## Lie algebras and representations

1.1 A Lie algebra is a vector space $𝔤$ over $k$ with a bracket $\left[,\right]:𝔤 × 𝔤\to 𝔤$ which satisfies The first relation is the skew-symmetric relation and is equivalent to $\left[x,y\right]=-\left[y,x\right]$ 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 $D ( [ x , y ] )= [ x , D(y) ]+ [ D(y) , y ]$

1.3 A $𝔤$-module or representation of $𝔤$ is a pair $\left(\mathrm{\rho ,V}\right)$ consisting of a vector space $V$ over $k$ and a linear map $\rho :𝔤\to End\left(V\right)$ such that $ρ ( [ x , y ] ) =ρ(x)ρ(y) -ρ(y)ρ(x),$ 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 $\left[x,y\right]v=xyv-yxv$ for all $x,y\in 𝔤$ and all $v\in V$.

1.4 If $\rho ,V$ and $\left(\tau ,W\right)$ are $𝔤$-modules then the tensor product $V\otimes W$ is a $𝔤$-module with action given by $(ρ⊗τ)(x)= ρ(x)⊗1+1⊗ τ(x),$ 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 $⟨ xv∗ , w ⟩= ⟨ v∗ , -w ⟩,$ 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 $(ad x)(y)= [ x , y ],$ 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 $\omega :{\bigwedge }^{p}𝔤\to V$ of ${C}^{p}\left(𝔤,V\right)=Hom\left({\bigwedge }^{p}𝔤,V\right)$ are called $p$-cochains. The maps ${d}_{p}:{C}^{p}\left(𝔤,V\right)\to {C}^{p+1}\left(𝔤,V\right)$ given by $dpω ( X1∧⋯∧Xp+1 ) = ∑ j=1 p+1 (-1)j+1 ρ(Xi) ω ( X1∧⋯∧ X ˆ j ∧⋯∧Xp+1 ) + ∑ r determine a cochain complex $0→V→ C1(𝔤,V)→⋯ Cp-1(𝔤,V)→ Cp(𝔤,V)→→ Cp+1(𝔤,V)→⋯.$ 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 $dv(X)=ρ (X)v, dω (X,Y)= ρ(X)ω(Y)- ρ(Y)ω(X) -ω ( [ X , Y ], )$ 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 $pp-1dp=0,$ for all positive integers $p$. Thus, if we define $Zp(𝔤,V)= ker dp, Bp(𝔤,V)= im dp-1,$ for all $p$, then $Hp(𝔤,V)= Zp(𝔤,V) Bp(𝔤V)$ 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 ${d}_{1}D\left(x\wedge y\right)=0=\left(ad x\right)D\left(y\right)-\left(ad y\right)D\left(x\right)-D\left(\left[x,y\right]\right)$, for all $x,y\in 𝔤$. Thus $1$-cocycles are the maps $D:𝔤\to 𝔤$ such that i.e., the derivations. For any two elements $x,y\in 𝔤$ $d0x(y)= ( ad y )(y) = [ y , x ]= - [ x , y ]= ad x(y).$ Thus ${d}_{0}x=-ad x$. 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 ${ad}^{\otimes 2}x=ad x\otimes 1+1\otimes ad x,$ for all $x\in 𝔤$. Then using the explicit formula for ${d}_{1}$, $dω(w,y) = ( ad⊗2x ) ω(y)- ( ad⊗2y ) ω(x) -ω ( [ x , y ] ) = [ x⊗1+1⊗x , ω(y) ] - [ y⊗1+1⊗y , ω(x) ]- ω ( [ x , y ] ).$ An element $\omega \in {C}^{1}\left(𝔤,𝔤\otimes 𝔤\right)$ is a $1$-cocycle if $d\omega =0$, i.e., if $ω ( [ x , y ] )= [ x⊗1+1⊗x , ω(y) ] - [ y⊗1+1⊗y , ω(x) ] .$ 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