nLab cyclic derivation

Contents

Idea

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

D(f 1f 2)(f)=D(f 1)(f 2f)+D(f 2)(ff 1) D(f_1 f_2)(f) = D(f_1)(f_2 f) + D(f_2)(f f_1)

Given a cyclic derivation DD a corresponding cyclic derivative δ:AA\delta:A\to A is defined by δf=(Df)(1)\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.

Literature

  • 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

Last revised on September 2, 2024 at 11:24:33. See the history of this page for a list of all contributions to it.