Tensor products of Poisson algebras
The author considers cohomologies, deformations, and homotopies of tensor products of Poisson algebras with applications to homogeneous (in particular quadratic) brackets. Let \((P_i, \mu_i, [ ,\;]_i)\), \(i=1,2\) be Poisson algebras, \(M_i\) modules over \(P_i\), and \(D^\bullet (P_i,M_i)\) the associated complexes. The author constructs a morphism of complexes \[ \iota: D^\bullet (P_1, M_1) \otimes D^\bullet (P_2, M_2)\to D^\bullet (P_1 \otimes P_2, M_1 \otimes M_2). \] He then associates to each bi-derivation \(\theta: P_1\otimes P_2\to P_1 \otimes P_2\) (subject to a symmetry condition) a new Poisson algebra \((P_1 \otimes P_2, \mu_\otimes, [ ,\;]_\theta)\) characterized by the requirements that \(P_1 \otimes 1\), \(1\otimes P_2\) be subalgebras and \([x \otimes 1, 1\otimes y]_\theta= \theta (x\otimes y)\). As an application, exact Poisson deformations of \(P_1\otimes P_2\) are constructed. Formulae for homotopies of \(P_1\otimes P_2\) by means of central elements (with applications to \(P_1\) homogeneous and \(P_2\) trivial) are given. The author then combines homotopies and the above \(\theta\)- construction to construct yet other classes of Poisson brackets.
scientific article; zbMATH DE number 687512