Full quivers of representations of algebras

Belov-Kanel, Alexei; Rowen, Louis; Vishne, Uzi

doi:10.1090/S0002-9947-2012-05565-6

Full quivers of representations of algebras
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by Alexei Belov-Kanel, Louis H. Rowen and Uzi Vishne PDF

Trans. Amer. Math. Soc. 364 (2012), 5525-5569 Request permission

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.

References

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, DOI 10.1017/CBO9780511614309
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, DOI 10.1070/SM2000v191n03ABEH000460
Belov, A., Algebras with polynomial identities, representations, and combinatorial methods, Doctoral dissertation (2001), Moscow State University.
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, DOI 10.1070/RM2007v062n02ABEH004400
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, DOI 10.1070/SM2004v195n12ABEH000862
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, DOI 10.1070/IM2010v074n01ABEH002481
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
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, DOI 10.1016/j.aam.2005.08.006
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, DOI 10.1090/S0002-9947-10-04993-7
Belov-Kanel, A., Rowen, L.H., and Vishne, U., Application of full quivers of representations of algebras, to polynomial identities, preprint.
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.
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
Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103; correction, ibid. 6 (1972), 309 (German, with English summary). MR 332887, DOI 10.1007/BF01298413
A. Giambruno and M. Zaicev, Minimal varieties of algebras of exponential growth, Adv. Math. 174 (2003), no. 2, 310–323. MR 1963697, DOI 10.1016/S0001-8708(02)00047-6
Antonio Giambruno and Mikhail Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR 2176105, DOI 10.1090/surv/122
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
V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), no. 1, 57–92. MR 557581, DOI 10.1007/BF01403155
V. G. Kac, Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), no. 1, 141–162. MR 677715, DOI 10.1016/0021-8693(82)90105-3
Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhiro Takeuchi, Unipotent algebraic groups, Lecture Notes in Mathematics, Vol. 414, Springer-Verlag, Berlin-New York, 1974. MR 0376696
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, DOI 10.1007/BF01978562
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
Jacques Lewin, A matrix representation for associative algebras. I, II, Trans. Amer. Math. Soc. 188 (1974), 293–308; ibid. 188 (1974), 309–317. MR 338081, DOI 10.1090/S0002-9947-1974-0338081-5
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 158891, DOI 10.1007/BF02412850
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
Rowen, L.H., “Ring Theory”, AK Peters, 1988.
Tits, J., Lectures on algebraic groups, Dept. of Math., Yale Univ. of New Haven, 1966/67.