Journal of the Mathematical Society of Japan

Minor degenerations of the full matrix algebra over a field

Hisaaki FUJITA, Yosuke SAKAI, and Daniel SIMSON

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Abstract

Given a positive integer n 2 , an arbitrary field K and an n -block q = [ q ( 1 ) | | q ( n ) ] of n × n square matrices q ( 1 ) , , q ( n ) with coefficients in K satisfying certain conditions, we define a multiplication $._q : \mathbf{M}_n(K) \otimes_K \mathbf{M}_n(K) \rightarrow \mathbf{M}_n(K)$ on the K -module M n ( K ) of all square n × n matrices with coefficients in K in such a way that q defines a K -algebra structure on M n ( K ) . We denote it by M n q ( K ) , and we call it a minor q -degeneration of the full matrix K -algebra M n ( K ) . The class of minor degenerations of the algebra M n ( K ) and their modules are investigated in the paper by means of the properties of q and by applying quivers with relations. The Gabriel quiver of M n q ( K ) is described and conditions for q to be M n q ( K ) a Frobenius algebra are given. In case K is an infinite field, for each n 4 a one-parameter K -algebraic family { C μ } μ K * of basic pairwise non-isomorphic Frobenius K -algebras of the form C μ = M n q μ ( K ) is constructed. We also show that if A q = M n q ( K ) is a Frobenius algebra such that J ( A q ) 3 = 0 , then A q is representation-finite if and only if n = 3 , and A q is tame representation-infinite if and only if n = 4 .

Article information

Source
J. Math. Soc. Japan, Volume 59, Number 3 (2007), 763-795.

Dates
First available in Project Euclid: 5 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1191591857

Digital Object Identifier
doi:10.2969/jmsj/05930763

Mathematical Reviews number (MathSciNet)
MR2344827

Zentralblatt MATH identifier
1155.16009

Subjects
Primary: 16G10: Representations of Artinian rings
Secondary: 16G60: Representation type (finite, tame, wild, etc.) 14R20: Group actions on affine varieties [See also 13A50, 14L30] 16S80: Deformations of rings [See also 13D10, 14D15]

Keywords
degeneration tame representation type bound quiver group action

Citation

FUJITA, Hisaaki; SAKAI, Yosuke; SIMSON, Daniel. Minor degenerations of the full matrix algebra over a field. J. Math. Soc. Japan 59 (2007), no. 3, 763--795. doi:10.2969/jmsj/05930763. https://projecteuclid.org/euclid.jmsj/1191591857


