I don't get a part of Bernstein's / Deligne-Morgan's proof of Poincaré-Birkhoff-Witt

Question

Question: I am talking about the proof given on pages 50-52 of Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten (editors), Quantum Fields and Strings: A Course for Mathematicians, Volume 1, AMS 1999 (on google books and in the usual internet sources).

The problem is easily described: In the middle of page 52, the authors say "and (1.3.7.7) gives that [...]". But I don't see how (1.3.7.7) gives the equation that follows.

Sidenotes: The proof was rather readable and well-written up to that point, so I assume the blindness is on my side. If anyone wishes to read the proof (or reprint the book ;) ), here are a few minor mistakes to watch out for:

• On page 51, $\left[xy\right]$ should be $\left[x,y\right]$ in "while the second term $\frac12\left[xy\right]$ is antisymmetric".

• On page 51, in the definition of the map $\left\lbrace x_1,...,x_{n+1}\right\rbrace$, all three terms on the right hand side should end with $x_{n+1}$ rather than $x_n$.

• On page 52, in the first formula of this page, the commutators $\left[x\left[y,z\right]\right]$ and $\left[z\left[x,y\right]\right]$ should be $\left[x,\left[y,z\right]\right]$ and $\left[z,\left[x,y\right]\right]$ instead.

• On page 52, in the middle of this page, "and the $\left\lbrace x_1,...,x_n\right\rbrace$ vanish" should probably be "and the $\left\lbrace x_1,...,x_{n+1}\right\rbrace$ vanish".

• On page 52, in the middle of this page, "1.3.7.4" should be "(1.3.7.4)".

Answer 1

Here is an explanation Pavel Etingof has given to me in email. Thanks Pavel!

Every $\sigma\in S_{n}$ satisfies

$\sum\limits_{i}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast\left[ x,y_{\sigma i}\right] \ast\cdots\ast y_{\sigma n}$

$=n\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}$

$+\sum\limits_{i>1}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast\left[ x,y_{\sigma i}\right] \ast\cdots\ast y_{\sigma n}$.

But since

$\sum\limits_{i>1}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast\underbrace{\displaystyle \left[ x,y_{\sigma i}\right] }_{\displaystyle\substack{\displaystyle =x\ast y_{\sigma i}-y_{\sigma i}\ast x\\\displaystyle \text{(since the inclusion of }\mathfrak{L}\\\displaystyle \text{into }\operatorname*{Sym} \nolimits^{\ast}\mathfrak{L}\text{ is a morphism}\\\displaystyle \text{of Lie algebras)} }}\ast\cdots\ast y_{\sigma n}$

$=\sum\limits_{i>1}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast\underbrace{\displaystyle \left( x\ast y_{\sigma i}-y_{\sigma i}\ast x\right) \ast\cdots\ast y_{\sigma n} }_{\substack{\displaystyle =x\ast y_{\sigma i}\ast\cdots\ast y_{\sigma n}-y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}\\\displaystyle \text{(by the induction hypothesis, since }i>1\text{)}}}$

$=\sum\limits_{i>1}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast x\ast y_{\sigma i}\ast\cdots\ast y_{\sigma n}$

$-\sum\limits_{i>1}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$

$= \sum\limits_{i>0}\left( n-\left( i+1\right) +1\right) \underbrace{\displaystyle y_{\sigma1}\ast\cdots\ast x\ast y_{\sigma \left(i+1\right)}\ast\cdots\ast y_{\sigma n}}_{\displaystyle =y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}}$

$-\sum\limits_{i>1}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$ (here we substituted $i+1$ for $i$ in the first sum)

$=\sum\limits_{i>0}\left( n-\left( i+1\right) +1\right) y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$

$-\sum\limits_{i>1}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$

$=\left( n-1\right) y_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}-\sum\limits _{i>1}y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$,

this becomes

$\sum\limits_{i}\left( n-i+1\right) y_{\sigma1}\ast\cdots\ast\left[ x,y_{\sigma i}\right] \ast\cdots\ast y_{\sigma n}$

$=n\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}+\left( n-1\right) y_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}$

$-\sum\limits_{i>1}y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$

$=n\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}+ny_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}$

$-y_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}-\sum\limits_{i>1}y_{\sigma1}\ast \cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$

$=n\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}+ny_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}$

$-\sum\limits_{i>0}y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$.

