Irreducible bimodules over alternative algebras and superalgebras

Shestakov, Ivan; Trushina, Maria

doi:10.1090/tran/6475

Irreducible bimodules over alternative algebras and superalgebras
HTML articles powered by AMS MathViewer

by Ivan Shestakov and Maria Trushina PDF

Trans. Amer. Math. Soc. 368 (2016), 4657-4684 Request permission

Abstract:

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.

References

Yuri Bahturin, Antonio Giambruno, and David M. Riley, Group-graded algebras with polynomial identity, Israel J. Math. 104 (1998), 145–155. MR 1622291, DOI 10.1007/BF02897062
George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218. MR 506890, DOI 10.1016/0001-8708(78)90010-5
L. A. Bokut′, Imbeddings into simple associative algebras, Algebra i Logika 15 (1976), no. 2, 117–142, 245 (Russian). MR 0506423
L. A. Bokut′ and P. S. Kolesnikov, Gröbner-Shirshov bases: from inception to the present time, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), no. Vopr. Teor. Predst. Algebr i Grupp. 7, 26–67, 345 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 116 (2003), no. 1, 2894–2916. MR 1811792, DOI 10.1023/A:1023490323855
R. H. Bruck and Erwin Kleinfeld, The structure of alternative division rings, Proc. Amer. Math. Soc. 2 (1951), 878–890. MR 45099, DOI 10.1090/S0002-9939-1951-0045099-9
M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), no. 1, 237–258. MR 728711, DOI 10.1090/S0002-9947-1984-0728711-4
Alberto Elduque and Ivan P. Shestakov, Irreducible non-Lie modules for Malcev superalgebras, J. Algebra 173 (1995), no. 3, 622–637. MR 1327872, DOI 10.1006/jabr.1995.1106
N. Jacobson, Structure of alternative and Jordan bimodules, Osaka Math. J. 6 (1954), 1–71. MR 64762
M. C. López-Díaz and Ivan P. Shestakov, Representations of exceptional simple alternative superalgebras of characteristic 3, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2745–2758. MR 1895201, DOI 10.1090/S0002-9947-02-02993-8
M. C. López-Díaz and Ivan P. Shestakov, Alternative superalgebras with DCC on two-sided ideals, Comm. Algebra 33 (2005), no. 10, 3479–3487. MR 2175446, DOI 10.1080/AGB-200058391
N. A. Pisarenko, The structure of alternative superbimodules, Algebra i Logika 33 (1994), no. 6, 689–707, 717 (Russian, with Russian summary); English transl., Algebra and Logic 33 (1994), no. 6, 386–397 (1995). MR 1347266, DOI 10.1007/BF00756352
S. V. Polikarpov and I. P. Shestakov, Nonassociative affine algebras, Algebra and Logic 29 (1990), no. 6, 458–466 (1991). MR 1159142, DOI 10.1007/BF01978558
Edward C. Posner, Differentiably simple rings, Proc. Amer. Math. Soc. 11 (1960), 337–343. MR 113908, DOI 10.1090/S0002-9939-1960-0113908-6
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
R. D. Schafer, Alternative algebras over an arbitrary field, Bull. Amer. Math. Soc. 49 (1943), 549–555. MR 8610, DOI 10.1090/S0002-9904-1943-07967-0
R. D. Schafer, Representations of alternative algebras, Trans. Amer. Math. Soc. 72 (1952), 1–17. MR 45101, DOI 10.1090/S0002-9947-1952-0045101-X
I. P. Šestakov, Absolute zero divisors and radicals of finitely generated alternative algebras, Algebra i Logika 15 (1976), no. 5, 585–602, 606 (Russian). MR 0506525
I. P. Shestakov, Superalgebras and counterexamples, Sibirsk. Mat. Zh. 32 (1991), no. 6, 187–196, 207 (Russian); English transl., Siberian Math. J. 32 (1991), no. 6, 1052–1060 (1992). MR 1156760, DOI 10.1007/BF00971214
I. P. Shestakov, Prime alternative superalgebras of arbitrary characteristic, Algebra i Logika 36 (1997), no. 6, 675–716, 722 (Russian, with Russian summary); English transl., Algebra and Logic 36 (1997), no. 6, 389–412. MR 1657313, DOI 10.1007/BF02671556
I. P. Shestakov, Alternative and Jordan superalgebras, Algebra, geometry, analysis and mathematical physics (Russian) (Novosibirsk, 1996) Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1997, pp. 157–169, 191 (Russian, with Russian summary). MR 1624131
Shuen Yuan, Differentiably simple rings of prime characteristic, Duke Math. J. 31 (1964), 623–630. MR 167499
A. M. Slin′ko and I. P. Šestakov, Right representations of algebras, Algebra i Logika 13 (1974), no. 5, 544–588, 605–606 (Russian). MR 0498761
Agata Smoktunowicz and MichałZiembowski, Differential polynomial rings over locally nilpotent rings need not be Jacobson radical, J. Algebra 412 (2014), 207–217. MR 3215954, DOI 10.1016/j.jalgebra.2014.04.022
M. N. Trushina, Irreducible alternative superbimodules over the simple alternative superalgebra $B(1,2)$, Fundam. Prikl. Mat. 7 (2001), no. 3, 897–908 (Russian, with English and Russian summaries). MR 1879305
M. N. Trushina and I. P. Shestakov, Representations of alternative algebras and superalgebras, Fundam. Prikl. Mat. 17 (2011/12), no. 1, 233–246 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 185 (2012), no. 3, 504–512. MR 2898228, DOI 10.1007/s10958-012-0932-y
C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/64), 187–199. MR 167498, DOI 10.1515/crll.1964.213.187
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
E. I. Zel′manov and I. P. Shestakov, Prime alternative superalgebras and the nilpotency of the radical of a free alternative algebra, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 676–693 (Russian); English transl., Math. USSR-Izv. 37 (1991), no. 1, 19–36. MR 1073082
K. A. Ževlakov, The radical and representations of alternative rings, Algebra i Logika 11 (1972), 162–173, 237 (Russian). MR 0325723
K. A. Ževlakov, A. M. Slin′ko, I. P. Šestakov, and A. I. Širšov, Kol′tsa, blizkie k assotsiativnym, Sovremennaya Algebra. [Modern Algebra], “Nauka”, Moscow, 1978 (Russian). MR 518614
Max Zorn, Theorie der alternativen ringe, Abh. Math. Sem. Univ. Hamburg 8 (1931), no. 1, 123–147 (German). MR 3069547, DOI 10.1007/BF02940993
Max Zorn, Alternative rings and related questions I: existence of the radical, Ann. of Math. (2) 42 (1941), 676–686. MR 5098, DOI 10.2307/1969256