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
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