lecture 4

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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

The third lecture


Contents

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Lifting the representation to the forms

Let A be a G-module. In order to give a general definition of a coboundary operator d^n, one defines first an induced representation on C^n (G,A) as follows.

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definition

Let \mathfrak{a} be an \mathfrak{l}-module. In order to give a general definition of a coboundary operator d^n, one defines first an induced representation on C^n (G,A) as follows. Let, for h\in G,

(d^{(n)}(h)a)(g_1,\cdots,g_n)=a(g_1,\cdots,g_n h)-a(g_1,\cdots,g_{n-1}, h)

This is indeed a representation. Let h,k\in G. Then

d^{(n)}(h)d^{(k)}(z)a(g_1,\cdots,g_n)=
=d^{(n)}(k)a(g_1,\cdots,g_n k)-d^{(n)}(k)a(g_1,\cdots,g_{n-1}, k)=
=a(g_1,\cdots,g_n kh)-a(g_1,\cdots,g_{n-1},h)-a(g_1,\cdots,g_{n-1} kh)+a(g_1,\cdots,g_{n-1},h)
=a(g_1,\cdots,g_n kh)-a(g_1,\cdots,g_{n-1} kh)=(d^{(n)}(hk)a)(g_1,\cdots,g_n).

By definition

d^{(0)}(h)a(g_1,\cdots,g_n)=ha(g_1,\cdots,g_n).

We can then write the coboundary operators as follows

(d^{1}b)(g_1,g_2)=gd^{(k)}(z)a(g,h)=ga(h)-d^{(1)}(h)a(g)-a(h)+a(g)=
=(ga(h)-a(h))-(d^{(1)}(h)a(g)-a(g))=
=d^{(0)}(g)a(h)-(d^{(1)}(h)a(g)-a(g))=d^{(0)}(g)a(h)-\Lambda^{1} (h)a(g),

where for convenience we define

\Lambda^{n} (h)


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example

Trivial module. Every abelian group A can be viewed as a G-module defining the (trivial) action ga=a, \forall g\in G. Such a G-module is called trivial G-module.

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example

Let A=\mathbb{Z} and let G=\mathbb{Z}_2=\{e,g\}, g^2=e, then defining the map gn=-n A becomes a G-module (nontrivial).

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remark

Any abelian group is a \mathbb{Z}-module.

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remark

Any G-module is a \mathbb{Z}-module as well. More examples:

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example

Let A=\{e,a,b,ab\} be the Klein's (Viergruppe) group, let G=\mathbb{Z}_2 and let \phi(g) be the homomorphism from G to Aut(A) which exchanges a and b and keeps ab fixed. This makes A into a G-module.

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example

Let G be a finite group. Let \mathcal{A}=\mathbb{C}(\lambda) be the field of rational functions in \lambda. Suppose we have a fixed representation \sigma: G \rightarrow PSL(2,\mathbb{C}). Then this induces a representation of G in \mathcal{A} by identifying PSL(2,\mathbb{C}) with the fractional linear transformations of \mathbb{C}. This makes \mathcal{A} into a G-module. Let, for instance, G=\mathbb{Z}_2=\{e,g\}, g^2=e, and identify the group with the group of fractional-linear transformation in the complex \lambda-plane generated by \sigma^g(\lambda)=\lambda^{-1}. This makes A into a G-module; indeed (1) and (2) are obvious while (3) can be verified as follows

\sigma^g(f_1 (\lambda)+f_2 (\lambda))=f_1 (\lambda^{-1})+f_2 (\lambda^{-1})= \sigma^g(f_1 (\lambda))+ \sigma^g(f_2 (\lambda^{-1}))\,.

In what follows we will consider G to be a finite group of order |G|=N.

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The group ring \mathbb{Z}[G]

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definition

The group Ring \mathbb{Z}[G] is the ring whose additive group is the abelian group of all formal sums

\left\{ \sum_{g\in G}n_{g}\,g\,\,n_{g}\in \mathbb{Z},\,g\in G\right\}\,,

and whose multiplication operation is defined by the multiplication in G, extended \mathbb{Z}-linearly to \mathbb{Z}[G]. Two such formal sums \sum_{g\in G}n_{g}\,g and \sum_{g\in G}m_{g}\,g are equal iff n_{g}=m_{g}. Addition in \mathbb{Z}[G] is componentwise, while, according to the definition, multiplication is determined in the natural way by the multiplication in G; more in detail

\sum_{g\in G}n_{g}\,g+\sum_{g\in G}m_{g}\,g=\sum_{g\in G}(n_{g}+m_{g})\,g\,,
\left(\sum_{g\in G}n_{g}\,g\right)\left(\sum_{g\in G}m_{g}\,g\right)=\sum_{g\in G}\left(\sum_{\tau \rho\in G}n_{\tau}m_{\rho}\right)\,g=\sum_{g\in G}\left(\sum_{\tau\in G}n_{\tau}m_{\tau^{-1}g}\right)\,g\,.

Sometimes the group Ring \mathbb{Z}[G] is identified with its additive group, that is the abelian group of formal integer linear combinations of elements of G. We may view G as imbedded in \mathbb{Z}[G] under the identification of g\in G with 1g\in \mathbb{Z}[G].

