.) London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 200-224. MR**94f:16026****[Du1]**J. Du,*Kazhdan-Lusztig bases and isomorphism theorems for -Schur algebras*, Contemp. Math.**139**(1991), 121-140. MR**94b:17019****[Du2]**J. Du,*Integral Schur algebras for*, Manuscripta Math.**75**(1992), 411-427. MR**93g:20086****[Du3]**J. Du,*Canonical basis for irreducible representations of quantum GL*, Bull. London Math. Soc.**24**(1992), 325-334. MR**93g:17023****[Du4]**J. Du,*IC bases and quantum linear groups*, Proc. Sympo. Pure.Math.**56**(1994), part 2, 135-148. MR**95d:17010****[Du5]**J. Du,*Canonical basis for irreducible representations of quantum GL, II*, J. London Math. Soc. (2)**51**(1995), 461-470. MR**96h:17016****[Du6]**J. Du,*A new proof for the canonical bases of type*, preprint, UNSW.**[DS]**J. Du and L. Scott,*Lusztig conjectures, old and new, I*, J. Reine Angew. Math.**455**(1994), 141-182. MR**95i:20062****[G1]**J. A. Green,*Polynomial Representations of*, Lecture Notes in Math.**830**, Springer-Verlag, Berlin, 1980. MR**83j:20003****[G2]**J. A. Green,*On certain subalgebras of the Schur algebras*, J. Algebra**131**(1990), 265-280. MR**91b:20019****[G3]**J. A. Green,*Combinatorics and the Schur algebra*, J. Pure Appl. Algebra**88**(1993), 89-106. MR**94g:05100****[Gr]**R. Green,*-Schur algebras and quantized enveloping algebras*, Thesis (1994).**[GL]**J. Graham and G. Lehrer,*Cellular algebras*, Invent. Math.**123**(1996), 1-34. MR**97h:20016****[K]**S. König,*A criterion for quasi-hereditary, and an abstract straightening formula*, Invent. Math.**127**(1997), 481-488. MR**97m:16019****[KL]**D. Kazhdan and G. Lusztig,*Representation of Coxeter groups and Hecke algebras*, Invent. Math.**53**(1979), 155-174. MR**81j:20006****[Lu]**G. Lusztig,*Introduction to quantum groups*, Birkhäuser, Boston, 1993. MR**94m:17016****[PW]**B. Parshall and J. Wang,*Quantum linear groups*, Memoirs Amer. Math. Soc.**89**(1991), No. 439. MR**91g:16028****[S]**L. Scott,*Simulating algebraic geometry with algebra, I*, Proc. Sympos. Pure Math.**47**(1987), part 2, 271-281. MR**89c:20062a**

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Additional Information

**Jie Du**

Affiliation:
Schools of Mathematics, University of New South Wales, Sydney, 2052, Australia

Email:
jied@maths.unsw.edu.au

**Hebing Rui**

Affiliation:
Department of Mathematics, University of Shanghai for Science & Technology, Shanghai, 200093, People’s Republic of China

Email:
hbruik@online.sh.cn

DOI:
https://doi.org/10.1090/S0002-9947-98-02305-8

Received by editor(s):
September 13, 1996

Additional Notes:
Both authors gratefully acknowledge support received from Australian Research Council under Large ARC Grant A69530243. The second author is partially supported by the National Natural Science Foundation, grant no. 19501016, in P.R. China. He wishes to thank the University of New South Wales for its hospitality during his visit.

Article copyright:
© Copyright 1998
American Mathematical Society