A characterisation of Lie algebras using ideals and subalgebras
Vladimir Dotsenko
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, Strasbourg, France
Search for more papers by this authorCorresponding Author
Xabier García-Martínez
Departamento de Matemáticas, Esc. Sup. de Enx. Informática, Campus de Ourense, CITMAga & Universidade de Vigo, Ourense, Spain
Correspondence
Xabier García-Martínez, CITMAga, and Universidade de Vigo, Departamento de Matemáticas, Esc. Sup. de Enx. Informática, Campus de Ourense, E–32004 Ourense, Spain.
Email: [email protected]
Search for more papers by this authorVladimir Dotsenko
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, Strasbourg, France
Search for more papers by this authorCorresponding Author
Xabier García-Martínez
Departamento de Matemáticas, Esc. Sup. de Enx. Informática, Campus de Ourense, CITMAga & Universidade de Vigo, Ourense, Spain
Correspondence
Xabier García-Martínez, CITMAga, and Universidade de Vigo, Departamento de Matemáticas, Esc. Sup. de Enx. Informática, Campus de Ourense, E–32004 Ourense, Spain.
Email: [email protected]
Search for more papers by this authorAbstract
We prove that if, for a non-trivial variety of non-associative algebras, every subalgebra of every free algebra is free and is an ideal whenever is an ideal, then this variety coincides with the variety of all Lie algebras.
REFERENCES
- 1M. Aguiar and S. Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series, vol. 29, American Mathematical Society, Providence, RI, 2010. With forewords by Kenneth Brown and Stephen Chase and André Joyal.
10.1090/crmm/029 Google Scholar
- 2T. Anderson, The Levitzki radical in varieties of algebras, Math. Ann. 194 (1971), 27–34.
- 3T. Anderson and E. Kleinfeld, On a class of 2-varieties, J. Algebra 51 (1978), no. 2, 367–374.
- 4F. Bergeron, G. Labelle, and P. Leroux, Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications, vol. 67, Cambridge University Press, Cambridge, 1998. Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota.
- 5W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3–4, 235–265. Computational algebra and number theory (London, 1993).
- 6M. R. Bremner and V. Dotsenko, Algebraic operads: an algorithmic companion, CRC Press, Boca Raton, FL, 2016.
10.1201/b20061 Google Scholar
- 7A. S. Cigoli, J. R. A. Gray, and T. Van der Linden, Algebraically coherent categories, Theory Appl. Categ. 30 (2015), Paper No. 54, 1864–1905.
- 8W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 4-3-0 — A computer algebra system for polynomial computations, 2022. http://www.singular.uni-kl.de.
- 9V. Dotsenko, Word operads and admissible orderings, Appl. Categ. Structures 28 (2020), no. 4, 595–600.
- 10V. Dotsenko and A. Khoroshkin, Gröbner bases for operads, Duke Math. J. 153 (2010), no. 2, 363–396.
- 11V. Dotsenko and U. Umirbaev, An effective criterion for Nielsen–Schreier varieties, 2022.
- 12X. García-Martínez, M. Tsishyn, T. Van der Linden, and C. Vienne, Algebras with representable representations, Proc. Edinb. Math. Soc. (2) 64 (2021), no. 3, 555–573.
- 13X. García-Martínez and T. Van der Linden, A characterisation of Lie algebras amongst anti-commutative algebras, J. Pure Appl. Algebra 223 (2019), no. 11, 4857–4870.
- 14X. García-Martínez and T. Van der Linden, A characterisation of Lie algebras via algebraic exponentiation, Adv. Math. 341 (2019), 92–117.
- 15J. R. A. Gray, Algebraic exponentiation for categories of Lie algebras, J. Pure Appl. Algebra 216 (2012), no. 8–9, 1964–1967.
- 16J. R. A. Gray, Algebraic exponentiation in general categories, Appl. Categ. Structures 20 (2012), no. 6, 543–567.
- 17A. Joyal, Une théorie combinatoire des séries formelles, Adv. in Math. 42 (1981), no. 1, 1–82.
- 18Y. A. Khashina, Schreier varieties of -Lie algebras, Sibirsk. Mat. Zh. 32 (1991), no. 2, 197–199.
- 19M. H. Kleinfeld, A generalization of anti-commutative rings arising from 2-varieties, Comm. Algebra 4 (1976), no. 10, 919–927.
- 20A. Kurosh, Non-associative free algebras and free products of algebras, Rec. Math. [Mat. Sbornik] N.S. 20 (1947), no. 62, 239–262.
- 21S. X. Liu and C. E. Tsai, Wedderburn theorem on varieties of algebras, J. Algebra 75 (1982), no. 2, 315–323.
- 22S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer, New York-Berlin, 1971.
- 23M. Markl and E. Remm, Algebras with one operation including Poisson and other Lie-admissible algebras, J. Algebra 299 (2006), no. 1, 171–189.
- 24J. Nielsen, Die Isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924), no. 3-4, 169–209.
10.1007/BF01556078 Google Scholar
- 25G. M. Piacentini Cattaneo, A special class of 2-varieties, J. Algebra 95 (1985), no. 1, 116–124.
- 26D. Pokrass, Some radical properties of rings with , Pacific J. Math. 76 (1978), no. 2, 479–483.
- 27M. Rich, Some radical properties of -rings, Proc. Amer. Math. Soc. 30 (1971), 40–42.
- 28O. Schreier, Die Untergruppen der freien Gruppen, Abh. Math. Sem. Univ. Hamburg 5 (1927), no. 1, 161–183.
10.1007/BF02952517 Google Scholar
- 29U. U. Umirbaev, On Schreier varieties of algebras, Algebra i Logika 33 (1994), no. 3, 317–340.
- 30A. I. Širšov, Subalgebras of free Lie algebras, Mat. Sbornik N.S. 33 (1953), no. 75, 441–452.
- 31A. I. Širšov, Subalgebras of free commutative and free anticommutative algebras, Mat. Sbornik N.S. 34 (1954), no. 76, 81–88.
- 32E. Witt, Die Unterringe der freien Lieschen Ringe, Math. Z. 64 (1956), 195–216.
10.1007/BF01166568 Google Scholar
- 33P. J. Zwier, Prime ideals in a large class of nonassociative rings, Trans. Amer. Math. Soc. 158 (1971), 257–271.
