An introduction to Lie algebra cohomology/saras/lecture 4
Contents |
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.
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 \[ \Lambda^{n} (h)\]
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.
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).
remark
Any abelian group is a \(\mathbb{Z}\)-module.
remark
Any \(G\)-module is a \(\mathbb{Z}\)-module as well. More examples:
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.
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 \(|G|=N\ .\)
The group ring \(\mathbb{Z}[G]\)
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 \(1g\in \mathbb{Z}[G]\ .\)
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.
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
definition
A \(G\)-module \(A\) is \(G\)-free (or \(\mathbb{Z}[G]\)-free) if it is the direct sum of \(G\)-modules isomorphic to \(\mathbb{Z}[G]\) \[A=\sum_{i}\bigoplus\Gamma_{i}, \quad\textrm{with}\,\,\,\Gamma_{i}\simeq \mathbb{Z}[G].\]
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 \[q=0\] \(d_{0}\, e=N_{G}=\sum_{g\in G}g;\) \[q=1\] \(d_{1}\,(g_{1})=g_{1}-e;\) \[q>1\]\(d_{q}\,(g_{1},g_{2},\ldots,g_{q}) = g_{1}(g_{2},\ldots,g_{q})+\sum_{i=1}^{q-1}(-1)^{i}(g_{1},\ldots,g_{i-1},g_{i}g_{i+1},g_{i+2},\ldots,g_{q})+\ :\)
- \[ +(-1)^{q}(g_{1},g_{2},\ldots,g_{q-1});\]
Let \(A\) be a \(G\)-module, we define \[A_{q}=Hom_{G}(X_{q},A).\]
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 \(\partial_{q}\)
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 \[\exists\,\, y\in A_{q-1}\,\,\,\,\textrm{s.t.}\,\,\,\,x=\partial_{q} y.\]
