Lie algebra coholomology

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 14 April 2011

Lie algebras and representations

1.1 A Lie algebra is a vector space $𝔤$ over $k$ with a bracket $[,]:𝔤\hspace{0.17em}\times \hspace{0.17em}𝔤\to 𝔤$ which satisfies $$\begin{array}{c}[x,x]=0,\text{for all} x\in 𝔤,\\ [x,[y,z\left]\right]+[z,[x,y\left]\right]+[y,[z,z\left]\right]=0,\text{for all} x,y,z\in 𝔤.\end{array}$$ The first relation is the skew-symmetric relation and is equivalent to $[x,y]=-[y,x]$ for all $x,y\in 𝔤$ provided $char\left(k\right)\ne 2$. The second relation is called the Jacobi identity.

1.2 A derivation of a Lie algebra $𝔤$ is a map $D:𝔤\to 𝔤$ such that $$D\left([x,y]\right)=[x,D(y\left)\right]+\left[D\right(y),y]$$

1.3 A $𝔤$-module or representation of $𝔤$ is a pair consisting of a vector space $V$ over $k$ and a linear map $\rho :𝔤\to End\left(V\right)$ such that $$\rho \left([x,y]\right)=\rho \left(x\right)\rho \left(y\right)-\rho \left(y\right)\rho \left(x\right),$$ for all $x,y\in 𝔤$. It is common to suppress the $\rho$ in writing the $𝔤$ action on $V$. With the suppressed notation the condition is $[x,y]v=xyv-yxv$ for all $x,y\in 𝔤$ and all $v\in V$.

1.4 If $\rho ,V$ and are $𝔤$-modules then the tensor product $V\otimes W$ is a $𝔤$-module with action given by $$\left(\rho \otimes \tau \right)\left(x\right)=\rho \left(x\right)\otimes 1+1\otimes \tau \left(x\right),$$ for all $x\in 𝔤$. In the suppressed notation the action of $𝔤$ on $V\otimes W$ is given by $x\left(v\otimes w\right)=xv\otimes \otimes w+v\otimes xw$ for all $x\in 𝔤$ and $v,w\in 𝔤$.

1.5 The trivial module for $𝔤$ is a $1$-dimensional vector space $V$ with $𝔤$-action given by $xv=0$.

1.6 The dual of a $𝔤$-module $\left(\rho ,V\right)$ is the vector space $V^{\ast }$ and the $𝔤$-action given by $$⟨x{v}^{\ast },w⟩=⟨v^{\ast },-w⟩,$$ for all $x\in 𝔤$, $v^{\ast }\in V^{\ast }$ and $w\in V$. Here $⟨v^{\ast },w⟩=v^{\ast }\left(w\right)$ denotes the evaluation of the linear functional $v^{\ast }\in V^{\ast }$ at the element $w\in V$.

1.7 The adjoint representation of $𝔤$ is the $𝔤$-module $\left(ad,𝔤\right)$ where the $𝔤$ action on $𝔤$ is given by for all $x,y\in 𝔤$.

Lie algebra cohomology

2.1 Let $𝔤$ be a Lie algebra and let $\left(\rho ,V\right)$ be a $𝔤$-module. Elements $\omega :{\bigwedge }^p𝔤\to V$ of $C^p\left(𝔤,V\right)=Hom\left({\bigwedge }^p𝔤,V\right)$ are called $p$-cochains. The maps $d_p:C^p\left(𝔤,V\right)\to C^{p+1}\left(𝔤,V\right)$ given by $$\begin{array}{rcl}{d}_p\omega \left(X_1\wedge \cdots \wedge X_{p+1}\right)& =& \sum _{j=1}^{p+1}{\left(-1\right)}^{j+1}\rho \left(X_i\right)\omega \left(X_1\wedge \cdots \wedge {\hat{X}}_j\wedge \cdots \wedge X_{p+1}\right)\\ & & +\sum _{r<s}{\left(-1\right)}^{r+s}\omega \left([X_r,X_s]\wedge X_1\wedge \cdots \wedge {\hat{X}}_r\wedge \cdots \wedge {\hat{X}}_s\wedge \cdots \wedge X_{p+1}\right)\end{array}$$ determine a cochain complex $$0\to V\to C^1\left(𝔤,V\right)\to \cdots C^{p-1}\left(𝔤,V\right)\to C^p\left(𝔤,V\right)\to \to C^{p+1}\left(𝔤,V\right)\to \cdots .$$ It is common to suppress the subscript on the map $d_p$ when the $p$ is irrelevant or the context is clear. In the cases $p=0,1$ the map $d$ is given explicitly by $$\begin{array}{c}dv\left(X\right)=\rho \left(X\right)v,\\ d\omega \left(X,Y\right)=\rho \left(X\right)\omega \left(Y\right)-\rho \left(Y\right)\omega \left(X\right)-\omega \left([X,Y],\right)\end{array}$$ for all $v\in V$, $\omega \in C^1\left(𝔤,V\right)$ and $X,Y\in 𝔤$.

