Given a field] , a cyclic derivation on an -algebra (example: algebra of formal noncommutative power series in -variables) is an -linear map satisfying
Given a cyclic derivation a corresponding cyclic derivative is defined by .
They are appearing in the definition of Jacobian algebra(also called Jacobi algebra) of a quiver with potential, see there.
Gian-Carlo Rota, Bruce Sagan, Paul R.Stein, A cyclic derivative in noncommutative algebra, J. Algebra 64:1 (1980) 54-75 doi MR575782
Christophe Reutenauer, Cyclic derivation of noncommutative algebraic power series, J. Alg. 85, 32-39 (1983)
Daniel Lopez-Aguayo, Cyclic derivations, species realizations and potentials, pdf
A class of identities involving multiple zeta functions is described using cyclic derivations in
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