Irreducible bimodules over alternative algebras and superalgebras. (English) Zbl 1397.17006
Summary: The aim of the paper is to describe irreducible birepresentations of alternative algebras and superalgebras. The complete classification is obtained for irreducible even bimodules of arbitrary dimension and characteristic and for finite-dimensional irreducible superbimodules over an algebraically closed field. We also describe irreducible superbimodules of any dimension and characteristic over the simple alternative superalgebras.
MSC:
| 17A60 | Structure theory for nonassociative algebras |
| 17A70 | Superalgebras |
| 17D05 | Alternative rings |
| 16D90 | Module categories in associative algebras |
| 16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
References:
| [1] | Bahturin, Yuri; Giambruno, Antonio; Riley, David M., Group-graded algebras with polynomial identity, Israel J. Math., 104, 145-155 (1998) · Zbl 0920.16010 · doi:10.1007/BF02897062 |
| [2] | Bergman, George M., The diamond lemma for ring theory, Adv. in Math., 29, 2, 178-218 (1978) · Zbl 0326.16019 · doi:10.1016/0001-8708(78)90010-5 |
| [3] | Bokut{\cprime }, L. A., Imbeddings into simple associative algebras, Algebra i Logika, 15, 2, 117-142, 245 (1976) · Zbl 0349.16007 |
| [4] | Bokut{\cprime }, L. A.; Kolesnikov, P. S., Gr\"obner-Shirshov bases: from inception to the present time, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). J. Math. Sci. (N. Y.), 272 116, 1, 2894-2916 (2003) · Zbl 1069.16026 · doi:10.1023/A:1023490323855 |
| [5] | Bruck, R. H.; Kleinfeld, Erwin, The structure of alternative division rings, Proc. Amer. Math. Soc., 2, 878-890 (1951) · Zbl 0044.02205 |
| [6] | Cohen, M.; Montgomery, S., Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc., 282, 1, 237-258 (1984) · Zbl 0533.16001 · doi:10.2307/1999586 |
| [7] | Elduque, Alberto; Shestakov, Ivan P., Irreducible non-Lie modules for Malcev superalgebras, J. Algebra, 173, 3, 622-637 (1995) · Zbl 0824.17005 · doi:10.1006/jabr.1995.1106 |
| [8] | Jacobson, N., Structure of alternative and Jordan bimodules, Osaka Math. J., 6, 1-71 (1954) · Zbl 0059.02902 |
| [9] | L{\'o}pez-D{\'{\i }}az, M. C.; Shestakov, Ivan P., Representations of exceptional simple alternative superalgebras of characteristic 3, Trans. Amer. Math. Soc., 354, 7, 2745-2758 (electronic) (2002) · Zbl 1063.17021 · doi:10.1090/S0002-9947-02-02993-8 |
| [10] | L{\'o}pez-D{\'{\i }}az, M. C.; Shestakov, Ivan P., Alternative superalgebras with DCC on two-sided ideals, Comm. Algebra, 33, 10, 3479-3487 (2005) · Zbl 1099.17016 · doi:10.1080/AGB-200058391 |
| [11] | Pisarenko, N. A., The structure of alternative superbimodules, Algebra i Logika. Algebra and Logic, 33 33, 6, 386-397 (1995) (1994) · Zbl 0840.17030 · doi:10.1007/BF00756352 |
| [12] | Polikarpov, S. V.; Shestakov, I. P., Nonassociative affine algebras, Algebra and Logic, 29, 6, 458-466 (1991) (1990) · Zbl 0786.17002 · doi:10.1007/BF01978558 |
| [13] | Posner, Edward C., Differentiably simple rings, Proc. Amer. Math. Soc., 11, 337-343 (1960) · Zbl 0103.26802 |
| [14] | Rowen, Louis Halle, Polynomial identities in ring theory, Pure and Applied Mathematics 84, xx+365 pp. (1980), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0461.16001 |
