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• Morita equivalence
  A Morita equivalence is an equivalence between [[algebra|algebras]]. Given a ring <math>R</math>, two <math>R</math>-algebras are de
  383 bytes (56 words) - 18:11, 3 January 2019
• Classification by p-group
  ...ath> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] ...math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]]
  33 KB (3,797 words) - 19:59, 10 January 2024
• Guide to contributing
  == Basic algebra == ...nd the structure of the projective indecomposable modules of a given group algebra (and hence of its blocks).
  3 KB (364 words) - 17:32, 9 December 2019
• References
  ...'', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234. ...esentations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.
  21 KB (2,957 words) - 12:29, 2 May 2024
• C2xC2
  ...In [[References|[CEKL11] ]] the CFSG was used to show that only one source algebra can occur for each Morita equivalence class.
  1 KB (218 words) - 09:04, 15 September 2018
• Open problems
  ...ibed in [[Blocks with basic algebras of low dimension]] occur as the basic algebra of a block of a finite group? - no (see [[References#L|[LM20]]] and [[Refer
  1 KB (178 words) - 12:30, 21 June 2021
• M(16,10,1)
  == Basic algebra ==
  2 KB (256 words) - 09:41, 4 December 2018
• M(3,1,1)
  == Basic algebra ==
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• M(3,1,2)
  == Basic algebra ==
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• M(2,1,1)
  == Basic algebra ==
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• M(4,1,1)
  == Basic algebra ==
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• M(4,2,1)
  == Basic algebra ==
  2 KB (253 words) - 14:30, 19 December 2018
• M(4,2,2)
  ...= <math>\mathcal{T}(B)=C_2</math><ref>Every Morita equivalence is a source algebra equivalence by [[References#C|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T} == Basic algebra ==
  2 KB (309 words) - 15:47, 4 January 2019
• Blocks
  <math>B_0(A)</math> will denote the principal block of an algebra <math>A</math>. However we often use the notation of [http://www.math.rwth-
  503 bytes (83 words) - 09:28, 21 September 2018
• M(4,2,3)
  ...= <math>\mathcal{T}(B)=S_3</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B == Basic algebra ==
  3 KB (348 words) - 16:29, 22 November 2018
• M(5,1,1)
  == Basic algebra ==
  1 KB (177 words) - 23:44, 2 January 2019
• M(25,1,1)
  == Basic algebra ==
  2 KB (269 words) - 16:24, 8 October 2018
• Generic classifications by p-group class
  ...vial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]]. Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].
  11 KB (1,772 words) - 12:15, 9 January 2024
• M(5,1,2)
  == Basic algebra ==
  2 KB (245 words) - 10:32, 4 September 2018
• M(5,1,3)
  == Basic algebra ==
  2 KB (252 words) - 10:31, 4 September 2018
• M(5,1,4)
  == Basic algebra ==
  2 KB (290 words) - 10:33, 4 September 2018
• M(5,1,5)
  == Basic algebra ==
  2 KB (288 words) - 10:20, 4 September 2018
• M(5,1,6)
  == Basic algebra ==
  2 KB (202 words) - 08:56, 28 September 2018
• Glossary
  ...<math>R</math> be a commutative ring and <math>A</math> an <math>R</math>-algebra. The Picard group <math>{\rm Pic}_R(A)</math> has elements isomorphism clas === Source algebra ===
  5 KB (841 words) - 23:02, 18 November 2020
• M(9,1,1)
  == Basic algebra ==
  1 KB (163 words) - 17:56, 1 February 2019
• M(9,1,2)
  == Basic algebra ==
  2 KB (265 words) - 21:43, 5 September 2018
• M(9,1,3)
  == Basic algebra ==
  2 KB (267 words) - 21:56, 5 September 2018
• M(27,1,1)
  == Basic algebra ==
  2 KB (254 words) - 13:30, 9 September 2018
• M(27,1,2)
  == Basic algebra ==
  2 KB (265 words) - 17:49, 6 September 2018
• M(27,1,3)
  == Basic algebra ==
  2 KB (270 words) - 17:54, 6 September 2018
• M(3^n,1,1)
  == Basic algebra ==
  2 KB (225 words) - 08:44, 3 December 2018
• M(3^n,1,2)
  == Basic algebra ==
  2 KB (290 words) - 15:22, 8 September 2018
• M(3^n,1,3)
  == Basic algebra ==
  2 KB (310 words) - 15:40, 8 September 2018
• M(8,1,1)
  == Basic algebra ==
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• M(2^n,1,1)
