Abstract
We show that if C is a finite split category, k is a field of characteristic 0, and α is a 2-cocycle of C with values in k × , then the twisted category algebra k α C is quasi-hereditary.
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Boltje R., Danz S.: A ghost algebra of the double Burnside algebra in characteristic zero. J. Pure Appl. Algebra. 217, 608–635 (2013)
C. Bowman A. Coxand M. De Visscher Decomposition numbers for the cyclotomic Brauer algebras in characteristic zero, preprint, 2012. arXiv:1205.3345v1 [math.RT].
Cline E., Parshall B., Scott L.: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)
Cox A., De Visscher M., Martin P.: The blocks of the Brauer algebra in characteristic zero. Represent. Theory 13, 272–308 (2009)
C. W. Curtis and I. Reiner, Methods of representation theory, vol. 1, J. Wiley & Sons, New York, 1981.
Dlab V., Ringel C.M.: Quasi-hereditary algebras. Illinois J. Math. 33, 280–291 (1989)
S. Donkin, The q-Schur algebra. London Mathematical Society Lecture Note Series, 253. Cambridge University Press, Cambridge, 1998.
Ganyushkin O., Mazorchuk V., Steinberg B.: On the irreducible representations of a finite semigroup. Proc. Amer. Math. Soc. 137, 3585–3592 (2009)
Graham J.J., Lehrer G.I.: Cellular algebras. Invent. Math. 123, 1–34 (1996)
Green J.A.: On the structure of semigroups. Ann. of Math. 54, 163–172 (1951)
J. A. Green, Polynomial representations of GL n . Second corrected and augmented edition. With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green, and M. Schocker. Lecture Notes in Mathematics, 830. Springer-Verlag, Berlin, 2007.
König S., Xi C.: When is a cellular algebra quasi-hereditary? Math. Ann. 315, 281–293 (1999)
Linckelmann M., Stolorz M.: On simple modules over twisted finite category algebras. Proc. Amer. Math. Soc. 140, 3725–3737 (2012)
M. Linckelmann and M. Stolorz, Quasi-hereditary twisted category algebras, Preprint, 2012.
Martin P.: The structure of the partition algebras. J. Algebra 183, 219–358 (1996)
Putcha M.S.: Complex representations of finite monoids. II. Highest weight categories and quivers. J. Algebra 205, 53–76 (1998)
Rui H., Xi C.: The representation theory of the cyclotomic Temperley–Lieb algebras. Comment. Math. Helvet. 79, 427–450 (2004)
Rui H., Yu W.: On the semi-simplicity of the cyclotomic Brauer algebras. J. Algebra 277, 187–221 (2004)
Webb P.: Stratifications and Mackey functors II: globally defined Mackey functors. J. K-Theory 6, 99–170 (2010)
Westbury B.W.: The representation theory of the Temperley–Lieb algebras. Math. Z. 219, 539–565 (1995)
Wilcox S.: Cellularity of diagram algebras as twisted semigroup algebras. J. Algebra 309, 10–31 (2007)
Xi C.: Partition algebras are cellular. Compositio Math. 119, 99–109 (1999)
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Boltje, R., Danz, S. Twisted split category algebras as quasi-hereditary algebras. Arch. Math. 99, 589–600 (2012). https://doi.org/10.1007/s00013-012-0448-1
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DOI: https://doi.org/10.1007/s00013-012-0448-1