## What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by $$d(a_1\wedge\cdots \wedge a_k)=\sum_{i,j}(-1)^{i+j-1}[a_i,a_j] a_1\wedge \cdots\wedge\hat{a_i}\wedge\cdots\wedge\hat{a}_j\wedge\cdots\wedge a_k.$$ The differential $d$ is not a derivation with respect to the exterior product $\wedge$, but the deviation from being a derivation is a binary operation which defines a graded Lie algebra structure on $\wedge^* \mathfrak g$: If $\underline{a},\underline{b}\in\wedge^*\mathfrak g$, let $$[\underline{a},\underline{b}]_{s}=d(\underline{a}\wedge\underline{b})-d\underline{a}\wedge b+\underline{a}\wedge d\underline{b}$$ (I'm omitting some signs.) This bracket operation vanishes once you take homology, since if $d\underline{a}=d\underline{b}=0$ then it is obvious that $d(\underline{a}\wedge\underline{b})=[\underline{a},\underline{b}]_s$. However, I was talking to Jim Stasheff several years ago, and he mentioned that the Schouten bracket doesn't necessarily vanish on Lie algebra homology if there are coefficients in a nontrivial $\mathfrak g$ module, $M$. However, I don't know what the definition of the Schouten bracket is in this case. The Chevalley-Eilenberg complex is easy enough to understand: $\wedge^*\mathfrak g\otimes M$, where the differential includes terms where the $a_i$ act on $M$, but the obvious generalization of the above construction fails since two elements of $M$ somehow need to get combined into one element. So my basic question is how you define a Schouten bracket on the Chevalley-Eilenberg complex with coefficients in a nontrivial $\mathfrak g$-module?

-
The analog thing for associative algebras, which is the Gerstenhaber bracket, is not defined for not-regular values (which is the analogue of trivial values) – Mariano Suárez-Alvarez Oct 4 2010 at 20:06
By "values" I mean "coefficients"... Sometime ago I decided that homology takes coefficients in the module, and cohomology takes values in the module :) – Mariano Suárez-Alvarez Oct 4 2010 at 20:06

1
A truly terrible way to get at this bracket is as follows. If $\mathfrak g$ acts on $M$, then it also acts on the dual space $M^*$, which you should think of as a geometric space, and so there is a map $\mathfrak g \to \Gamma(\rm TM^*)$ (sections of tangent bundle). The Schouten bracket on $\wedge^\bullet\mathfrak g\otimes M$ is the pullback of said bracket on $\wedge^\bullet\Gamma(\rm TM^*)$ to $\mathfrak g$, and restricted to those sections that are linear in the base $M^*$. As I said, this is a terrible way to get at this bracket. – Theo Johnson-Freyd Oct 5 2010 at 2:46

## 1 Answer

Theo Johnson-Freyd:

A truly terrible way to get at this bracket is as follows. If $g$ acts on $M$, then it also acts on the dual space `$M^*$`, which you should think of as a geometric space, and so there is a map `$g\to\Gamma(TM^*)$` (sections of tangent bundle). The Schouten bracket on `$\wedge^*  g\otimes M$` is the pullback of said bracket on `$\wedge^*\Gamma(TM^*)$` to $g$ , and restricted to those sections that are linear in the base $M^*$ . As I said, this is a terrible way to get at this bracket.

(JC: I'm trying to clear the unanswered question backlog.)

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 28, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9303131103515625, "perplexity_flag": "head"}