ON SPLIT LEIBNIZ TRIPLE SYSTEMS
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In order to study the structure of arbitrary split Leibniz triple systems, we introduce the class of split Leibniz triple systems as the natural extension of the class of split Lie triple systems and split Leibniz algebras. By developing techniques of connections of roots for this kind of triple systems, we show that any of such Leibniz triple systems $T$ with a symmetric root system is of the form $T=U+\sum_{[j]\in \Lambda^{1}/\sim} I_{[j]}$ with $U$ a subspace of $T_{0}$ and any $I_{[j]}$ a well described ideal of $T$, satisfying $\{I_{[j]},T,I_{[k]}\} =\{I_{[j]},I_{[k]},T\}=\{T,I_{[j]},I_{[k]}\}=0$ if $[j]\neq [k]$.
Abstract
In order to study the structure of arbitrary split Leibniz triple systems, we introduce the class of split Leibniz triple systems as the natural extension of the class of split Lie triple systems and split Leibniz algebras. By developing techniques of connections of roots for this kind of triple systems, we show that any of such Leibniz triple systems $T$ with a symmetric root system is of the form $T=U+\sum_{[j]\in \Lambda^{1}/\sim} I_{[j]}$ with $U$ a subspace of $T_{0}$ and any $I_{[j]}$ a well described ideal of $T$, satisfying $\{I_{[j]},T,I_{[k]}\} =\{I_{[j]},I_{[k]},T\}=\{T,I_{[j]},I_{[k]}\}=0$ if $[j]\neq [k]$.
참고문헌 (15)
- S. Albeverio, "On nilpotent and simple Leibniz algebras", Comm. Algebra, 33(1), pp.159~172 (2005)
- S. Albeverio, "Cartan subalgebras, weight spaces, and criterion of solvability of finite dimen-sional Leibniz algebras", Rev. Mat. Complut, 19(1), pp.183~195 (2006)
- S. Ayupov, "On Leibniz algebras, Algebra and operator theory", pp.1~12 (1998), Kluwer Acad. Publ
- D. W. Barnes, "On Engel’s theorem for Leibniz algebras", Comm. Algebra, 40(4), pp.1388~1389 (2012)
- D. W. Barnes, "On Levi’s theorem for Leibniz algebras", Bull. Aust. Math. Soc, 86(2), pp.184~185 (2012)
- M. Bremner, "Leibniz triple systems", Commun. Contemp. Math, 16(1), pp.19~ (2014)
- A. J. Calderon Martın, "On split Lie algebras with symmetric root systems", Proc. Indian Acad. Sci. Math. Sci, 118(3), pp.351~356 (2008)
- A. J. Calderon Martın, "On split Lie triple systems", Proc. Indian Acad. Sci. Math. Sci, 119(2), pp.165~177 (2009)
- A. J. Calderon Martın, "On simple split Lie triple systems", Algebr. Represent. Theory, 12(2-5), pp.401~415 (2009)
- A. J. Calderon Martın, "On split Leibniz algebras", Linear Algebra Appl, 436(6), pp.1651~1663 (2012)
- A. J. Calderon Martın, "On split Lie triple systems. II", Proc. Indian Acad. Sci. Math. Sci, 120(2), pp.185~198 (2010)
- Y. Cao, "On the structure of graded Leibniz triple systems", Linear Algebra Appl, 496(5), pp.496~509 (2016)
- P. Kolesnikov, "Varieties of dialgebras, and conformal algebras", Sib. Math. J, 49(2), pp.257~272 (2008)
- J.-L. Loday, "Une version non commutative des algebres de Lie: les algebres de Leibniz", (French) Enseign. Math, 39(3-4), pp.269~293 (1993)
- Y. Ma, "Some structures of Lebniz triple systems", arXiv:1407.3978, ~ (2014)