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  "body": "\n## Summary\n\nA deformation of an [[associative algebra|3CTW1P4HBP]] $A$ is a family of associative products on $A$ depending on a parameter $t$.\nAt $t = 0$ it gives back the original product.\nIn the formal theory of Murray Gerstenhaber, the parameter is a formal variable and the products are power series in $t$.\nThe first-order terms are classified by the second Hochschild cohomology of $A$, and the obstructions to extending them lie in the third.\nDeformations of algebras are the algebraic core of [[deformation quantization|3MKSJ1J04K]].\n\n## Explanations\n\n### L1 human\n\nImagine changing the multiplication of an algebra a little bit, in a way controlled by a small parameter $t$.\nAt $t = 0$ one has the old algebra, and for small $t$ a nearby algebra.\nFor example, the commutative polynomial ring in $x$ and $y$ can be bent into an algebra where $yx = xy - t$, which is a Weyl algebra.\nSome algebras cannot be bent at all except in trivial ways; they are called rigid.\nThe theory asks which bendings exist and when two of them are really the same after a change of coordinates.\n",
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