Lie algebra coholomology

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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×𝔤𝔤 which satisfies [ x , x ]=0,for all x𝔤, [ x , [ y , z ] ]+ [ z , [ x , y ] ]+ [ y , [ z , z ] ]=0, for all x,y,z𝔤. The first relation is the skew-symmetric relation and is equivalent to [ x , y ]= - [ y , x ] for all x,y𝔤 provided chark2. The second relation is called the Jacobi identity.

1.2 A derivation of a Lie algebra 𝔤 is a map D:𝔤𝔤 such that D ( [ x , y ] )= [ x , D(y) ]+ [ D(y) , y ]

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

1.4 If ρ,V and (τ,W) are 𝔤-modules then the tensor product VW is a 𝔤-module with action given by (ρτ)(x)= ρ(x)1+1 τ(x), for all x𝔤. In the suppressed notation the action of 𝔤 on VW is given by x(vw)= xvw+vxw for all x𝔤 and v,w𝔤.

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 (ρ,V) is the vector space V and the 𝔤-action given by xv , w = v , -w , for all x𝔤, vV and wV. Here v , w = v(w) denotes the evaluation of the linear functional vV at the element wV.

1.7 The adjoint representation of 𝔤 is the 𝔤-module ( ad,𝔤 ) where the 𝔤 action on 𝔤 is given by (adx)(y)= [ x , y ], for all x,y𝔤.

Lie algebra cohomology

2.1 Let 𝔤 be a Lie algebra and let (ρ,V) be a 𝔤-module. Elements ω: p𝔤V of Cp(𝔤,V)= Hom(p𝔤,V) are called p-cochains. The maps dp: Cp(𝔤,V) Cp+1 (𝔤,V) given by dpω ( X1Xp+1 ) = j=1 p+1 (-1)j+1 ρ(Xi) ω ( X1 X ˆ j Xp+1 ) + r<s (-1)r+s ω ( [ Xr , Xs ] X1 X ˆ r X ˆ s Xp+1 ) determine a cochain complex 0V C1(𝔤,V) Cp-1(𝔤,V) Cp(𝔤,V) Cp+1(𝔤,V). It is common to suppress the subscript on the map dp 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 vV, ω C1 (𝔤,V) and X,Y𝔤.

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)= kerdp, Bp(𝔤,V)= imdp-1, for all p, then Hp(𝔤,V)= Zp(𝔤,V) Bp(𝔤V) is well defined. The elements of Zp(𝔤,V) are p-cocycles, the elements of Bp(𝔤,V) are p-coboundaries, and Hp(𝔤,V) is the pth cohomology group of 𝔤 with coefficients in V.

2.3 Consider the complex C(𝔤,ad) where ad is the adjoint representation of the Lie algebra 𝔤. In this case the 1-cocycles are maps D:𝔤𝔤 such that d1D(xy)=0= (adx)D(y)- (ady)D(x)- D ( [ x , y ] ) , for all x,y𝔤. Thus 1-cocycles are the maps D:𝔤𝔤 such that D ( [ x , y ] )= [ x , D(y) ]+ [ D(x) , y ], for all x,y𝔤, i.e., the derivations. For any two elements x,y𝔤 d0x(y)= ( ady )(y) = [ y , x ]= - [ x , y ]= adx(y). Thus d0x=-adx. Thus the 1-coboundaries are the inner derivations.

2.4 The case where (ρ,V) is the second tensor power of the adjoint representation ( ad,𝔤 ) of 𝔤 is useful for understanding Lie bialgebras. Let ( ad2,𝔤𝔤 ) be the tensor product of the adjoint representation with itself; specifically 𝔤 acts on 𝔤𝔤 by ad2x= adx1+1adx, for all x𝔤. Then using the explicit formula for d1, dω(w,y) = ( ad2x ) ω(y)- ( ad2y ) ω(x) -ω ( [ x , y ] ) = [ x1+1x , ω(y) ] - [ y1+1y , ω(x) ]- ω ( [ x , y ] ). An element ω C1 ( 𝔤,𝔤𝔤 ) is a 1-cocycle if dω=0, i.e., if ω ( [ x , y ] )= [ x1+1x , ω(y) ] - [ y1+1y , ω(x) ] . This is the origin of the terminology 1-cocycle condition in the definition of a Lie bialgebra.


[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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