I am learning about deformation theory, e.g. through The unbearable lightness of deformation theory by Szendröi. There the standard example of deformations of a structure of associative algebra, leading to the Hochschild complex, is presented. Personally, I am more interested to the case of Lie algebras.
Let $V$ be a vector space, let's say finite dimensional, and $[\cdot,\cdot]\in\hom(V^{\otimes2},V)$ a Lie bracket. The group $G=GL(V)$ acts on $V$ in the obvious way, and the structures that are isomorphic to the original one are those of the form $$[x,y]_g = g[g^{-1}x,g^{-1}y].$$ Let's look at infinitesimal deformations (i.e. we work on $k[\epsilon]/(\epsilon^2)$), so consider $$[x,y]' = [x,y] + \epsilon f(x,y)$$ with $f\in\hom(V^{\otimes2},V)$ antisymmetric. Then $[\cdot,\cdot]'$ satisfies the Jacobi rule if, and only if $$[x,f(y,z)] + f(x,[y,z]) + \text{cyclic} = 0.$$ The infinitesimal action of $G$ on $V$ is given by elements of the type $$T(x) = x + \epsilon g(x)$$ with $g\in\hom(V,V)$ (that is, in the Lie algebra of $G$). Then it can be checked that $$[x,y]'_T := T[T^{-1}x,T^{-1}y] = [x,y] + \epsilon f_T(x,y)$$ with $$f_T(x,y) = f(x,y) + g([x,y]) - [x,g(y)] - [g(x),y].$$ Therefore, the deformation complex should be a vector space containing the possible $f$ and $g$ (so I would say maybe the space of completely antisymmetric $\hom(V^{\otimes n},V)$) with a differential such that \begin{align*} (dg)(x,y) = & g([x,y]) - [x,g(y)] - [g(x),y],\\ (df)(x,y,z) = & [x,f(y,z)] + f(x,[y,z]) + \text{cyclic}. \end{align*} Currently, I have no idea how to define such a differential. Does anybody know, have an idea, or know a reference where this is done? Also, what is the good structure to put on the deformation complex to make it into a dgla?
For future reference, as it is remarked in Gerstenhaber's On the Deformation of Rings and Algebras (but not proved, as it is analogous to the associative case; thanks to @DietrichBurde for the reference, really nice paper, by the way) the correct complex is the Chevalley-Eilenberg complex of the Lie algebra $\mathfrak{g}$ (with coefficients in itself), which is defined as $$C^k(\mathfrak{g},\mathfrak{g}) = \hom(\Lambda^{k+1}\mathfrak{g},\mathfrak{g})$$ with differential $\delta$ defined by \begin{align*} (\delta f)(x_0,\ldots,x_n) = & \sum_{i}(-1)^i[x_i,f(x_0,\ldots,\widehat{x_i},\ldots,x_n)]\\ & + \sum_{i<j}(-1)^{i+j}f([x_i,x_j],x_0,\ldots,\widehat{x_i},\ldots,\widehat{x_j},\ldots,x_n). \end{align*} A possible reference for this is the original paper by Chevalley and Eilenberg (you'll have to get used to their notation, though).