American Institute of Mathematical Sciences

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doi: 10.3934/era.2020086

Some properties for almost cellular algebras

Yongjie Wang 1, and  Nan Gao 2,,

1. 

School of Mathematics, Hefei University of Technology, Hefei, Anhui 230009, China
2. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

* Corresponding author: Nan Gao

Received  April 2020 Revised  June 2020 Published  August 2020

Fund Project: The first author is supported by National Natural Science Foundation of China Project (No. 11901146), and the second author is supported by National Natural Science Foundation of China Project (No.11771272)

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.

Keywords: Cellular algebra, quasi-hereditary algebra, iterated inflations, involution, filtration.
Mathematics Subject Classification: 16W70, 20C08.
Citation: Yongjie Wang, Nan Gao. Some properties for almost cellular algebras. Electronic Research Archive, doi: 10.3934/era.2020086
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Ehrig and D. Tubbenhauer, Relative cellular algebras, Transform. Groups, (2019). doi: 10.1007/s00031-019-09544-5.  Google Scholar

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F. M. Goodman, Cellularity of cyclotomic Birman-Wenzl-Murakami algebras, J. Algebra, 321 (2009), 3299-3320.  doi: 10.1016/j.jalgebra.2008.05.017.  Google Scholar

[6]

J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math., 123 (1996), 1-34.  doi: 10.1007/BF01232365.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. König and C. Xi, When is a cellular algebra quasi-hereditary?, Math. Ann., 315 (1999), 281-293.  doi: 10.1007/s002080050368.  Google Scholar

[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.  Google Scholar

[13]

S. König and C. Xi, Affine cellular algebras, Adv. Math., 229 (2012), 139-182.  doi: 10.1016/j.aim.2011.08.010.  Google Scholar

[14]

C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math., 154 (2000), 280-298.  doi: 10.1006/aima.2000.1919.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Ehrig and D. Tubbenhauer, Relative cellular algebras, Transform. Groups, (2019). doi: 10.1007/s00031-019-09544-5.  Google Scholar

[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.  Google Scholar

[6]

J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math., 123 (1996), 1-34.  doi: 10.1007/BF01232365.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. König and C. Xi, When is a cellular algebra quasi-hereditary?, Math. Ann., 315 (1999), 281-293.  doi: 10.1007/s002080050368.  Google Scholar

[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.  Google Scholar

[13]

S. König and C. Xi, Affine cellular algebras, Adv. Math., 229 (2012), 139-182.  doi: 10.1016/j.aim.2011.08.010.  Google Scholar

[14]

C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math., 154 (2000), 280-298.  doi: 10.1006/aima.2000.1919.  Google Scholar

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