On the projective-injective modules over cellular algebras

Author:
Yongzhi Cao

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1613-1619

MSC (2000):
Primary 16G30; Secondary 18G05

DOI:
https://doi.org/10.1090/S0002-9939-03-07268-X

Published electronically:
November 25, 2003

MathSciNet review:
2051121

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the projective module over a cellular algebra is injective if and only if the socle of coincides with the top of , and this is also equivalent to the condition that the th socle layer of is isomorphic to the th radical layer of for each positive integer . This eases the process of determining the Loewy series of the projective-injective modules over cellular algebras.

**[BGS]**A. Beilinson, V. Ginsburg, and W. Soergel,*Koszul duality patterns in representation theory*, J. Amer. Math. Soc.**9**(1996), 473-527. MR**96k:17010****[BGG]**J. Bernstein, I. Gelfand, and S. Gelfand,*Category of**-modules*, Funct. Anal. Appl.**10**(1976), 87-92. MR**53:10880****[FNP]**V. Futorny, D. K. Nakano, and R. D. Pollack,*Representation type of the blocks of category*, Quart. J. Math.**52**(2001), 285-305. MR**2002h:17006****[GL]**J. Graham and G. Lehrer,*Cellular algebras*, Invent. Math.**123**(1996), 1-34. MR**97h:20016****[G]**J. A. Green,*Polynomial Representations of*, Lecture Notes in Mathematics**830**, Springer-Verlag, Berlin, Heidelberg, New York, 1980. MR**83j:20003****[I1]**R. S. Irving,*Projective modules in the category**: Loewy series*, Trans. Amer. Math. Soc.**291**(1985), 733-754. MR**87h:17007****[I2]**R. S. Irving,*A filtered category**and applications*, Mem. Amer. Math. Soc.**83**(1990), no. 419. MR**90f:17011****[KM]**O. Khomenko and V. Mazorchuk,*Rigidity of generalized Verma modules*, Colloq. Math.**92**(2002), 45-57. MR**2003c:17008****[KX1]**S. König and C. C. Xi,*On the structure of cellular algebras*in ``Algebras and Modules II'' (I. Reiten, S. Smalø and Ø. Solberg, Eds.), Canadian Mathematical Society Conference Proceedings, Vol. 24, pp. 365-386, 1998. MR**2000a:16011****[KX2]**S. König and C. C. Xi,*A self-injective cellular algebra is weakly symmetric*, J. Algebra**228**(2000), 51-59. MR**2001d:16029****[M1]**P. Martin,*Temperley-Lieb algebras for nonplanar statistical mechanics-the partition algebra construction*, J. Knot Theory Ramifications**3**(1994), 51-82.MR**95a:82022****[M2]**P. Martin,*The structure of the partition algebras*, J. Algebra**183**(1996), 319-358. MR**98g:05152****[W]**B. W. Westbury,*The representation theory of the Temperley-Lieb algebra*, Math. Z.**219**(1995), 539-565. MR**96h:20029****[X1]**C. C. Xi,*The structure of Schur algebras**for*, Canad. J. Math.**44**(1992), 665-672. MR**93j:20031****[X2]**C. C. Xi,*On representation types of q-Schur algebras*, J. Pure Appl. Algebra**84**(1993), 73-84. MR**94g:16012****[X3]**C. C. Xi,*On the quasi-heredity of Birman-Wenzl algebras*, Adv. Math.**154**(2000), 280-298. MR**2001g:20008**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16G30,
18G05

Retrieve articles in all journals with MSC (2000): 16G30, 18G05

Additional Information

**Yongzhi Cao**

Affiliation:
Department of Mathematics, Beijing Normal University, 100875 Beijing, People’s Republic of China

Address at time of publication:
State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, People’s Republic of China

Email:
yongzhic@263.net

DOI:
https://doi.org/10.1090/S0002-9939-03-07268-X

Received by editor(s):
November 11, 2002

Received by editor(s) in revised form:
February 23, 2003

Published electronically:
November 25, 2003

Communicated by:
Martin Lorenz

Article copyright:
© Copyright 2003
American Mathematical Society