  == Basic algebra ==
  2 KB (240 words) - 16:03, 8 September 2018
• M(8,2,1)
  == Basic algebra ==
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• M(8,5,1)
  == Basic algebra ==
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• M(8,5,3)
  == Basic algebra ==
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• Miscallaneous results
  ...th the exception of one case of dimension 9 where no block with that basic algebra is identified. This final case was ruled out by Linckelmann and Murphy in [
  2 KB (289 words) - 14:51, 4 August 2022
• M(8,3,1)
  == Basic algebra ==
  1 KB (153 words) - 22:47, 5 October 2018
• M(8,4,3)
  == Basic algebra ==
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• M(8,5,2)
  == Basic algebra ==
  2 KB (326 words) - 21:49, 5 December 2018
• Blocks with basic algebras of low dimension
  ...th the exception of one case of dimension 9 where no block with that basic algebra was identified. This final case was ruled out by Linckelmann and Murphy in
  4 KB (528 words) - 14:33, 13 December 2023
• M(8,5,4)
  == Basic algebra ==
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• M(8,5,5)
  == Basic algebra ==
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• M(8,3,4)
  == Basic algebra ==
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• M(8,3,2)
  == Basic algebra ==
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• M(8,3,3)
  == Basic algebra ==
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• M(8,5,7)
  == Basic algebra == The basic algebra for the block defined over <math>\mathcal{O}</math> is described in [[Refer
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• M(16,2,1)
  == Basic algebra ==
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• M(16,2,2)
  == Basic algebra ==
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• M(8,3,6)
  == Basic algebra ==
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• M(8,3,5)
  == Basic algebra ==
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• M(16,1,1)
  == Basic algebra ==
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• M(8,5,6)
  == Basic algebra ==
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• M(8,5,8)
  == Basic algebra ==
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• M(8,4,1)
  == Basic algebra ==
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• M(8,4,2)
  == Basic algebra ==
  2 KB (265 words) - 09:40, 24 May 2022
• C3xC3
  Source algebra equivalence classes of principal blocks with this defect group have been cl
  6 KB (781 words) - 10:45, 24 May 2022
• M(9,2,1)
  == Basic algebra ==
  2 KB (185 words) - 18:16, 21 October 2018
• M(9,2,2)
  == Basic algebra ==
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• M(9,2,3)
  == Basic algebra ==
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• M(9,2,4)
  == Basic algebra ==
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• Picard groups
  The Picard group of an algebra is related to its automorphism group. Chapter 55 of [[References|[CuRe81b]] ...)</math> with finite index. There is equality if <math>A</math> is a basic algebra.<ref>For detail see [[References#C|[CuRe81b,Chapter 55]]]</ref>
  2 KB (316 words) - 12:05, 25 May 2021
• Known Picard groups
  ...m Out}_D(A) \times {\rm Aut}(D)/E</math>, where <math>A</math> is a source algebra for <math>B</math>.<ref>See Theorem 1.4 of [[References #B|[BKL18]]]</ref> Note that since any block with this defect group is source algebra equivalent to one of these three blocks<ref>See [[References#C|[CEKL11]]]</
  3 KB (577 words) - 12:04, 25 May 2021
• M(16,3,2)
  == Basic algebra ==
  1 KB (100 words) - 10:51, 15 August 2019
• M(16,10,2)
  == Basic algebra ==
  2 KB (254 words) - 09:31, 5 December 2018
• M(16,10,3)
  == Basic algebra ==
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• M(9,2,16)
  == Basic algebra ==
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• Nilpotent blocks
  ...nces#L|[Li18d,8.11.5]]]. Note that this does not in general imply a source algebra equivalence.</ref> *If the principal block is nilpotent, then it is source algebra equivalent to <math>kP</math>.
