Faithful representations of Leibniz algebras. (English) Zbl 1280.17003
The paper is devoted to Leibniz algebras which present a non anti-symmetric generalization of Lie algebras. Equivalently, a (left) Leibniz algebra can be defined as a linear algebra \(L\) whose left multiplication operators \(d_a:L\rightarrow L\), defined by \(d_a(x)=ax\) for all \(a,x\in L\), are derivations.
A module for the Leibniz algebra \(L\) is a vector space \(V\) with two bilinear compositions \(xv\), \(vx\) for \(x\in L\) and \(v\in V\) such that \[ x(yv)=(xy)v+y(xv),\quad x(vy)=(xv)y+v(xy),\quad v(xy)=(vx)y+x(vy) \] for all \(x, y\in L\) and \(v\in V\). The main result of the present paper asserts that if \(L\) is an \(n\)-dimensional Leibniz algebra then there exists a faithful \(L\)-module of dimension less than or equal to \(n+1\) with some additional properties.
A module for the Leibniz algebra \(L\) is a vector space \(V\) with two bilinear compositions \(xv\), \(vx\) for \(x\in L\) and \(v\in V\) such that \[ x(yv)=(xy)v+y(xv),\quad x(vy)=(xv)y+v(xy),\quad v(xy)=(vx)y+x(vy) \] for all \(x, y\in L\) and \(v\in V\). The main result of the present paper asserts that if \(L\) is an \(n\)-dimensional Leibniz algebra then there exists a faithful \(L\)-module of dimension less than or equal to \(n+1\) with some additional properties.
Reviewer: Sh. A. Ayupov (Tashkent)
MSC:
| 17A32 | Leibniz algebras |
References:
| [1] | Sh. A. Ayupov and B. A. Omirov, On Leibniz algebras, Algebra and operator theory (Tashkent, 1997) Kluwer Acad. Publ., Dordrecht, 1998, pp. 1 – 12. · Zbl 0928.17001 |
| [2] | Donald W. Barnes, Ado-Iwasawa extras, J. Aust. Math. Soc. 78 (2005), no. 3, 407 – 421. · Zbl 1075.17006 · doi:10.1017/S1446788700008600 |
| [3] | D. W. Barnes, Schunck classes of soluble Leibniz algebras, arXiv:1101.3046 (2011). To appear in Comm. Algebra. · Zbl 1307.17001 |
| [4] | Donald W. Barnes and Humphrey M. Gastineau-Hills, On the theory of soluble Lie algebras, Math. Z. 106 (1968), 343 – 354. · Zbl 0164.03701 · doi:10.1007/BF01115083 |
| [5] | G. Hochschild, An addition to Ado’s theorem, Proc. Amer. Math. Soc. 17 (1966), 531 – 533. · Zbl 0143.05205 |
| [6] | Kenkichi Iwasawa, On the representation of Lie algebras, Jap. J. Math. 19 (1948), 405 – 426. · Zbl 0038.01301 |
| [7] | Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. · Zbl 0121.27504 |
| [8] | Jean-Louis Loday and Teimuraz Pirashvili, Leibniz representations of Lie algebras, J. Algebra 181 (1996), no. 2, 414 – 425. · Zbl 0855.17018 · doi:10.1006/jabr.1996.0127 |
| [9] | Alexandros Patsourakos, On nilpotent properties of Leibniz algebras, Comm. Algebra 35 (2007), no. 12, 3828 – 3834. · Zbl 1130.17002 · doi:10.1080/00927870701509099 |
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