The hom-associative Weyl algebras in prime characteristic

Year 2022, , 203 - 229, 17.01.2022

Per Back Johan Rıchter

https://doi.org/10.24330/ieja.1058430

Abstract

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general ``twisting'' procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket.

Keywords

Hom-associative Ore extension, hom-associative Weyl algebra, formal multi-parameter hom-associative deformation, formal multi-parameter hom-Lie deformation

References

• G. Benkart, S. A. Lopes and M. Ondrus, Derivations of a parametric family of subalgebras of the Weyl algebra, J. Algebra, 424 (2015), 46-97.
• P. Back, Multi-parameter Formal Deformations of Ternary Hom-Nambu-Lie Algebras, Dobrev V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 335, Springer, Singapore, 2020.
• P. Back and J. Richter, Hilbert's basis theorem for non-associative and hom-associative Ore extensions, arXiv:1804.11304.
• P. Back and J. Richter, On the hom-associative Weyl algebras, J. Pure Appl. Algebra, 224(9) (2020) 106368 (12 pp).
• P. Back, J. Richter and S. Silvestrov, Hom-associative Ore extensions and weak unitalizations, Int. Electron. J. Algebra, 24 (2018), 174-194.
• S. C. Coutinho, A Primer of Algebraic D-modules, Cambridge University Press, Cambridge, 1995.
• J. Dixmier, Sur les algebres de Weyl, Bull. Soc. Math. France, 96 (1968), 209- 242.
• Y. Fregier and A. Gohr, On unitality conditions for hom-associative algebras, arXiv:0904.4874.
• K. R. Goodearl and R. B. Warfield, An Introduction to Noncommutative NoetherianRings, Second edition, Cambridge University Press, Cambridge U. K.,2004.
• J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra, 295 (2006), 314-361.
• A. Kanel-Belov and M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Mosc. Math. J., 7(2) (2007), 209-218.
• E. Lucas, Sur les congruences des nombres euleriens et des coeffcients differentiels des fonctions trigonometriques suivant un module premier, Bull. Soc. Math. France, 6 (1878), 49-54.
• L. Makar-Limanov, On automorphisms of Weyl algebra, Bull. Soc. Math. France, 112 (1984), 359-363.
• A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl., 2 (2008), 51-64.
• A. Makhlouf and S. Silvestrov, Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras, Forum Math., 22(4) (2010), 715-739.
• J. C. McConnell, J. C. Robson. With the cooperation of L. W. Small. Revised edition. Noncommutative Noetherian Rings, American Mathematical Society, Providence, R. I., 2001.
• P. Nystedt, J.  Oinert and J. Richter, Non-associative Ore extensions, Israel J. Math., 224(1) (2018), 263-292.
• O. Ore, Theory of Non-Commutative Polynomials, Ann. of Math., 34(3) (1933), 480-508.
• M. P. Revoy, Algebres de Weyl en caracteristique p, C. R. Acad. Sci. Paris Ser. A-B, 276 (1973), A225A228.
• R. Sridharan, Filtered Algebras and Representations of Lie Algebras, Trans. Amer. Math. Soc., 100 (1961), 530-550.
• Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math., 42(2) (2005), 435-452.
• Y. Tsuchimoto, Preliminaries on Dixmier conjecture, Mem. Fac. Sci. Kochi Univ. Ser. A Math., 24 (2003), 43-59.
• D. Yau, Hom-algebras and Homology, J. Lie Theory, 19(2) (2009), 409-421.