Chevalley-Eilenberg complex: Difference between revisions

Latest revision as of 18:18, 14 August 2013

Definition

Suppose $L$ is a Lie algebra over a base field $K$ . The Chevalley-Eilenberg complex of $L$ is a particular chain complex of $L$ -modules that can serve as a projective resolution of $K$ as a trivial $L$ -module.

It can be thought of as a kind of Lie algebra analogue of the bar resolution for groups.

The Chevalley-Eilenberg complex can be defined as follows. Denote by $U$ the universal enveloping algebra of $L$ . Denote by $⋀^{n} L$ the $n$ -fold exterior power of $L$ as a $K$ -vector space. For $n \in N_{0}$ , set $B_{n} = U \otimes ⋀^{n} L$ and define $d : B_{n} \to B_{n - 1}$ by:

$d (u \otimes g_{1} \land g_{2} \land \dots \land g_{n}) = \sum_{i = 1}^{n} (- 1)^{i + 1} u g_{i} \otimes g_{1} \land \dots \land \hat{g_{i}} \land \dots \land g_{n} + \sum_{i < j} (- 1)^{i + j} u \otimes [g_{i}, g_{j}] \land g_{1} \land \dots \land \hat{g_{i}} \land \dots \land \hat{g_{j}} \land \dots \land g_{n}$

$d^{2} = 0$ , so we get a complex. In fact, this is an exact sequence, and serves as a projective resolution of $K$ as a trivial $L$ -module.

Related notions

Bar resolution is the analogous construction for groups.

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 ==Definition==
-Suppose <math>L</math> is a [[Lie algebra]] over a base field <math>K</math>. The '''Chevalley-Eilenberg complex''' of <matH>L</math>, also called the '''standard complex''' of <math>L</math>, is a particular chain complex of <math>L</math>-modules that can serve as a projective resolution of <math>K</math> as a trivial <math>L</math>-module.
+Suppose <math>L</math> is a [[Lie algebra]] over a base field <math>K</math>. The '''Chevalley-Eilenberg complex''' of <math>L</math> is a particular chain complex of <math>L</math>-modules that can serve as a projective resolution of <math>K</math> as a trivial <math>L</math>-module.
 It can be thought of as a kind of Lie algebra analogue of the [[bar resolution]] for groups.
+The Chevalley-Eilenberg complex can be defined as follows. Denote by <math>U</math> the [[universal enveloping algebra]] of <math>L</math>. Denote by <math>\bigwedge^n L</math> the <math>n</math>-fold exterior power of <math>L</math> as a <math>K</math>-vector space. For <math>n \in \mathbb{N}_0</math>, set <math>B_n = U \otimes \bigwedge^nL</math> and define <math>d: B_n \to B_{n-1}</math> by:
+<math>d(u \otimes g_1 \wedge g_2 \wedge \dots \wedge g_n) = \sum_{i=1}^n (-1)^{i+1} ug_i \otimes g_1 \wedge \dots \wedge \hat{g_i} \wedge \dots \wedge g_n + \sum_{i < j} (-1)^{i+j} u \otimes [g_i,g_j] \wedge g_1 \wedge \dots \wedge \hat{g_i} \wedge \dots \wedge \hat{g_j} \wedge \dots \wedge g_n</math>
+<math>d^2 = 0</math>, so we get a complex. In fact, this is an exact sequence, and serves as a projective resolution of <math>K</math> as a trivial <math>L</math>-module.
+==Related notions==
+* [[Bar resolution]] is the analogous construction for groups.