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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)".

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

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".