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Year 2018, , 68 - 72, 05.07.2018
https://doi.org/10.24330/ieja.440216

Abstract

References

  • R. Boltje and B. Kulshammer, On the depth 2 condition for group algebra and Hopf algebra extensions, J. Algebra, 323(6) (2010), 1783-1796.
  • R. Boltje and B. Kulshammer, Group algebra extensions of depth one, Algebra Number Theory, 5(1) (2011), 63-73.
  • C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. II, with applications to nite groups and orders, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.
  • L. Kadison and K. Szlachanyi, Bialgebroid actions on depth two extensions, Adv. Math., 179(1) (2003), 75-121.
  • M. Linckelmann, On splendid derived and stable equivalences between blocks of nite groups, J. Algebra, 242(2) (2001), 819-843.
  • L. Puig, Nilpotent blocks and their source algebras, Invent. Math., 93(1) (1988), 77-116.
  • L. Puig, Pointed groups and construction of modules, J. Algebra, 116(1) (1988), 7-129.
  • [L. Puig, The hyperfocal subalgebra of a block, Invent. Math., 141(2) (2000), 365-397.
  • J.-P. Serre, Corps Locaux, Deuxieme edition, Publications de l'Universite de Nancago, No. VIII, Hermann, Paris, 1968.
  • J. Thevenaz, G-Algebras and Modular Representation Theory, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  • A. Watanabe, Note on hyperfocal subalgebras of blocks of nite groups, J. Algebra, 322(2) (2009), 449-452.

A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP

Year 2018, , 68 - 72, 05.07.2018
https://doi.org/10.24330/ieja.440216

Abstract

By results of Boltje and Kulshammer, if a source algebra A of a
principal p-block of a nite group with a defect group P with inertial quotient
E is a depth two extension of the group algebra of P, then A is isomorphic
to a twisted group algebra of the group P o E. We show in this note that
this is true for arbitrary blocks. We observe further that the results of Boltje
and Kulshammer imply that A is a depth two extension of its hyperfocal
subalgebra, with a criterion for when this is a depth one extension. By a
result of Watanabe, this criterion is satised if the defect groups are abelian.

References

  • R. Boltje and B. Kulshammer, On the depth 2 condition for group algebra and Hopf algebra extensions, J. Algebra, 323(6) (2010), 1783-1796.
  • R. Boltje and B. Kulshammer, Group algebra extensions of depth one, Algebra Number Theory, 5(1) (2011), 63-73.
  • C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. II, with applications to nite groups and orders, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.
  • L. Kadison and K. Szlachanyi, Bialgebroid actions on depth two extensions, Adv. Math., 179(1) (2003), 75-121.
  • M. Linckelmann, On splendid derived and stable equivalences between blocks of nite groups, J. Algebra, 242(2) (2001), 819-843.
  • L. Puig, Nilpotent blocks and their source algebras, Invent. Math., 93(1) (1988), 77-116.
  • L. Puig, Pointed groups and construction of modules, J. Algebra, 116(1) (1988), 7-129.
  • [L. Puig, The hyperfocal subalgebra of a block, Invent. Math., 141(2) (2000), 365-397.
  • J.-P. Serre, Corps Locaux, Deuxieme edition, Publications de l'Universite de Nancago, No. VIII, Hermann, Paris, 1968.
  • J. Thevenaz, G-Algebras and Modular Representation Theory, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  • A. Watanabe, Note on hyperfocal subalgebras of blocks of nite groups, J. Algebra, 322(2) (2009), 449-452.
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Markus Linckelmann This is me

Publication Date July 5, 2018
Published in Issue Year 2018

Cite

APA Linckelmann, M. (2018). A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. International Electronic Journal of Algebra, 24(24), 68-72. https://doi.org/10.24330/ieja.440216
AMA Linckelmann M. A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. IEJA. July 2018;24(24):68-72. doi:10.24330/ieja.440216
Chicago Linckelmann, Markus. “A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP”. International Electronic Journal of Algebra 24, no. 24 (July 2018): 68-72. https://doi.org/10.24330/ieja.440216.
EndNote Linckelmann M (July 1, 2018) A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. International Electronic Journal of Algebra 24 24 68–72.
IEEE M. Linckelmann, “A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP”, IEJA, vol. 24, no. 24, pp. 68–72, 2018, doi: 10.24330/ieja.440216.
ISNAD Linckelmann, Markus. “A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP”. International Electronic Journal of Algebra 24/24 (July 2018), 68-72. https://doi.org/10.24330/ieja.440216.
JAMA Linckelmann M. A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. IEJA. 2018;24:68–72.
MLA Linckelmann, Markus. “A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP”. International Electronic Journal of Algebra, vol. 24, no. 24, 2018, pp. 68-72, doi:10.24330/ieja.440216.
Vancouver Linckelmann M. A NOTE ON THE DEPTH OF A SOURCE ALGEBRA OVER ITS DEFECT GROUP. IEJA. 2018;24(24):68-72.