lecture 1

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< An introduction to Lie algebra cohomology | saras

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Author: Dr. Jan A. Sanders, Vrije Universiteit Amsterdam
Author: Dr. Sara Lombardo, Vrije Universiteit Amsterdam

1 Introduction
- 1.1 Morphism (Group homomorphism)
- 1.2 Linear forms
2 Representations
- 2.1 Example of a representation
3 The coboundary operator
- 3.1 Exercise

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Introduction

Let $G$ be a group with identity $e$ .

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Morphism (Group homomorphism)

Let $G$ , $H$ be two groups; Let $\phi:G\rightarrow H$ be a map. If $\phi(g_1\cdot g_2)=\phi(g_1)\circ\phi(g_2)$ , where $\cdot$ is the operation on G and $\circ$ is the operation on H, then $\phi$ is a group homorphism.

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Linear forms

Let $A$ be a module. The space of $n$ -linear forms, with arguments in $G$ and values in $A$ , is denoted by $C^n(G,A)$ .

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Representations

Let $A$ be a module or a vector space; $d^{(0)}:G\rightarrow End(A)$ is a representation of $G$ in $A$ if

$d^{(0)}(g_1 \cdot g_2)=d^{(0)}(g_1),d^{(0)}(g_2),\quad g_1, g_2\in G$ .

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Example of a representation

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The coboundary operator

We now define the coboundary operators $d^n$ : Let $a\in A=C^0(G,A)$ . Then define $d^0 a\in C^1(G,A)$ by

$(d^0 a) (g)=ga-a$ , $g\in G$ .

Thus $d^0 :C^0(G,A)\rightarrow C^1(G,A)$ . Let $b\in C^1(G,A)$ . Then define $d^1 b\in C^2(G,A)$ by* $(d^1 b)(g_1,g_2)=g_1 b(g_2)-b(g_1 g_2)+b(g_1)$ . Thus $d^1:C^1(G,A)\rightarrow C^2(G,A)$ . One checks that $d^1d^0=0$ :