An introduction to Lie algebra cohomology/saras/lecture 1

Introduction

Let \(G\) be a group with identity \(e\ .\)

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.

Linear forms

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

Representations

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

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

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 \(d^1d^0=0\ :\) \[ d^1d^0 a(g_1,g_2)=g_1d^0 a(g_2)-d^0 a(g_1g_2)+d^0 a(g_1)=g_1g_2 a- g_1a-g_1g_2 a+a+g_1 a-a=0. \] Let \(c\in C^2(G,A)\) be a two-form. Then define \[\tag{1} d^2 c(g_1,g_2,g_3)=g_1c(g_2,g_3)-c(g_1g_2,g_3)+c(g_1,g_2g_3)-c(g_2,g_3)\ .\]

In general, when one has defined \(d^i:C^i(G,A)\rightarrow C^{i+1}(G,A)\) such that \(d^{i+1}d^i=0\ ,\) then one calls \(d^\cdot\) a coboundary operator.

Exercise

Show that \(d^2 d^1=0\ .\)

To the second lecture