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 Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 1, Pages 183–206 (Mi izv470)

Prime alternative algebras that are nearly commutative

S. V. Pchelintsev

Abstract: We prove that by deforming the multiplication in a prime commutative alternative algebra using a C-operation we obtain a prime non-commutative alternative algebra. Under certain restrictions on non-commutative algebras this relation between algebras is reversible. Isotopes are special cases of deformations. We introduce and study a linear space generated by the Bruck C-operations. We prove that the Bruck space is generated by operations of rank 1 and 2 and that “general” Bruck operations of rank 2 are independent in the following sense: a sum of $n$ operations of rank 2 cannot be written as a linear combination of $(n-1)$ operations of rank 2 and an arbitrary operation of rank 1. We describe infinite series of non-isomorphic prime non-commutative algebras of bounded degree that are deformations of a concrete prime commutative algebra.

DOI: https://doi.org/10.4213/im470

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English version:
Izvestiya: Mathematics, 2004, 68:1, 181–204

Bibliographic databases:

UDC: 512.554.5
MSC: 17D05

Citation: S. V. Pchelintsev, “Prime alternative algebras that are nearly commutative”, Izv. RAN. Ser. Mat., 68:1 (2004), 183–206; Izv. Math., 68:1 (2004), 181–204

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv470
• https://doi.org/10.4213/im470
• http://mi.mathnet.ru/eng/izv/v68/i1/p183

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. V. Pchelintsev, “Structure of finitely generated commutative alternative algebras and special Moufang loops”, Math. Notes, 80:3 (2006), 396–402
2. S. V. Pchelintsev, “Exceptional prime alternative algebras”, Siberian Math. J., 48:6 (2007), 1060–1073
3. S. V. Pchelintsev, “Degenerate alternative algebras”, Siberian Math. J., 55:2 (2014), 323–335
4. A. S. Panasenko, “Just Infinite Alternative Algebras”, Math. Notes, 98:5 (2015), 805–812
5. S. V. Pchelintsev, “Isotopes of the alternative monster and the Skosyrsky algebra”, Siberian Math. J., 57:4 (2016), 666–678
6. V. N. Zhelyabin, A. S. Panasenko, “Nil Ideals of Finite Codimension in Alternative Noetherian Algebras”, Math. Notes, 101:3 (2017), 460–466
7. Pchelintsev S.V., “Proper identities of finitely generated commutative alternative algebras”, J. Algebra, 470 (2017), 425–440
8. S. V. Pchelintsev, O. V. Shashkov, “Simple $5$-dimensional right alternative superalgebras with trivial even part”, Siberian Math. J., 58:6 (2017), 1078–1089
9. S. V. Pchelintsev, “Prime algebras connected with monsters”, Siberian Math. J., 59:2 (2018), 341–356
10. V. N. Zhelyabin, A. S. Panasenko, “Nearly finite-dimensional Jordan algebras”, Algebra and Logic, 57:5 (2018), 336–352
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