Enveloping algebras of Lie algebras

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

Last updates: 10 April 2011

Basic properties

1.1 A Lie algebra is a vector space 𝔤 over k with a bracket [ , ]: 𝔤𝔤𝔤 which satisfies [ x , x ]=0, for all x𝔤, [ x , [ y , z ] ]+ [ z , [ x , y ] ]+ [ y , [ z , x ] ]=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 Let 𝔤 be a Lie algebra. Let T𝔤 be the tensor algebra of 𝔤 and let J be the ideal of T𝔤 generated by the tensors xy-yx- [ x , y ], where x,y𝔤. The enveloping algebra of 𝔤, 𝔘𝔤 is the associative algebra 𝔘𝔤= T𝔤 J . There is a canonical map α0: 𝔤 𝔘𝔤 x x+J. The algebra 𝔘𝔤 can be given the following universal property:

  1. Let α:𝔤A be a mapping of 𝔤 to an associative algebra A such that α [ x , y ] = αxαy- αyαx for all x,y𝔤 and let 1 and 1A denote the identities in 𝔘𝔤 and A respectively. Then there exists a unique mapping τ:𝔘𝔤A such that τ1=1A and α=τα0 , i.e., the following diagram commutes.

    𝔤 𝔘𝔤 A α τ α0

1.3 The following statement is the Poincaré-Birkoff-Witt theorem. See [Bou] or [J] for an exposition and a proof.

Suppose that 𝔤 has a totally ordered basis (xi)iΛ . Then the elements xi1xi2 xin of the enveloping algebra 𝔘𝔤, where i1i2in is an arbitrary increasing finite seqeuce of elements of Λ, form a basis of 𝔘𝔤.

1.4 It follows from the Poincaré-Birkoff-Witt theorem that the canonical map of 𝔤 into 𝔘𝔤 is injective. Thus we can view 𝔤 as a subspace of 𝔘𝔤. Also it is natural to give 𝔘𝔤 a filtraition by defining 𝔘n= span- xi1 xi2 xik kn, xij𝔤 . Then 𝔘𝔤=𝔘m, 𝔘m𝔘n, if mn, and 𝔘m𝔘n 𝔘m+n.

1.5 The enveloping algebra of 𝔤 is made into a Hopf algebra by defining a comultiplication Δ:𝔘𝔤𝔘𝔤𝔘𝔤 by Δx= 1x+ x1, for all x𝔤, a counit ϵ:𝔘𝔤k by ϵx=0, for allx𝔤, and an antipode S:𝔘𝔤𝔘𝔤 by Sx=-x, for all x𝔤.

1.6 The algebra 𝔘𝔤𝔘𝔤 also has a filtration given by (𝔘𝔤𝔘𝔤)n= r+s=n 𝔘r𝔘s. If y𝔘m then Δy (𝔘𝔤𝔘𝔤)m since if x𝔤 then Δx= 1x+ x1 (𝔘𝔤𝔘𝔤)1 .

1.7 An element x of a Hopf algebra A is primitive if Δx= 1x+ x1 . All elements of 𝔤 are primitive elements of 𝔘𝔤. In fact, the following Proposition shows that if chark=0 then the elements of 𝔤 are all the primitive elements of 𝔘𝔤.

If chark=0 then the subspace 𝔤 of 𝔘𝔤 is the set of primitive elements of 𝔘𝔤.



Chapter I §2 of [Bou] gives a nice, in depth, exposition of enveloping algebras and the Poincaré-Birkoff-Witt theorem. Chapter II §1 discusses the bialgebra structure on the universal enveloping algebra. In particular [Bou] Chapt II. §1.5 Corollary to Proposition 9 gives a very slick proof of Proposition 1.2. The reason it is so slick is because of the marvelous combinatorial fact Bourbacki, Algebra Chapt. I §8.2 Prop. 2.

[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

[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

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.

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

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