## The Lie algebra structure of tangent cohomology and deformation theory.(English)Zbl 0576.17008

The main goal of the paper is to present Harrison cohomology in characteristic zero as the homology of a natural differential graded Lie algebra associated to an augmented commutative algebra $$A$$. It turns out that for the algebra $$A$$ Harrison and Quillen cohomologies agree for the coefficient in any $$A$$-module $$M$$. This approach elucidates the differential graded Lie structure of the cohomology and substantiates the existence of Massey-Lie brackets in the cohomology. The basic point of view is that of multiplicative resolutions of $$A$$, i.e. a differential graded commutative algebra $${\mathcal A}\to A$$ inducing $$H(\mathcal A)\approx A$$. For such resolutions, the deformations of A correspond to changes in the differential of $$\mathcal A$$.
The authors exhibit a particular resolution which makes the comparison with Harrison cohomology quite transparent if considering Lie coalgebras. Then the Harrison complex is precisely $$\operatorname{Hom}(\Gamma A,A)$$ where $$\Gamma A$$ is the free graded Lie coalgebra on the augmentation ideal of A considered to have degree 1. This construction gives a computation for the tangent cohomology as the cohomology of the Lie algebra of “outer derivations” of the construction $$\Gamma A$$.

### MSC:

 17B55 Homological methods in Lie (super)algebras 17B56 Cohomology of Lie (super)algebras 55P62 Rational homotopy theory 55S30 Massey products
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### References:

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