  2 KB (457 words) - 10:41, 19 December 2018
• Tame blocks
  A finite dimensional <math>k</math>-algebra <math>A</math> is said to have ''finite representation type'' if there are ...f papers and her book [[References#E|[Er90]]], Erdmann describes the basic algebra of tame type (see page vi of [[References#E|[Er90]]] for a definition), and
  3 KB (427 words) - 19:28, 9 November 2022
• Morita invariants
  * Isomorphism type of centre of algebra, and so dimension of centre (hence, in the case of <math>k</math>-blocks of * [[Glossary#Source algebra|Source algebra]]
  2 KB (267 words) - 12:40, 2 May 2024
• M(16,3,3)
  == Basic algebra ==
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• M(16,14,1)
  == Basic algebra ==
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• M(16,14,2)
  == Basic algebra ==
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• M(16,14,4)
  == Basic algebra ==
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• M(16,14,3)
  == Basic algebra ==
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• M(16,14,5)
  == Basic algebra ==
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• M(16,14,6)
  == Basic algebra ==
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• M(16,14,7)
  == Basic algebra ==
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• M(16,14,8)
  == Basic algebra ==
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• M(16,14,9)
  == Basic algebra ==
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• M(16,14,10)
  == Basic algebra ==
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• M(16,14,11)
  == Basic algebra ==
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• M(16,14,12)
  == Basic algebra ==
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• M(16,14,13)
  == Basic algebra ==
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• M(16,14,14)
  == Basic algebra ==
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• M(16,14,15)
  == Basic algebra ==
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• M(16,14,16)
  == Basic algebra ==
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• M(32,51,6)
  == Basic algebra ==
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• M(32,51,1)
  == Basic algebra ==
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• M(32,51,2)
  == Basic algebra ==
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• M(32,51,3)
  == Basic algebra ==
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• M(32,51,4)
  == Basic algebra ==
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• M(32,51,5)
  == Basic algebra ==
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• M(32,51,7)
  == Basic algebra ==
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• M(32,51,8)
  == Basic algebra ==
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• M(32,51,9)
  == Basic algebra ==
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• M(32,51,10)
  == Basic algebra ==
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• M(32,51,11)
  == Basic algebra ==
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• M(32,51,12)
  == Basic algebra ==
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• M(32,51,13)
  == Basic algebra ==
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• M(32,51,14)
  == Basic algebra ==
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• M(32,51,15)
  == Basic algebra ==
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• M(32,51,16)
  == Basic algebra ==
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• M(32,51,17)
  == Basic algebra ==
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• M(32,51,18)
  == Basic algebra ==
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• M(32,51,19)
  == Basic algebra ==
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• M(32,51,20)
  == Basic algebra ==
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• M(32,51,21)
  == Basic algebra ==
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• M(32,51,22)
  == Basic algebra ==
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• M(32,51,23)
  == Basic algebra ==
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• M(32,51,24)
  == Basic algebra ==
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• M(32,51,25)
  == Basic algebra ==
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• M(32,51,26)
  == Basic algebra ==
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• M(32,51,27)
  == Basic algebra ==
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• M(32,51,28)
  == Basic algebra ==
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• M(32,51,29)
  == Basic algebra ==
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• M(32,51,30)
  == Basic algebra ==
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• M(32,51,31)
  == Basic algebra ==
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• M(32,51,32)
  == Basic algebra ==
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• M(32,51,33)
  == Basic algebra ==
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• M(32,51,34)
  == Basic algebra ==
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• (Non-)Morita invariance of the isomorphism type of a defect group
  ...a negative answer . Pairs of nonisomorphic 2-groups are given whose group algebra over a field of two elements are isomorphic, hence their group algebras ove
  1 KB (204 words) - 12:30, 21 June 2021