nLab necklace Lie algebra

A necklace in a quiver is a closed oriented path in the quiver up to cyclic permutation of the arrows making up the cycle. The space of necklaces in the path algebra of a quiver has a natural Lie algebra structure, which is a noncommutative analogue of a Poisson bracket on the sympletic reduction. In the case of one vertex and n loops the path algebra is the free associative algebra on n generators. In that case the necklace Lie algebra is introduced in

• M. Kontsevich, Formal noncommutative sympletic geometry, The Gelfand mathematical seminars, 1990–1992, Springer 1993, 173–187 pdf

• related notions: quiver with potential, formal noncommutative symplectic geometry

• R. Bocklandt, L. Le Bruyn, Necklace Lie algebras and noncommutative symplectic geometry, Math Z 240, 141–167 (2002) doi

• Lieven Le Bruyn, http://www.neverendingbooks.org/quiver-superpotentials

• Wee Liang Gan, Travis Schedler, The necklace Lie coalgebra and renormalization algebras, J. Noncommut. Geom. 2 (2008) no. 2, 195–214 doi

Quantization of the necklace Lie bialgebra

• Victor Ginzburg, Travis Schedler, Moyal quantization of necklace Lie algebras, arXiv:0503405
• T. Schedler, A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver, Int. Math. Res. Not. 2005 (12) (2005) 725–760

category: algebra