| [15] | Schafer, R. D., Alternative algebras over an arbitrary field, Bull. Amer. Math. Soc., 49, 549-555 (1943) · Zbl 0061.05201 |
| [16] | Schafer, R. D., Representations of alternative algebras, Trans. Amer. Math. Soc., 72, 1-17 (1952) · Zbl 0046.03503 |
| [17] | Shestakov, I. P., Absolute zero divisors and radicals of finitely generated alternative algebras, Algebra i Logika, 15, 5, 585-602, 606 (1976) · Zbl 0369.17011 |
| [18] | Shestakov, I. P., Superalgebras and counterexamples, Sibirsk. Mat. Zh.. Siberian Math. J., 32 32, 6, 1052-1060 (1992) (1991) · Zbl 0777.17004 · doi:10.1007/BF00971214 |
| [19] | Shestakov, I. P., Prime alternative superalgebras of arbitrary characteristic, Algebra i Logika. Algebra and Logic, 36 36, 6, 389-412 (1997) · Zbl 0904.17025 · doi:10.1007/BF02671556 |
| [20] | Shestakov, I. P., Alternative and Jordan superalgebras. Algebra, geometry, analysis and mathematical physics (Russian) , Novosibirsk, 1996, 157-169, 191 (1997), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk · Zbl 0902.17017 |
| [21] | Yuan, Shuen, Differentiably simple rings of prime characteristic, Duke Math. J., 31, 623-630 (1964) · Zbl 0145.27701 |
| [22] | Slin{\cprime }ko, A. M.; Shestakov, I. P., Right representations of algebras, Algebra i Logika, 13, 5, 544-588, 605-606 (1974) |
| [23] | Smoktunowicz, Agata; Ziembowski, Micha\l, Differential polynomial rings over locally nilpotent rings need not be Jacobson radical, J. Algebra, 412, 207-217 (2014) · Zbl 1303.16024 · doi:10.1016/j.jalgebra.2014.04.022 |
| [24] | Trushina, M. N., Irreducible alternative superbimodules over the simple alternative superalgebra \(B(1,2)\), Fundam. Prikl. Mat., 7, 3, 897-908 (2001) · Zbl 1033.17005 |
| [25] | Trushina, M. N.; Shestakov, I. P., Representations of alternative algebras and superalgebras, Fundam. Prikl. Mat.. J. Math. Sci. (N. Y.), 17 185, 3, 504-512 (2012) · Zbl 1279.17006 · doi:10.1007/s10958-012-0932-y |
| [26] | Wall, C. T. C., Graded Brauer groups, J. Reine Angew. Math., 213, 187-199 (1963/1964) · Zbl 0125.01904 |
| [27] | Zariski, Oscar; Samuel, Pierre, Commutative algebra, Volume I, The University Series in Higher Mathematics, xi+329 pp. (1958), D. Van Nostrand Company, Inc., Princeton, New Jersey · Zbl 0081.26501 |
| [28] | Zel{\cprime }manov, E. I.; Shestakov, I. P., Prime alternative superalgebras and the nilpotency of the radical of a free alternative algebra, Izv. Akad. Nauk SSSR Ser. Mat.. Math. USSR-Izv., 54 37, 1, 19-36 (1991) · Zbl 0724.17022 |
| [29] | Zevlakov, K. A., The radical and representations of alternative rings, Algebra i Logika, 11, 162-173, 237 (1972) · Zbl 0262.17010 |
| [30] | Zevlakov, K. A.; Slin{\cprime }ko, A. M.; Shestakov, I. P.; Sir{\v{s}}ov, A. I., Koltsa, blizkie k assotsiativnym, 431 pp. (1978), “Nauka”, Moscow · Zbl 0445.17001 |
| [31] | Zorn, Max, Theorie der alternativen ringe, Abh. Math. Sem. Univ. Hamburg, 8, 1, 123-147 (1931) · JFM 56.0140.01 · doi:10.1007/BF02940993 |
| [32] | Zorn, Max, Alternative rings and related questions I: existence of the radical, Ann. of Math. (2), 42, 676-686 (1941) · Zbl 0025.30203 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
