A quantum version of van Est's theorem (Q1970727)

The author introduces the notion of ``quantum Lie algebra'' as a triple \((T, \sigma, C)\), where \(T\) is a vector space, \(\sigma: T\otimes T \to T\otimes T\) is a solution of the braid equation and \(C: T\otimes T \to T\) is a linear map satisfying a number of compatibility conditions. Several examples are given. A cohomology theory for these ``quantum Lie algebras'' is developed; it extends the classical Lie algebra cohomology. If \((\Gamma, d)\) is a differential calculus in the sense of S. Woronowicz and if \(\Gamma^{\text{inv}}\) is the space of its left coinvariant elements, then \(T= (\Gamma^{\text{inv}})^*\) has a natural structure of quantum Lie algebra, provided that \(\Gamma^{\text{inv}}\) is finite dimensional. A quantum version of a theorem of van Est is established, allowing to compare the cohomology of this quantum Lie algebra with the cohomology of the quantum de Rham complex associated to \((\Gamma, d)\).