The Laplacian of a Lie quasi-algebra (Q2516273)

Lie quasi-bialgebras were introduced by Drinfeld as generalizations of Lie bialgebras [\textit{V. G. Drinfel'd}, Leningr. Math. J. 1, No. 6, 1419--1457 (1990; Zbl 0718.16033); translation from Algebra Anal. 1, No. 6, 114--148 (1989)], and were studied by \textit{M. Bangoura} [Quasi-bigèbres jacobiennes et généralisations des groupes de Lie-Poisson. Thesis, Université de Lille (1995)]. They are quadruples \((G,m,c,f)\), \(G\) a Lie algebra with bracket \(m\), \(c\) a cobracket of \(G\) which is a 1-cocycle of \(G\) with values in the the second exterior product \(E^2\) of \(G\) under the adjoint action, and \(f\) is in \(E^3\) satisfying some alternating conditions. \(c\) does not satisfy the coJacobi identity, but if \(f=0\), then \((G,m,c)\) is a Lie bialgebra. The authors of the paper under review work over a field \(K\) either the real or complex numbers with \(G\) finite-dimensional over \(K\). A Drinfeld double \(D\) of \(G\) was defined by \textit{M. Bangoura} and \textit{Y. Kosmann-Schwarzbach} [Lett. Math. Phys. 28, No. 1, 13--29 (1993; Zbl 0796.17008)]. \(D\) is a coboundary Lie quasi-bialgebra, i.e., its cobracket comes from an element \(r\) in its \(E^2\). The authors first define a Laplacian operator for a trivial \(D\)-module \(M\), i.e., \(M=K\). It is a degree zero derivation of \(E^2\). Then they define a Laplacian operator \(L_M\) for an arbitrary \(D\)-module \(M\). \(L_M\) is a degree zero derivation of the tensor product of \(E^2\) and \(M\). An explicit formula for \(L_M\) is given by means of adjoint characters of \(G\) and \(G^*\). A parametrized Laplacian operator is also defined. Finally, in an appendix, the authors recall the Laplacian operator of a Lie bialgebra defined and studied by \textit{J. H. Lu} [``Lie bialgebras and Lie algebra cohomology'', preprint (1996)] and develop its properties.