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  "body": "\n## Summary\n\nA Gerstenhaber algebra is a graded commutative algebra with a Lie bracket of degree $-1$ that acts on products by the Leibniz rule.\nMurray Gerstenhaber found this structure on the Hochschild cohomology of an associative algebra.\nPolyvector fields on a manifold with the [[Schouten–Nijenhuis bracket|FJ1MDBZ680]] form another basic example.\nIt is the graded analogue of a Poisson algebra, with the bracket shifted by one degree.\n\n## Explanations\n\n### L1 human\n\nA Gerstenhaber algebra carries two operations: a product that is commutative up to signs, and a bracket like the commutator of vector fields.\nThe bracket lowers the total degree by one.\nThe two operations fit together by a product rule, just as the derivative of a product follows the Leibniz rule.\nFor polyvector fields on a manifold, the product is the wedge product and the bracket extends the Lie bracket of vector fields.\nThe same structure on Hochschild cohomology governs the deformations of an associative algebra.\n",
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