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Some properties for almost cellular algebras
1. | School of Mathematics, Hefei University of Technology, Hefei, Anhui 230009, China |
2. | Department of Mathematics, Shanghai University, Shanghai 200444, China |
In this paper, we will investigate some properties for almost cellular algebras. We compare the almost cellular algebras with quasi-hereditary algebras, which are known to carry any homological and categorical structures. We prove that any almost cellular algebra is the iterated inflation and obtain some sufficient and necessary conditions for an almost cellular algebra $ \mathrm{A} $ to be quasi-hereditary.
References:
[1] |
G. Benkart, N. Guay, J. H. Jung, S.-J. Kang and S. Wilcox,
Quantum walled Brauer-Clifford superalgebras, J. Algebra, 454 (2016), 433-474.
doi: 10.1016/j.jalgebra.2015.04.038. |
[2] |
Z. Chang and Y. Wang,
Howe duality for quantum queer superalgebras, J. Algebra, 547 (2020), 358-378.
doi: 10.1016/j.jalgebra.2019.11.023. |
[3] |
J. Du and H. Rui,
Based algebras and standard bases for quasi-hereditary algebras, Trans. Amer. Math. Soc., 350 (1998), 3207-3235.
doi: 10.1090/S0002-9947-98-02305-8. |
[4] |
M. Ehrig and D. Tubbenhauer, Relative cellular algebras, Transform. Groups, (2019).
doi: 10.1007/s00031-019-09544-5. |
[5] |
F. M. Goodman,
Cellularity of cyclotomic Birman-Wenzl-Murakami algebras, J. Algebra, 321 (2009), 3299-3320.
doi: 10.1016/j.jalgebra.2008.05.017. |
[6] |
J. J. Graham and G. I. Lehrer,
Cellular algebras, Invent. Math., 123 (1996), 1-34.
doi: 10.1007/BF01232365. |
[7] |
N. Guay and S. Wilcox,
Almost cellular algebras, J. Pure Appl. Algebra, 219 (2015), 4105-4116.
doi: 10.1016/j.jpaa.2015.02.010. |
[8] |
H. Rui and C. Xi,
The representation theory of cyclotomic Temperley-Lieb algebras, Comment. Math. Helv., 79 (2004), 427-450.
doi: 10.1007/s00014-004-0800-6. |
[9] |
A. S. Kleshchev,
Affine highest weight categories and affine quasihereditary algebras, Proc. Lond. Math. Soc. (3), 110 (2015), 841-882.
doi: 10.1112/plms/pdv004. |
[10] |
S. König and C. Xi, On the structure of cellular algebras, in Algebras and Modules, II, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998,365–385. |
[11] |
S. König and C. Xi,
When is a cellular algebra quasi-hereditary?, Math. Ann., 315 (1999), 281-293.
doi: 10.1007/s002080050368. |
[12] |
S. König and C. Xi,
Cellular algebras: Inflations and Morita equivalences, J. London Math. Soc. (2), 60 (1999), 700-722.
doi: 10.1112/S0024610799008212. |
[13] |
S. König and C. Xi,
Affine cellular algebras, Adv. Math., 229 (2012), 139-182.
doi: 10.1016/j.aim.2011.08.010. |
[14] |
C. Xi,
On the quasi-heredity of Birman-Wenzl algebras, Adv. Math., 154 (2000), 280-298.
doi: 10.1006/aima.2000.1919. |
show all references
References:
[1] |
G. Benkart, N. Guay, J. H. Jung, S.-J. Kang and S. Wilcox,
Quantum walled Brauer-Clifford superalgebras, J. Algebra, 454 (2016), 433-474.
doi: 10.1016/j.jalgebra.2015.04.038. |
[2] |
Z. Chang and Y. Wang,
Howe duality for quantum queer superalgebras, J. Algebra, 547 (2020), 358-378.
doi: 10.1016/j.jalgebra.2019.11.023. |
[3] |
J. Du and H. Rui,
Based algebras and standard bases for quasi-hereditary algebras, Trans. Amer. Math. Soc., 350 (1998), 3207-3235.
doi: 10.1090/S0002-9947-98-02305-8. |
[4] |
M. Ehrig and D. Tubbenhauer, Relative cellular algebras, Transform. Groups, (2019).
doi: 10.1007/s00031-019-09544-5. |
[5] |
F. M. Goodman,
Cellularity of cyclotomic Birman-Wenzl-Murakami algebras, J. Algebra, 321 (2009), 3299-3320.
doi: 10.1016/j.jalgebra.2008.05.017. |
[6] |
J. J. Graham and G. I. Lehrer,
Cellular algebras, Invent. Math., 123 (1996), 1-34.
doi: 10.1007/BF01232365. |
[7] |
N. Guay and S. Wilcox,
Almost cellular algebras, J. Pure Appl. Algebra, 219 (2015), 4105-4116.
doi: 10.1016/j.jpaa.2015.02.010. |
[8] |
H. Rui and C. Xi,
The representation theory of cyclotomic Temperley-Lieb algebras, Comment. Math. Helv., 79 (2004), 427-450.
doi: 10.1007/s00014-004-0800-6. |
[9] |
A. S. Kleshchev,
Affine highest weight categories and affine quasihereditary algebras, Proc. Lond. Math. Soc. (3), 110 (2015), 841-882.
doi: 10.1112/plms/pdv004. |
[10] |
S. König and C. Xi, On the structure of cellular algebras, in Algebras and Modules, II, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998,365–385. |
[11] |
S. König and C. Xi,
When is a cellular algebra quasi-hereditary?, Math. Ann., 315 (1999), 281-293.
doi: 10.1007/s002080050368. |
[12] |
S. König and C. Xi,
Cellular algebras: Inflations and Morita equivalences, J. London Math. Soc. (2), 60 (1999), 700-722.
doi: 10.1112/S0024610799008212. |
[13] |
S. König and C. Xi,
Affine cellular algebras, Adv. Math., 229 (2012), 139-182.
doi: 10.1016/j.aim.2011.08.010. |
[14] |
C. Xi,
On the quasi-heredity of Birman-Wenzl algebras, Adv. Math., 154 (2000), 280-298.
doi: 10.1006/aima.2000.1919. |
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