Left-symmetric algebroids (Q2830672)

A \textit{left-symmetric algebra} is a vector space \({\mathfrak g}\) together with a bilinear product \(\cdot_{\mathfrak g}\) such that the corresponding associator of the product, defined by \((x,y,z):=(x\cdot_{\mathfrak g} y)\cdot_{\mathfrak g} z - x\cdot_{\mathfrak g}(y\cdot_{\mathfrak g}z)\), is symmetric in the first two entries, i.e. it satisfies for all \(x,y,z\in{\mathfrak g}\) NEWLINE\[NEWLINE(x,y,z)=(y,x,z).NEWLINE\]NEWLINE Left-symmetric algebras are also called pre-Lie or Vinberg algebras in the literature, and they are special cases of Lie-admissible algebras, i.e. the commutator corresponding to the product, defined by \([x,y]_{\mathfrak g}:=x\cdot_{\mathfrak g}y-y\cdot_{\mathfrak g}x\), gives \({\mathfrak g}\) the structure of a Lie algebra.NEWLINENEWLINEA \textit{Lie algebroid} is a vector bundle \(A\to M\) on a manifold \(M\) together with a map of vector bundles \(a:A\to TM\), the \textit{anchor}, such that the space of sections \(\Gamma(A)\) is a real Lie algebra whose bracket \([,]_A\) satisfies for all \(x,y\in\Gamma(A)\) and \(f\in{\mathcal C}^{\infty}(M)\) NEWLINE\[NEWLINE[x,fy]_A=f[x,y]_A+a_A(x)(f) y.NEWLINE\]NEWLINE The main point of the article under review is to bring together these two concepts to define \textit{left-symmetric algebroids} on a manifold \(M\). The data is similar as for a Lie algebroid, but the above property becomes replaced by NEWLINE\[NEWLINEx\cdot_A(fy)=f(x\cdot_A y)+a_A(x)(f)y,NEWLINE\]NEWLINE while \(\cdot_A\) is \({\mathcal C}^{\infty}(M)\)-linear in the first variable.NEWLINENEWLINEA meaningful example of a left-symmetric algebroid is the tangent bundle \(TM\) of \(M\) together a flat torsion free connection \(\nabla\) which defines then the product, the anchor being the identity. The authors prove basic properties about left-symmetric algebroids, for example with respect to their phase spaces: For quadratic left-symmetric algebroids, these admit complex structures. They furthermore investigate representations and deformation cohomology, the main result being perhaps the property \(d^2=0\) in the deformation complex.