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remark

A G-module structure on A is equivalent to a \mathbb{Z}[G]-module structure via

\mathbb{Z}[G]\rightarrow Aut(A)
\left(\sum_{g\in G}n_{g}\,g\right)a=\sum_{g\in G}n_{g}\,ga,\,\,
\forall a\in A.

As abelian group, \mathbb{Z}[G] is itself a G-module.


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The G-module X_q

For each q\ge 1 we define a q-cell, this is a set

\{g_{1},g_{2},\ldots,g_{q}\,\,|\,\,g_{i}\in G\}\,.

g_{1},g_{2},\ldots,g_{q} are elements of the group G. We use the q-cells as free generators of our G-modules X_{q}, we namely define

X_{q}
=\sum_{g_{1},\ldots,g_{q}\in G}\oplus\mathbb{Z}[G](g_{1},g_{2},\ldots,g_{q}).

For q=0 we set

X_{0}=
\mathbb{Z}[G],

considering the 0-cell generated by e\in \mathbb{Z}[G]. By construction the modules

X_{0},\,\,X_{1},X_{-2},\ldots

are G-free, where

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definition

A G-module A is G-free (or \mathbb{Z}[G]-free) if it is the direct sum of G-modules isomorphic to \mathbb{Z}[G]

A=\sum_{i}\bigoplus\Gamma_{i}, \quad\textrm{with}\,\,\,\Gamma_{i}\simeq \mathbb{Z}[G].
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example

Let G=\mathbb{Z}_{2}=\{e,g\}; then

X_{1}=X_{-2}=\sum_{g\in G}\oplus\mathbb{Z}[G](g)=\mathbb{Z}[G]\oplus\mathbb{Z}[G](g).
X_{2}=X_{-3}=\sum_{g_{1},g_{2}\in G}\oplus\mathbb{Z}[G](g_{1},g_{2})=
\mathbb{Z}[G](e,e)\oplus\mathbb{Z}[G](e,g)\oplus\mathbb{Z}[G](g,e)\oplus\mathbb{Z}[G](g,g).

Let us now define the G-homomorphisms d_{q}\,:\;X_{q+1}\to X_{q}; since Z_q are G-free, to define the G-homomorphisms d_{q} it is sufficient to specify the value they take on the generators (g_{1},g_{2},\ldots,g_{q}). We set

q=0 d_{0}\, e=N_{G}=\sum_{g\in G}g;
q=1 d_{1}\,(g_{1})=g_{1}-e;
q>1d_{q}\,(g_{1},g_{2},\ldots,g_{q}) = g_{1}(g_{2},\ldots,g_{q})+\sum_{i=1}^{q-1}(-1)^{i}(g_{1},\ldots,g_{i-1},g_{i}g_{i+1},g_{i+2},\ldots,g_{q})+
+(-1)^{q}(g_{1},g_{2},\ldots,g_{q-1});

Let A be a G-module, we define

A_{q}=Hom_{G}(X_{q},A).
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definition

The elements of A_{q} are called \textbf{q-cochains} of A. They are the G-homomorphisms

x\,\,:\,\,\,X_{q}\to A

Recall that the cochains group A_{q}=A_{-q-1}=Hom_{G}(X_{q},A), q\ge 1 is the group of all G-homomorphisms x\,\,:\,\,X_{q}\to A. Recall also that, by definition, X_{q} is free generated by the q-cells (g_{1},\ldots,g_{q}), g_{i}\in G. So one can uniquely specify the G-homomorphism x through its value on the q-cells (g_{1},\ldots,g_{q}). Each cochain can be interpret as a function

x\,\,:\,\,\underbrace{G\times G\times\ldots\times G}_{q-times}\,\to\, A

so that we can identify

A_{q}=
\{x\,\,:\,\,\underbrace{G\times G\times\ldots\times G}_{q-times}\,\to\,A\},\quad q\ge 1

with

A_{0}=Hom_{G}(\mathbb{Z}[G],A)\simeq A.

From the definition of G-homomorphisms d_{q} in the standard complex we obtain the following for \partial_{q}

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itemize

q=0 (\partial_{0} x)e=N_{G}x=\sum_{g\in G}g x,\,\,\,\textrm{for}\,\,\, x\in A_{-1}=A;
q=1 (\partial_{1}x)(g_{1})=g_{1}x-ex,\,\,\,\textrm{for}\,\,\, x\in A_{0}=A;
q\ge 1 (\partial_{q}x)\,(g_{1},g_{2},\ldots,g_{q})=g_{1}x(g_{2},\ldots,g_{q})+\sum_{i=1}^{q-1}(-1)^{i}x(g_{1},\ldots,g_{i-1},g_{i}g_{i+1},g_{i+2},\ldots,g_{q})+
+(-1)^{q}x(g_{1},g_{2},\ldots,g_{q-1}),\,\,\,\textrm{for}\,\,\,x\in A_{q-1}

In this setting, we can define cocycles as maps

x\,\,:\,\,\underbrace{G\times G\times\ldots\times G}_{q-times}\,\to\, A
\,\,\,\,\textrm{s.t.}\,\,\,\,\partial_{q+1}x=0.

Coboundaries are those cocycles for which

\exists\,\, y\in A_{q-1}\,\,\,\,\textrm{s.t.}\,\,\,\,x=\partial_{q} y.

The fourth lecture

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