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

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Summary

The pre-Lie algebra of graphs

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

The canonical solution
Conclusions and further developments
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