On the Koszul map of Lie algebras (Q901399)

Let \({\mathfrak g}\) be a Lie algebra. The Lie algebras considered in this article are vector spaces over a field of characteristic zero or sometimes even modules over an arbitrary commutative ring \(R\). The author introduces the \textit{Killing module} \(\text{Kill}({\mathfrak g})\) as the cokernel of the map \({\mathfrak g}^{\otimes 3}\to S^2{\mathfrak g}\) given by \[ x\otimes y\otimes z\mapsto [x,y]\circledcirc z- x\circledcirc[y,z]. \] He further introduces the \textit{raw Koszul map} \[ \check{\eta}:{\mathfrak g}^{\otimes 3}\to S^2{\mathfrak g} \] with \(\check{\eta}(x\otimes y\otimes z)=[x,y]\circledcirc z\). The raw Koszul map induces the \textit{Koszul map} \(\eta:{\mathfrak g}^{\otimes 3}\to\text{Kill} ({\mathfrak g})\) and the \textit{reduced Koszul map} \(\bar{\eta}:H_3({\mathfrak g})\to\text{Kill}({\mathfrak g})\). These are the main objects of study in the present article. There are two types of results. On the one hand, the author shows vanishing theorems for the reduced Koszul map. On the other hand, the author constructs examples of nilpotent and solvable Lie algebras \({\mathfrak g}\) for which the reduced Koszul map is non-zero. Going into details, the main tools for the first kind of results are gradings on the Lie algebra \({\mathfrak g}\) on the one hand, and the filtration on \(\text{Kill}({\mathfrak g})\) which is induced by the \textit{lower central series filtration} of \({\mathfrak g}\) on the other hand, i.e. \[ {\mathfrak g}^{(1)}\,=\,{\mathfrak g},\quad\text{and}\quad {\mathfrak g}^{(i+1)}\,=\,[{\mathfrak g},{\mathfrak g}^{(i)}]. \] The induced filtration is then given by defining \(\text{Kill}^{(i)}({\mathfrak g})\) to be the image of \({\mathfrak g}\otimes {\mathfrak g}^{(i-1)}\) in \(\text{Kill}({\mathfrak g})\) for \(i\geq 2\). Theorem 3.1 reads Let \({\mathfrak g}\) be a Lie algebra over \(R\). Suppose that \(\text{Tor}_1^{R}([{\mathfrak g},{\mathfrak g}],{\mathfrak g}\,/\, [{\mathfrak g},{\mathfrak g}])=0\). Then the image of \(3\check{\eta}\) is contained in \(\text{Kill}^{(4)}({\mathfrak g})\). In particular, if 3 is invertible in \(R\), then the image of the reduced Koszul map \(\bar{\eta}\) is contained in \(\text{Kill}^{(4)}({\mathfrak g})\). Section 3 discusses further the existence of 2-nilpotent Lie algebras over \(R\) whose reduced Koszul map is zero for rings \(R\) containing a non-zero nilpotent element and such that 2 is invertible, and gives a counter-example in characteristic 3 of rank 7. Section 4 studies the vanishing of the Koszul homomorphism for graded Lie algebras. Theorem 4.1 reads Assume that 6 is invertible in the ground ring \(R\). Let \({\mathfrak g}\) be graded in an abelian group \(A\) which embeds into \(R\). Then the Koszul homomorphism of \({\mathfrak g}\) is concentrated in degree 0. Sections 5, 6 and 7 are concerned with the construction of examples of nilpotent and solvable Lie algebras with zero or non-zero reduced Koszul map. Section 5 culminates in Theorem 5.1: For every nilpotent Lie algebra \({\mathfrak g}\) over a field of characteristic zero of dimension \(\leq 9\), the reduced Koszul map vanishes. For the proof of this theorem, the author extends the classification of real \textit{quadrable} (i.e. admitting an invariant non-degenerate symmetric bilinear form) Lie algebras of dimension \(\leq 9\) of \textit{I. Kath} [J. Lie Theory 17, No. 1, 41--61 (2007; Zbl 1135.17004)] to complex quadrable Lie algebras. By Lemma 2.6, for the property of having a non-zero reduced Koszul map, one can always assume the Lie algebra to be quadrable. Section 6 constructs a 12-dimensional example of a nilpotent Lie algebra with non-zero reduced Koszul map. Section 7 constructs a 9-dimensional example of a solvable Lie algebra with non-zero reduced Koszul map. This example is minimal as all solvable Lie algebras of dimension \(\leq 8\) over a field of characteristic zero have zero reduced Koszul map (Theorem 7.2). The paper closes with two appendices reinterpreting Neeb-Wagemann's results [\textit{K.-H. Neeb} and \textit{F. Wagemann}, Can. J. Math. 60, No. 4, 892--922 (2008; Zbl 1162.17019)].