2.2 One can prove directly (see [Kn] Lemma 4.5 p. 172) or by using some tricks (see [HGW] Lemma 4.1.4) tha $$p_{p-1}{d}_p=0,$$ for all positive integers $p$. Thus, if we define $$\begin{array}{c}{Z}^p\left(𝔤,V\right)=ker\hspace{0.17em}{d}_p,\\ B^p\left(𝔤,V\right)=im\hspace{0.17em}{d}_{p-1},\end{array}$$ for all $p$, then $$H^p\left(𝔤,V\right)=\frac{Z^p\left(𝔤,V\right)}{B^p\left(𝔤V\right)}$$ is well defined. The elements of $Z^p\left(𝔤,V\right)$ are $p$-cocycles, the elements of $B^p\left(𝔤,V\right)$ are $p$-coboundaries, and $$ H^p\left(𝔤,V\right)$is\; the$ p^{th}$$cohomology group of $𝔤$ with coefficients in $V$.

2.3 Consider the complex $C^{\ast }\left(𝔤,ad\right)$ where $ad$ is the adjoint representation of the Lie algebra $𝔤$. In this case the $1$-cocycles are maps $D:𝔤\to 𝔤$ such that $d_{1}D\left(x\wedge y\right)=0=\left(ad\hspace{0.17em}x\right)D\left(y\right)-\left(ad\hspace{0.17em}y\right)D\left(x\right)-D\left([x,y]\right)$, for all $x,y\in 𝔤$. Thus $1$-cocycles are the maps $D:𝔤\to 𝔤$ such that $$D\left([x,y]\right)=[x,D(y\left)\right]+\left[D\right(x),y],\text{for all} x,y\in 𝔤,$$ i.e., the derivations. For any two elements $x,y\in 𝔤$ $$d_{0}x\left(y\right)=\left(ad\hspace{0.17em}y\right)\left(y\right)=[y,x]=-[x,y]=ad\hspace{0.17em}x\left(y\right).$$ Thus $d_{0}x=-ad\hspace{0.17em}x$. Thus the $1$-coboundaries are the inner derivations.

2.4 The case where $\left(\rho ,V\right)$ is the second tensor power of the adjoint representation $\left(ad,𝔤\right)$ of $𝔤$ is useful for understanding Lie bialgebras. Let $\left({ad}^{\otimes 2},𝔤\otimes 𝔤\right)$ be the tensor product of the adjoint representation with itself; specifically $𝔤$ acts on $𝔤\otimes 𝔤$ by ${ad}^{\otimes 2}x=ad\hspace{0.17em}x\otimes 1+1\otimes ad\hspace{0.17em}x,$ for all $x\in 𝔤$. Then using the explicit formula for $d_1$, $$\begin{array}{rcl}d\omega \left(w,y\right)& =& \left({ad}^{\otimes 2}x\right)\omega \left(y\right)-\left({ad}^{\otimes 2}y\right)\omega \left(x\right)-\omega \left([x,y]\right)\\ & =& [x\otimes 1+1\otimes x,\omega (y\left)\right]-[y\otimes 1+1\otimes y,\omega (x\left)\right]-\omega \left([x,y]\right).\end{array}$$ An element $\omega \in C^1\left(𝔤,𝔤\otimes 𝔤\right)$ is a $1$-cocycle if $d\omega =0$, i.e., if $$\omega \left([x,y]\right)=[x\otimes 1+1\otimes x,\omega (y\left)\right]-[y\otimes 1+1\otimes y,\omega (x\left)\right].$$ This is the origin of the terminology $1$-cocycle condition in the definition of a Lie bialgebra.

References

[Bou] N. Bourbaki, Lie groups and Lie algebras, Chapters I-III, Springer-Verlag, 1989.

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

The theorem that a Lie bialgebra structure determinse a co-Poisson Hopf algebra structure on its enveloping algebra is due to Drinfel'd and appears as Theorem 1 in the following article.

[D1] V.G. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258. MR0802128

[DHL] H.-D. Doebner, Hennig, J. D. and W. Lücke, Mathematical guide to quantum groups, Quantum groups (Clausthal, 1989), Lecture Notes in Phys., 370, Springer, Berlin, 1990, pp. 29–63. MR1201823

There is a very readable and informative chapter on Lie algebra cohomolgy in the forthcoming book

[HGW] R. Howe, R. Goodman, and N. Wallach, Representations and Invariants of the Classical Groups, manuscript, 1993.

The following book is a standard introductory book on Lie algebras; a book which remains a standard reference. This book has been released for a very reasonable price by Dover publishers. There is a short description of Lie algebra cohomology in Chapt. III §11, pp.93-96.

[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.

A very readable and complete text on Lie algebra cohomology is

[Kn] A. Knapp, Lie groups, Lie algebras and cohomology, Mathematical Notes 34, Princeton University Press, 1998. MR0938524

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