lecture 6

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< An introduction to Leibniz algebra cohomology

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

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1 The Serre-Hochschild spectral sequence

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The Serre-Hochschild spectral sequence

Let $\mathfrak{h}$ be a subalgebra or an ideal in $\mathfrak{g}$ . Define a filtration on $C^n(\mathfrak{g},\mathfrak{a})$ by

$F^pC^n(\mathfrak{g},\mathfrak{a})=\{a^n\in C^n(\mathfrak{g},\mathfrak{a})| a^n(x_1,\cdots,x_n)=0 \ \mathrm{if}\ n-p+1 \ \mathrm{of\ its \ variables\ are\ in\ } \mathfrak{h}\}.$

Then

$C^n(\mathfrak{g},\mathfrak{a})=F^0 C^n(\mathfrak{g},\mathfrak{a})\supset\cdots\supset F^nC^n(\mathfrak{g},\mathfrak{a})\supset F^{n+1}C^n(\mathfrak{g},\mathfrak{a})=0$

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remark

Since, when $\mathfrak{h}$ is an ideal,

$F^n C^n (\mathfrak{g},\mathfrak{a})\simeq C^n (\mathfrak{g}/\mathfrak{h},\mathfrak{a})$

one can see this as an approximation scheme to go from $C^n (\mathfrak{g},\mathfrak{a})$ to $C^n (\mathfrak{g}/\mathfrak{h},\mathfrak{a})$ .

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lemma

$d^{(n)} (x) F^pC^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1}C^n(\mathfrak{g},\mathfrak{a})$ .

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proof

Let $a^n\in F^pC^n(\mathfrak{g},\mathfrak{a})$ . That means that $a^n$ will be zero if $n-p+1$ of its variables are in $\mathfrak{h}$ . Since

$(d^{(n)}(y)a^n)(x_1,\cdots,x_n)=d_+^{(0)}(y)a^n(x_1,\cdots,x_n)-\sum_{i=1}^n a^n(x_1,\cdots, [y,x_i],\cdots,x_n),$

it is clear that $(d^{(n)}(y)a^n)(x_1,\cdots,x_n)=0$ if $n-p+2$ of its variables are in $\mathfrak{h}$ , that is, $d^{(n)}(y)a^n\in F^{p-1}C^n(\mathfrak{g},\mathfrak{a})$ .

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lemma

For $x\in \mathfrak{g}$ that

$\iota^n (x) F^pC^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1}C^{n-1}(\mathfrak{g},\mathfrak{a})$ .

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lemma

$d^n F^p C^n(\mathfrak{g},\mathfrak{a})\subset F^{p} C^{n+1}(\mathfrak{g},\mathfrak{a})$

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proof

For $n=0$ this is clear, since $d^0 C^0(\mathfrak{g},\mathfrak{a})\subset C^1(\mathfrak{g},\mathfrak{a})$ . Suppose the statement holds for all $k< n$ . Then, since

$\iota^{n+1}(x)d^n+d^{n-1}\iota^n(x)=d^{(n)}(x)$ ,

the statement holds by induction for all $n\in\N$ . Indeed,

$d^{(n)}(x)F^p C^n(\mathfrak{g},\mathfrak{a})\subset F^{p-1} C^n(\mathfrak{g},\mathfrak{a})$

and, using the induction hypothesis,

$d^{n-1}\iota^n(x)F^p C^n(\mathfrak{g},\mathfrak{a})\subset d^{n-1}F^{p-1} C^{n-1}(\mathfrak{g},\mathfrak{a}) \subset F^{p-1} C^{n}(\mathfrak{g},\mathfrak{a})$ .