Abstract
We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controling the deformation theory of associative algebras and prove that there is a canonical solution within the class of graphs without circuits, without assuming the Jacobi identity. The proof is based on the unique factorization property of graph insertions.
Highlights
The main result is an explicit solution of the Maurer-Cartan equation in the differential graded Lie algebra of graphs which controlls the deformation theory of associative algebras (Theorem 3.1)
Denote by H = kGn,m/ ∼ the quotient modulo the equivalence relation generated by Γ′ ∼ −Γ, where Γ′ is obtained from Γ by “negation” of the left/right labeling at one internal vertex only. This completes the process of taking orientation classes of edge-labeled graphs and will be considered independently of the Jacobi identity
We proved the existence of solutions of Maurer-Cartan equation, without assuming the Jacobi identity holds, implying that the primary obstruction to have a full deformation vanishes anyway
Summary
The combinatorial problem regarding the coefficients of a star-product is captured by the “graphical calculus” we will call Kontsevich rule, a sort of a “dual Feynman rule”:. The corresponding arrows will be labeled left/right, defining the orientation class of the graph Γ up to a “negation” of the edge labeling in any two internal vertices [12]. Denote by H = kGn,m/ ∼ the quotient modulo the equivalence relation generated by Γ′ ∼ −Γ , where Γ′ is obtained from Γ by “negation” (switching) of the left/right labeling at one internal vertex only. This completes the process of taking orientation classes of edge-labeled graphs and will be considered independently of the Jacobi identity (compare [12], p.5). We are ready to prove that the “sum of all graphs” is a solution
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