## Basic properties

1.1 A Lie algebra is a vector space $𝔤$ over $k$ with a bracket $\left[,\right]:𝔤\otimes 𝔤\to 𝔤$ 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 Let $𝔤$ be a Lie algebra. Let $T\left(𝔤\right)$ be the tensor algebra of $𝔤$ and let $J$ be the ideal of $T\left(𝔤\right)$ generated by the tensors $x⊗y-y⊗x- [ x , y ],$ where $x,y\in 𝔤$. 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 $\alpha :𝔤\to A$ be a mapping of $𝔤$ to an associative algebra $A$ such that $α [ x , y ] = αxαy- αyαx$ for all $x,y\in 𝔤$ and let $1$ and ${1}_{A}$ denote the identities in $𝔘𝔤$ and $A$ respectively. Then there exists a unique mapping $\tau :𝔘𝔤\to A$ such that $\tau \left(1\right)={1}_{A}$ and $\alpha =\tau \circ {\alpha }_{0}$, i.e., the following diagram commutes.

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 $\left({x}_{i}{\right)}_{i\in \Lambda }$. Then the elements ${x}_{{i}_{1}}{x}_{{i}_{2}}\cdots {x}_{{i}_{n}}$ of the enveloping algebra $𝔘𝔤$, where $\left({i}_{1}\le {i}_{2}\le \cdots \le {i}_{n}\right)$ is an arbitrary increasing finite seqeuce of elements of $\Lambda$, 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 $𝔘n= span- xi1 xi2 ⋯ xik ∣ k≤n, xij∈𝔤 .$ Then

1.5 The enveloping algebra of $𝔤$ is made into a Hopf algebra by defining a comultiplication $\Delta :𝔘𝔤\to 𝔘𝔤\otimes 𝔘𝔤$ by a counit $ϵ:𝔘𝔤\to k$ by $ϵx=0, for allx∈𝔤,$ and an antipode $S:𝔘𝔤\to 𝔘𝔤$ by

1.6 The algebra $𝔘𝔤\otimes 𝔘𝔤$ also has a filtration given by $(𝔘𝔤⊗𝔘𝔤)n= ∑ r+s=n 𝔘r⊗𝔘s.$ If $y\in {𝔘}_{m}$ then $\Delta \left(y\right)\in \left(𝔘𝔤\otimes 𝔘𝔤{\right)}_{m}$ since if $x\in 𝔤$ then $\Delta \left(x\right)=1\otimes x+x\otimes 1\in \in \left(𝔘𝔤\otimes 𝔘𝔤{\right)}_{1}$.

1.7 An element $x$ of a Hopf algebra $A$ is primitive if $\Delta \left(x\right)=1\otimes x+x\otimes 1$. All elements of $𝔤$ are primitive elements of $𝔘𝔤$. In fact, the following Proposition shows that if $char\left(k\right)=0$ then the elements of $𝔤$ are all the primitive elements of $𝔘𝔤$.

If $char\left(k\right)=0$ then the subspace $𝔤$ of $𝔘𝔤$ is the set of primitive elements of $𝔘𝔤$.

 Proof: Suppose that ${x}_{1},{x}_{2},\dots$ is a basis of $𝔤$. Then the monomials ${x}_{1}^{{k}_{1}}{x}_{1}^{{k}_{2}}\cdots {x}_{1}^{{k}_{m}}$ form a basis of $𝔘𝔤$ and the tensors $x1k1 x2k2 ⋯ xmkm ⊗ x1l1 x2l2 ⋯ xnln$ form a basis of $𝔘𝔤\otimes 𝔘𝔤$. Then $Δ x1k1 x2k2 ⋯ xmkm = x1⊗1⊗1⊗x1 k1 ⋯ xm⊗1⊗1⊗xm km = x1k1 x2k2 ⋯ xmkm ⊗1 + k1 x1k1-1 x2k2 ⋯ xmkm ⊗x1 + k2 x1k1 x2k2-1 ⋯ xmkm ⊗x2 +⋯+km x1k1 x2k2 ⋯ xmkm-1 ⊗ xm +⋯+ 1⊗ x1k1 x1k2 ⋯ x1km .$ Note that The term ${x}_{1}^{{k}_{1}-1}{x}_{2}^{{k}_{2}}\cdots {x}_{m}^{{k}_{m}}\otimes {x}_{1}$ cannot appear with non-xero coefficeint in $\Delta \left({x}_{1}^{{l}_{1}}{x}_{2}^{{l}_{2}}\cdots {x}_{n}^{{l}_{n}}\right)$ for any basis element ${x}_{1}^{{l}_{1}}{x}_{2}^{{l}_{2}}\cdots {x}_{n}^{{l}_{n}}$ different from $\otimes {x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\cdots {x}_{m}^{{k}_{m}}$ The coproduct $\Delta \left({x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\cdots {x}_{m}^{{k}_{m}}\right)$ is a linear combination of terms of the form ${x}_{1}^{{j}_{1}}{x}_{2}^{{j}_{2}}\cdots {x}_{m}^{{j}_{m}}\otimes 1$ and $1\otimes {x}_{1}^{{l}_{1}}{x}_{2}^{{l}_{2}}\cdots {x}_{m}^{{l}_{m}}$ only if all the ${k}_{i}$ are $0$ except for one, and this one is $1$, i.e., only if $x1k1 x2k2 ⋯ xmkm =xi∈𝔤$ for some $i$. It follows from 1) that if $a\in 𝔘𝔤$ and the expansion of $\Delta \left(a\right)$ contains only terms of the form ${x}_{1}^{{j}_{1}}{x}_{2}^{{j}_{2}}\cdots {x}_{m}^{{j}_{m}}\otimes 1$ and $1\otimes {x}_{1}^{{l}_{1}}{x}_{2}^{{l}_{2}}\cdots {x}_{m}^{{l}_{m}}$ then $a$ is a linear combination of terms ${x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\cdots {x}_{m}^{{k}_{m}}$ such that $\Delta \left({x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\cdots {x}_{m}^{{k}_{m}}\right)$ is a linear combination of terms of the form ${x}_{1}^{{j}_{1}}{x}_{2}^{{j}_{2}}\cdots {x}_{m}^{{j}_{m}}\otimes 1$ and $1\otimes {x}_{1}^{{l}_{1}}{x}_{2}^{{l}_{2}}\cdots {x}_{m}^{{l}_{m}}$. It now follows from 2) that $a$ is a linear combination of elements ${x}_{i}\in 𝔤$. Thus, if $a$ is a primitive element of $𝔘𝔤$, then $a\in 𝔤$. $\square$

## References

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.