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July 2014 Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula
Li Guo, Fang Li
Asian J. Math. 18(3): 545-572 (July 2014).

Abstract

This article studies the Lie algebra $\mathrm{Der}(\mathrm{k}\Gamma)$ of derivations on the path algebra $\mathrm{k}\Gamma$ of a quiver $\Gamma$ and the Lie algebra on the first Hochschild cohomology group $HH1(\mathrm{k}\Gamma)$. We relate these Lie algebras to the algebraic and combinatorial properties of the path algebra. Characterizations of derivations on a path algebra are obtained, leading to a canonical basis of $\mathrm{Der}(\mathrm{k}\Gamma)$ and its Lie algebra properties. Special derivations are associated to the vertices, arrows and faces of a quiver, and the concepts of a connection matrix and boundary matrix are introduced to study the relations among these derivations, giving rise to an interpretation of Euler's polyhedron formula in terms of derivations. By taking dimensions, this relation among spaces of derivations recovers Euler's polyhedron formula. This relation also leads to a combinatorial construction of a canonical basis of the Lie algebra $HH1(\mathrm{k}\Gamma)$, together with a new semidirect sum decomposition of $HH1(\mathrm{k}\Gamma)$.

Citation

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Li Guo. Fang Li. "Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula." Asian J. Math. 18 (3) 545 - 572, July 2014.

Information

Published: July 2014
First available in Project Euclid: 8 September 2014

zbMATH: 1322.16005
MathSciNet: MR3257840

Subjects:
Primary: 05C25 , 05E15 , 12H05 , 16E40 , 16G20 , 16S32 , 16W25

Keywords: connection matrix , differential algebra , Euler’s polyhedron formula , graph , Hochschild cohomology , Lie algebra , path algebra , Quiver

Rights: Copyright © 2014 International Press of Boston

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Vol.18 • No. 3 • July 2014
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