nLab cyclic derivation

Given a field $F$, a cyclic derivation on an $F$-algebra $A$ (example: algebra of formal noncommutative power series in $n$-variables) is an $F$-linear map $D:A\to End_F(A)$ satisfying

$$D(f_1 f_2)(f) = D(f_1)(f_2 f)+D(f_2)(f f_1)$$

Given a cyclic derivation $D$ a corresponding cyclic derivative $\delta:A\to A$ is defined by $\delta f = (D f)(1)$.

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

• Michael E.Hoffman, Yasuo Ohno, Relations of multiple zeta values and their algebraic expression, J. Algebra 262:2 (2003) 332-347 doi

category: algebra