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References

  • I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 1. Techniques of Representation Theory, London Math. Soc. Student Texts, 65, Cambridge Univ. Press, Cambridge-New York, 2006.
  • M. Auslander, I. Reiten and S. Smal\o, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, 1995.
  • P. Dowbor and A. Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv., 62 (1987), 311–337.
  • Ju. A. Drozd, Tame and wild matrix problems, Representations and Quadratic Forms, Akad. Nauk USSR, Inst. Matem., Kiev 1979, pp.,39–74.
  • J. A. Drozd and V. V. Kirichenko, Finite Dimensional Algebras, Springer-Verlag, Berlin, Heidelberg, New York, 1994.
  • M. A. Dukuchaev, V. V. Kirichenko and Ż. T. Chernousova, Tiled orders and Frobenius rings, Matem. Zametki, 72 (2002), 468–471.
  • H. Fujita, Full matrix algebras with structure systems, Colloq. Math., 98 (2003), 249–258.
  • H. Fujita and Y. Sakai, Frobenius full matrix algebras and Gorenstein tiled orders, Comm. Algebra, 34 (2006), 1181–1203.
  • H. Fujita, Y. Sakai and D. Simson, On Frobenius full matrix algebras with structure systems, Algebra Discrete Math., 1 (2007), in press.
  • P. Gabriel, Indecomposable representations II, Symposia Mat. Inst. Naz. Alta Mat., 11 (1973), 81–104.
  • P. Gabriel, Finite representation type is open, In: Proceedings of ICRA I, Ottawa, 1974, Lecture Notes in Math., 488, Springer-Verlag, 1975, pp.,132–155.
  • C. Geiss, On the degenerations of tame and wild algebras, Arch. Math. (Basel), 64 (1995), 11–16.
  • M. Gerstenhaber, On the deformations of rings and algebras, Ann. Math., 79 (1964), 59–103.
  • K. R. Goodearl and B. Huisgen-Zimmermann, Repetitive resolutions over classical orders and finite dimensional algebras, In: Algebras and Modules II, Proceedings of CMS Conference, Geiranger, 1996, 24, AMS, 1998, pp.,205–225.
  • E. L. Green and W. H. Gustafson, Pathological quasi-Frobenius algebras of finite type, Comm. Algebra, 2 (1974), 233–260.
  • V. A. Jategaonkar, Global dimension of tiled orders over discrete valuation rings, Trans. Amer. Math. Soc., 196 (1974), 313–330.
  • V. V. Kirichenko and T. I. Tsypiy, Tiled orders and their quivers, In: Abstracts of the Conference Representation Theory and Computer Algebra, Kiev, 1997, pp.,20–22.
  • H. Kraft, Geometric methods in representation theory, Lecture Notes in Math., 944 (1982), 180–258.
  • H. Kupisch, Über eine Klasse von Ringen mit Minimalbedingung I, Archiv Math., (Basel), 17 (1966), 20–35.
  • H. Kupisch, Über eine Klasse von Artin-Ringen II, Archiv Math., (Basel), 26 (1975), 23–35.
  • S. Montgomery, Hopf Algebras and Their Actions on Rings, CMBS, 82, AMS, 1993.
  • K. Oshiro and S. H. Rim, On QF-rings with cyclic Nakayama permutation, Osaka J. Math., 34 (1997), 1–19.
  • R. S. Pierce, Associative Algebras, Springer-Verlag, New York, Heidelberg, Berlin, 1982.
  • M. Ramras, Maximal orders over regular local rings of dimension two, Trans. Amer. Math. Soc., 142 (1969), 457–479.
  • K. W. Roggenkamp, V. V. Kirichenko, M. A. Khibina and V. N. Zhuravlev, Gorenstein tiled orders, Comm. Algebra, 29 (2001), 4231–4247.
  • D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Applications, 4, Gordon & Breach Science Publishers, 1992.
  • D. Simson, On Corner type Endo-Wild algebras, J. Pure Appl. Algebra, 202 (2005), 118–132.
  • Siu-Hung Ng, Non-semisimple Hopf algebras of dimension $p^2$, J. Algebra, 255 (2002), 182–197.
  • D. Simson and A. Skowroński, Extensions of artinian rings by hereditary injective modules, Lecture Notes in Math., 903 (1981), 315–330.
  • D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, 2, Tubes and Concealed Algebras of Euclidean Type, London Math. Soc. Student Texts, 71, Cambridge Univ. Press, Cambridge-New York, 2007.
  • D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, 3, Representation-Infinite Tilted Algebras, London Math. Soc. Student Texts, 72, Cambridge Univ. Press, Cambridge-New York, 2007.
  • A. Skowroński and J. Waschbüsch, Representation-finite biserial algebras, J. reine angew. Math., 345 (1985), 480–500.
  • A. Skowroński and K. Yamagata, A general form of non-Frobenius self-injective algebras, Colloq. Math., 105 (2006), 135–141.
  • R. B. Tarsy, Global dimension of orders, Trans. Amer. Math. Soc., 151 (1970), 335–340.
  • K. Yamagata, Frobenius algebras, (ed. M. Hazewinkel), Handbook of Algebra, 1, North-Holland Elsevier, Amsterdam, 1996, pp.,841–887.