Journal of the Mathematical Society of Japan

Minor degenerations of the full matrix algebra over a field

Abstract

Given a positive integer $n\geq 2$, an arbitrary field $K$ and an $n$-block $q = [ q^{(1)}| \cdots |q^{(n)} ]$ of $n\times n$ square matrices $q^{(1)}, \dots, q^{(n)}$ 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 $\bm{M}_n(K)$ of all square $n\times n$ matrices with coefficients in $K$ in such a way that $\cdot_{q}$ defines a $K$-algebra structure on $\bm{M}_n(K)$. We denote it by $\bm{M}^{q}_n(K)$, and we call it a minor $q$-degeneration of the full matrix $K$-algebra $\bm{M}_n(K)$. The class of minor degenerations of the algebra $\bm{M}_n(K)$ 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 $\bm{M}^{q}_n(K)$ is described and conditions for $q$ to be $\bm{M}^{q}_n(K)$ a Frobenius algebra are given. In case $K$ is an infinite field, for each $n\geq 4$ a one-parameter $K$-algebraic family $\{C_\mu\}_{\mu \in K^*}$ of basic pairwise non-isomorphic Frobenius $K$-algebras of the form $C_\mu=\bm{M}^{q_\mu}_n(K)$ is constructed. We also show that if $A_q = \bm{M}^{q}_n(K)$ 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

https://projecteuclid.org/euclid.jmsj/1191591857

Digital Object Identifier
doi:10.2969/jmsj/05930763

Mathematical Reviews number (MathSciNet)
MR2344827

Zentralblatt MATH identifier
1155.16009

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

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.