Leibniz algebra
In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product [ _ , _ ] satisfying the Leibniz identity
- [math]\displaystyle{ a,b],c] = [a,[b,c+ [[a,c],b]. \, }[/math]
In other words, right multiplication by any element c is a derivation. If in addition the bracket is alternating ([a, a] = 0) then the Leibniz algebra is a Lie algebra. Indeed, in this case [a, b] = −[b, a] and the Leibniz's identity is equivalent to Jacobi's identity ([a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0). Conversely any Lie algebra is obviously a Leibniz algebra.
In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. The investigation of which theorems and properties of Lie algebras are still valid for Leibniz algebras is a recurrent theme in the literature.[1] For instance, it has been shown that Engel's theorem still holds for Leibniz algebras[2][3] and that a weaker version of Levi-Malcev theorem also holds.[4]
The tensor module, T(V) , of any vector space V can be turned into a Loday algebra such that
- [math]\displaystyle{ [a_1\otimes \cdots \otimes a_n,x]=a_1\otimes \cdots a_n\otimes x\quad \text{for }a_1,\ldots, a_n,x\in V. }[/math]
This is the free Loday algebra over V.
Leibniz algebras were discovered in 1965 by A. Bloh, who called them D-algebras. They attracted interest after Jean-Louis Loday noticed that the classical Chevalley–Eilenberg boundary map in the exterior module of a Lie algebra can be lifted to the tensor module which yields a new chain complex. In fact this complex is well-defined for any Leibniz algebra. The homology HL(L) of this chain complex is known as Leibniz homology. If L is the Lie algebra of (infinite) matrices over an associative R-algebra A then Leibniz homology of L is the tensor algebra over the Hochschild homology of A.
A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. It has defining identity:
- [math]\displaystyle{ ( a \circ b ) \circ c = a \circ (b \circ c) + a \circ (c \circ b) . }[/math]
Notes
- ↑ Barnes, Donald W. (July 2011). "Some Theorems on Leibniz Algebras". Communications in Algebra 39 (7): 2463–2472. doi:10.1080/00927872.2010.489529.
- ↑ Patsourakos, Alexandros (26 November 2007). "On Nilpotent Properties of Leibniz Algebras". Communications in Algebra 35 (12): 3828–3834. doi:10.1080/00927870701509099.
- ↑ Sh. A. Ayupov; B. A. Omirov (1998). "On Leibniz Algebras". in Khakimdjanov, Y.; Goze, M.; Ayupov, Sh.. Algebra and Operator Theory Proceedings of the Colloquium in Tashkent, 1997. Dordrecht: Springer. pp. 1–13. ISBN 9789401150729.
- ↑ Barnes, Donald W. (30 November 2011). "On Levi's theorem for Leibniz algebras". Bulletin of the Australian Mathematical Society 86 (2): 184–185. doi:10.1017/s0004972711002954.
References
- Kosmann-Schwarzbach, Yvette (1996). "From Poisson algebras to Gerstenhaber algebras". Annales de l'Institut Fourier 46 (5): 1243–1274. doi:10.5802/aif.1547.
- Loday, Jean-Louis (1993). "Une version non commutative des algèbres de Lie: les algèbres de Leibniz". Enseign. Math.. Series 2 39 (3–4): 269–293. http://irma.math.unistra.fr/~loday/PAPERS/93Loday(Leibniz).pdf.
- Loday, Jean-Louis; Teimuraz, Pirashvili (1993). "Universal enveloping algebras of Leibniz algebras and (co)homology". Mathematische Annalen 296 (1): 139–158. doi:10.1007/BF01445099.
- Bloh, A. (1965). "On a generalization of the concept of Lie algebra". Dokl. Akad. Nauk SSSR 165: 471–3.
- Bloh, A. (1967). "Cartan-Eilenberg homology theory for a generalized class of Lie algebras". Dokl. Akad. Nauk SSSR 175 (8): 824–6.
- Dzhumadil'daev, A.S.; Tulenbaev, K.M. (2005). "Nilpotency of Zinbiel algebras". J. Dyn. Control Syst. 11 (2): 195–213. doi:10.1007/s10883-005-4170-1.
- Ginzburg, V.; Kapranov, M. (1994). "Koszul duality for operads". Duke Math. J. 76: 203–273. doi:10.1215/s0012-7094-94-07608-4.
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