Generalized omni-Lie algebras

Abstract

We introduce the notion of generalized omni-Lie algebras from omni-Lie algebras constructed by Weinstein. We prove that there is a one-to-one correspondence between Dirac structures of a generalized omni-Lie algebra and Lie structures on its linear space.

Keywords

• omni-Lie algebras,generalized omni-Lie algebras,Dirac structures

References

1. [1] Z. Chen and Z. Liu, Omni-Lie algeboids, J. Geom. Phys. 60 (5), 799–808, 2010.
2. [2] Z. Chen, Z. Liu and Y. Sheng, Dirac structures of omni-Lie algeboids, Int. J. Math. 22 (8), 1163–1185, 2008.
3. [3] Z. Liu, Some remarks on Dirac structures and Possion reductions, Poisson Geometry Banach Center Publ. 51, 165–173, 2000.
4. [4] J. Loday, Une version non commutative des alg`ebres de Lie: les alg`ebres de Leibniz, Enseign. Math. 39 (2), 269–293, 1993.
5. [5] D. Roytenberg and A. Weinstein, Courant algeboids and strongly homotopy Lie algebras, Lett. Math. Phys. 46 (1), 81–93, 1998.
6. [6] Y. Sheng, Z. Liu and C. Zhu, Omni-Lie 2-algebras and their Dirac structures, J. Geom. Phys. 61 (2), 560–575, 2010.
7. [7] A. Weinstein, Omni-Lie algebras, RIMS Kokyuroku, 1176, 95–102, 2000.
8. [8] T. Zhang and Z. Liu, Omni-Lie superalgebras and Lie 2-superalgebras, Front Math. China, 9 (5), 1195–1210, 2014.