Thus, (1.3.7.7) rewrites as

$\dfrac{1}{n!}x\ast\sum\limits_{\sigma}y_{\sigma1}\ast\cdots\ast y_{\sigma n}=\left( \text{symmetrized product of }x,y_{1},...,y_{n}\right) $

$+\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}n\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}+\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}ny_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}$

$-\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}\sum\limits_{i>0}y_{\sigma1}\ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}$.

Since

$\left( \text{symmetrized product of }x,y_{1},...,y_{n}\right) $

$=\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}\left( \sum\limits_{i>0}y_{\sigma1} \ast\cdots\ast y_{\sigma i}\ast x\ast\cdots\ast y_{\sigma n}+x\ast y_{\sigma 1}\ast\cdots\ast y_{\sigma n}\right) $,

this simplifies to

$\dfrac{1}{n!}x\ast\sum\limits_{\sigma}y_{\sigma1}\ast\cdots\ast y_{\sigma n} =\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}x\ast y_{\sigma1}\ast\cdots\ast y_{\sigma n}$

$+\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}n\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}+\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}ny_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}$.

Thus

$\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}n\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}+\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}ny_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}$

$=\dfrac{1}{n!}x\ast\sum\limits_{\sigma}y_{\sigma1}\ast\cdots\ast y_{\sigma n} -\dfrac{1}{\left( n+1\right) !}\sum\limits_{\sigma}x\ast y_{\sigma1}\ast\cdots\ast y_{\sigma n}$

$=\underbrace{\displaystyle \left( \dfrac{1}{n!}-\dfrac{1}{\left( n+1\right) !}\right) }_{\displaystyle =\dfrac{n}{\left( n+1\right) !}}\sum\limits_{\sigma}x\ast y_{\sigma1}\ast \cdots\ast y_{\sigma n}=\dfrac{n}{\left( n+1\right) !}\sum\limits_{\sigma}x\ast y_{\sigma1}\ast\cdots\ast y_{\sigma n}$.

Divide this by $\dfrac{n}{\left( n+1\right) !}$ to obtain

$\sum\limits_{\sigma}\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}+\sum\limits_{\sigma}y_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}$

$=\sum\limits_{\sigma}x\ast y_{\sigma1}\ast\cdots\ast y_{\sigma n}$.

In other words,

$0=\sum\limits_{\sigma}\left( x\ast y_{\sigma1}\ast\cdots\ast y_{\sigma n} -y_{\sigma1}\ast x\ast\cdots\ast y_{\sigma n}-\left[ x,y_{\sigma1}\right] \ast y_{\sigma2}\ast\cdots\ast y_{\sigma n}\right) $

$=\sum\limits_{\sigma}\left\lbrace x,y_{\sigma1},...,y_{\sigma n}\right\rbrace =\left( n-1\right) !\sum\limits_{i}\left\lbrace x,y_{i},y_{1},...,\widehat{y_{i}},...,y_{n} \right\rbrace $

(here we used that $\left\lbrace x_{1},...,x_{n+1}\right\rbrace $ is symmetric in the last $n-1$ variables, so that every $\sigma\in S_{n}$ satisfies $\left\lbrace x,y_{\sigma1},...,y_{\sigma n}\right\rbrace =\left\lbrace x,y_{i},y_{1} ,...,\widehat{y_{i}},...,y_{n}\right\rbrace $ for $i=\sigma1$).

Thus, $\sum\limits_{i}\left\lbrace x,y_{i},y_{1},...,\widehat{y_{i}},...,y_{n}\right\rbrace =0$, qed.