Full quivers of representations of algebras

Authors:
Alexei Belov-Kanel, Louis H. Rowen and Uzi Vishne

Journal:
Trans. Amer. Math. Soc. **364** (2012), 5525-5569

MSC (2010):
Primary 16R99, 16G99

Published electronically:
May 24, 2012

MathSciNet review:
2931338

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Abstract: We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study subtle combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations apply especially well to Zariski closed algebras, which have properties very like those of finite dimensional algebras over fields. By choosing the representation appropriately, one can restrict the gluing to two main types: *Frobenius* (along the diagonal) and, more generally, *proportional* Frobenius gluing (above the diagonal), and our main result is that any representable algebra has a faithful representation described completely by such a full quiver. Further reductions are considered, which bear on the polynomial identities.

**1.**Ibrahim Assem, Daniel Simson, and Andrzej Skowroński,*Elements of the representation theory of associative algebras. Vol. 1*, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR**2197389****2.**A. Ya. Belov,*Counterexamples to the Specht problem*, Mat. Sb.**191**(2000), no. 3, 13–24 (Russian, with Russian summary); English transl., Sb. Math.**191**(2000), no. 3-4, 329–340. MR**1773251**, 10.1070/SM2000v191n03ABEH000460**3.**Belov, A., Algebras with polynomial identities, representations, and combinatorial methods, Doctoral dissertation (2001), Moscow State University.**4.**A. Ya. Belov,*On varieties generated by a ring that is finite-dimensional over a centroid*, Uspekhi Mat. Nauk**62**(2007), no. 2(374), 171–172 (Russian); English transl., Russian Math. Surveys**62**(2007), no. 2, 379–381. MR**2352369**, 10.1070/RM2007v062n02ABEH004400**5.**A. Ya. Belov,*The Gel′fand-Kirillov dimension of relatively free associative algebras*, Mat. Sb.**195**(2004), no. 12, 3–26 (Russian, with Russian summary); English transl., Sb. Math.**195**(2004), no. 11-12, 1703–1726. MR**2138478**, 10.1070/SM2004v195n12ABEH000862**6.**A. Ya. Belov,*Local finite basis property and local representability of varieties of associative rings*, Izv. Ross. Akad. Nauk Ser. Mat.**74**(2010), no. 1, 3–134 (Russian, with Russian summary); English transl., Izv. Math.**74**(2010), no. 1, 1–126. MR**2655238**, 10.1070/IM2010v074n01ABEH002481**7.**Alexei Kanel-Belov and Louis Halle Rowen,*Computational aspects of polynomial identities*, Research Notes in Mathematics, vol. 9, A K Peters, Ltd., Wellesley, MA, 2005. MR**2124127****8.**Alexei Kanel-Belov, Louis H. Rowen, and Uzi Vishne,*Normal bases of PI-algebras*, Adv. in Appl. Math.**37**(2006), no. 3, 378–389. MR**2261179**, 10.1016/j.aam.2005.08.006**9.**Alexei Belov-Kanel, Louis Rowen, and Uzi Vishne,*Structure of Zariski-closed algebras*, Trans. Amer. Math. Soc.**362**(2010), no. 9, 4695–4734. MR**2645047**, 10.1090/S0002-9947-10-04993-7**10.**Belov-Kanel, A., Rowen, L.H., and Vishne, U., Application of full quivers of representations of algebras, to polynomial identities, preprint.**11.**Belov-Kanel, A., Rowen, L.H., and Vishne, U., Classification of full quivers of representations of algebras, in terms of their polynomial identities, in preparation.**12.**I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev,*Coxeter functors, and Gabriel’s theorem*, Uspehi Mat. Nauk**28**(1973), no. 2(170), 19–33 (Russian). MR**0393065****13.**Peter Gabriel,*Unzerlegbare Darstellungen. I*, Manuscripta Math.**6**(1972), 71–103; correction, ibid. 6 (1972), 309 (German, with English summary). MR**0332887****14.**A. Giambruno and M. Zaicev,*Minimal varieties of algebras of exponential growth*, Adv. Math.**174**(2003), no. 2, 310–323. MR**1963697**, 10.1016/S0001-8708(02)00047-6**15.**Antonio Giambruno and Mikhail Zaicev,*Polynomial identities and asymptotic methods*, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR**2176105****16.**A. V. Grishin,*Examples of 𝑇-spaces and 𝑇-ideals of characteristic 2 without the finite basis property*, Fundam. Prikl. Mat.**5**(1999), no. 1, 101–118 (Russian, with English and Russian summaries). MR**1799541****17.**V. G. Kac,*Infinite root systems, representations of graphs and invariant theory*, Invent. Math.**56**(1980), no. 1, 57–92. MR**557581**, 10.1007/BF01403155**18.**V. G. Kac,*Infinite root systems, representations of graphs and invariant theory. II*, J. Algebra**78**(1982), no. 1, 141–162. MR**677715**, 10.1016/0021-8693(82)90105-3**19.**Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhiro Takeuchi,*Unipotent algebraic groups*, Lecture Notes in Mathematics, Vol. 414, Springer-Verlag, Berlin-New York, 1974. MR**0376696****20.**A. R. Kemer,*Representability of reduced-free algebras*, Algebra i Logika**27**(1988), no. 3, 274–294, 375 (Russian); English transl., Algebra and Logic**27**(1988), no. 3, 167–184 (1989). MR**997959**, 10.1007/BF01978562**21.**Alexander R. Kemer,*Identities of associative algebras*, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 351–359. MR**1159223****22.**Jacques Lewin,*A matrix representation for associative algebras. I, II*, Trans. Amer. Math. Soc.**188**(1974), 293–308; ibid. 188 (1974), 309–317. MR**0338081**, 10.1090/S0002-9947-1974-0338081-5**23.**Maxwell Rosenlicht,*Questions of rationality for solvable algebraic groups over nonperfect fields*, Ann. Mat. Pura Appl. (4)**61**(1963), 97–120 (English, with Italian summary). MR**0158891****24.**Louis Halle Rowen,*Polynomial identities in ring theory*, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**576061****25.**Rowen, L.H., ``Ring Theory'', AK Peters, 1988.**26.**Tits, J.,*Lectures on algebraic groups*, Dept. of Math., Yale Univ. of New Haven, 1966/67.

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Additional Information

**Alexei Belov-Kanel**

Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Email:
belova@macs.biu.ac.il

**Louis H. Rowen**

Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Email:
rowen@macs.biu.ac.il

**Uzi Vishne**

Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Email:
vishne@macs.biu.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-2012-05565-6

Received by editor(s):
May 31, 2010

Received by editor(s) in revised form:
January 27, 2011

Published electronically:
May 24, 2012

Additional Notes:
This research was supported by the Israel Science Foundation (grant No. 1178/06).

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.