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David White
David White
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Note: the paper https://arxiv.org/abs/hep-th/9411113 by Braverman and Gaitsgory was already mentionnedmentioned in the comments by Vladimir Dotsenko and DamienC but I wanted to write down a sketch of the argument (at least for my own understanding).

Note: the paper https://arxiv.org/abs/hep-th/9411113 by Braverman and Gaitsgory was already mentionned in the comments by Vladimir Dotsenko and DamienC but I wanted to write down a sketch of the argument (at least for my own understanding).

Note: the paper https://arxiv.org/abs/hep-th/9411113 by Braverman and Gaitsgory was already mentioned in the comments by Vladimir Dotsenko and DamienC but I wanted to write down a sketch of the argument (at least for my own understanding).

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user25309
user25309
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The obstruction to lift such flat deformation to the second order (over $k[t]/(t^3)$) lives in the degree $-2$ part of $HH^3(S(\mathfrak{g}), S(\mathfrak{g}))=\wedge^3 \mathfrak{g}^* \otimes S(\mathfrak{g})$, i.e. in $\wedge^3 \mathfrak{g}^* \otimes \mathfrak{g}$ and the vanishing of this obstruction is exactly the Jacobi identity for $[-,-]$. The set of possible lifts is then parametrized by the degree $-2$ part of $HH^2$, i.e. $\wedge^2 \mathfrak{g}^*$. Choosing the trivial lift (a non-trivial lift would correspond to a further deformation $x \otimes y -y \otimes x - t [x,y] -t^2 \phi(x,y)$ for some $\phi \in \wedge^2 \mathfrak{g}^*$), the obstruction to lift to the third order, living in the degree $-3$ part of $HH^3$, i.e. $\wedge^3 \mathfrak{g}^*$, automatically vanishvanishes.

The obstruction to lift such flat deformation to the second order (over $k[t]/(t^3)$) lives in the degree $-2$ part of $HH^3(S(\mathfrak{g}), S(\mathfrak{g}))=\wedge^3 \mathfrak{g}^* \otimes S(\mathfrak{g})$, i.e. in $\wedge^3 \mathfrak{g}^* \otimes \mathfrak{g}$ and the vanishing of this obstruction is exactly the Jacobi identity for $[-,-]$. The set of possible lifts is then parametrized by the degree $-2$ part of $HH^2$, i.e. $\wedge^2 \mathfrak{g}^*$. Choosing the trivial lift (a non-trivial lift would correspond to a further deformation $x \otimes y -y \otimes x - t [x,y] -t^2 \phi(x,y)$ for some $\phi \in \wedge^2 \mathfrak{g}^*$), the obstruction to lift to the third order, living in the degree $-3$ part of $HH^3$, i.e. $\wedge^3 \mathfrak{g}^*$, automatically vanish.

The obstruction to lift such flat deformation to the second order (over $k[t]/(t^3)$) lives in the degree $-2$ part of $HH^3(S(\mathfrak{g}), S(\mathfrak{g}))=\wedge^3 \mathfrak{g}^* \otimes S(\mathfrak{g})$, i.e. in $\wedge^3 \mathfrak{g}^* \otimes \mathfrak{g}$ and the vanishing of this obstruction is exactly the Jacobi identity for $[-,-]$. The set of possible lifts is then parametrized by the degree $-2$ part of $HH^2$, i.e. $\wedge^2 \mathfrak{g}^*$. Choosing the trivial lift (a non-trivial lift would correspond to a further deformation $x \otimes y -y \otimes x - t [x,y] -t^2 \phi(x,y)$ for some $\phi \in \wedge^2 \mathfrak{g}^*$), the obstruction to lift to the third order, living in the degree $-3$ part of $HH^3$, i.e. $\wedge^3 \mathfrak{g}^*$, automatically vanishes.

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user25309
user25309
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Note: the paper https://arxiv.org/abs/hep-th/9411113 by Braverman and Gaitsgory was already mentionned in the comments by Vladimir Dotsenko and DamienC but I wanted to write down a sketch of the argument (at least for my own understanding).

Let $\mathfrak{g}$ be a Lie algebra over a field $k$, let $S(\mathfrak{g})$ be the symmetric algebra (it is a graded commutative associative algebra) and let $U(\mathfrak{g})$ be the universal enveloping algebra (it is a filtered associative algebra). There is a natural surjective morphism of algebras $S(\mathfrak{g}) \rightarrow \text{gr} U(\mathfrak{g})$ and PBW is the statement that it is an isomorphism.

Consider $A$ the quotient of the tensor algebra $T(\mathfrak{g}[t])$ by the 2-sided ideal generated by the elements of the form $x \otimes y -y \otimes x - t[x,y]$. Considering $t$ as a variable of degree 1, $A$ is a graded $k[t]$-algebra, which specializes to $S(\mathfrak{g})$ for $t=0$ and $U(\mathfrak{g})$ for $t=1$. The introduction of the parameter $t$ makes precise the idea that $U(\mathfrak{g})$ is a deformation of $S(\mathfrak{g})$ by $[-,-]$, and PBW is equivalent to the fact that $A$ is free as $k[t]$-module, i.e. that there is no jump in the dimensions of the graded pieces when going from $t \neq 0$ to $t=0$.

From this point of view, it is clear that the natural setting for PBW is the deformation theory of $S(\mathfrak{g})$ as a graded associative algebra, and the key point is that such deformation problem has a cohomological description. First order (i.e. over $k[t]/(t^2)$) flat deformations of $S(\mathfrak{g})$ as associative algebra are parametrized by the Hochschild cohomology group $HH^2(S(\mathfrak{g}), S(\mathfrak{g}))=\wedge^2 \mathfrak{g}^* \otimes S(\mathfrak{g})$ (computing this $HH^\bullet$ is easy: it is equivalent to writing down a resolution of the diagonal of the affine space $\mathfrak{g}^*$, i.e. a Koszul resolution). For graded first order flat deformations, with the deformation parameter $t$ of degree $1$, we want the degree $-1$ part of $HH^2$, i.e. $\wedge^2 \mathfrak{g}^* \otimes \mathfrak{g}$, i.e. some $[-,-] \colon \wedge^2 \mathfrak{g} \rightarrow \mathfrak{g}$.

The obstruction to lift such flat deformation to the second order (over $k[t]/(t^3)$) lives in the degree $-2$ part of $HH^3(S(\mathfrak{g}), S(\mathfrak{g}))=\wedge^3 \mathfrak{g}^* \otimes S(\mathfrak{g})$, i.e. in $\wedge^3 \mathfrak{g}^* \otimes \mathfrak{g}$ and the vanishing of this obstruction is exactly the Jacobi identity for $[-,-]$. The set of possible lifts is then parametrized by the degree $-2$ part of $HH^2$, i.e. $\wedge^2 \mathfrak{g}^*$. Choosing the trivial lift (a non-trivial lift would correspond to a further deformation $x \otimes y -y \otimes x - t [x,y] -t^2 \phi(x,y)$ for some $\phi \in \wedge^2 \mathfrak{g}^*$), the obstruction to lift to the third order, living in the degree $-3$ part of $HH^3$, i.e. $\wedge^3 \mathfrak{g}^*$, automatically vanish.

Now the key point is that the degree $i$ part of $HH^2$ vanishes for $i<-2$ and the degree $i$ part of $HH^3$ vanishes for $i<-3$ so our flat deformation automatically lifts to all orders in a unique way. We get some graded $k[t]$-algebra $B$, locally free as $k[t]$-module, which specializes to $S(\mathfrak{g})$ at $t=0$. Because the relations defining $U(\mathfrak{g})$ are satisfied in the specialization $B_1$ of $B$ at $t=1$, we have a natural surjective map $U(\mathfrak{g}) \rightarrow B_1$ of filtered associative algebras. So, we have obtained natural surjective maps $S(\mathfrak{g}) \rightarrow \text{gr} U(\mathfrak{g}) \rightarrow \text{gr} B_1=S(\mathfrak{g})$ whose composition is easily seen to be the identity, and so are isomorphisms, proving PBW.

This proof is not simple in the sense of elementary but is simple in the sense of conceptual. The thing to realize is that it is a deformation question and then one can apply the cohomological machinery without thinking. In particular, there is no computational trick and it is clear from this proof that the Jacobi identity is exactly the right hypothesis for a result such as PBW because it is the vanishing of some cohomological obstruction. The same approach gives PBW for Lie superalgebras, Weyl algebras or Clifford algebras, and again the cohomological machinery will tell us directly what to do, without having to invent a new trick for each special case.