Enveloping algebras of Lie algbras

Enveloping algebras of Lie algebras

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

Last updates: 10 April 2011

Basic properties

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

1.2 Let $𝔤$ be a Lie algebra. Let $T (𝔤)$ be the tensor algebra of $𝔤$ and let $J$ be the ideal of $T (𝔤)$ generated by the tensors $x \otimes y - y \otimes x - [x, y],$ where $x, y \in 𝔤$ . The enveloping algebra of $𝔤$ , $𝔘 𝔤$ is the associative algebra $𝔘 𝔤 = \frac{T (𝔤)}{J} .$ There is a canonical map $\begin{matrix} α_{0} : & 𝔤 & \to & 𝔘 𝔤 \\ x & \mapsto & x + J & . \end{matrix}$ The algebra $𝔘 𝔤$ can be given the following universal property:

Let $α : 𝔤 \to A$ be a mapping of $𝔤$ to an associative algebra $A$ such that $α ([x, y]) = α (x) α (y) - α (y) α (x)$ for all $x, y \in 𝔤$ and let $1$ and $1_{A}$ denote the identities in $𝔘 𝔤$ and $A$ respectively. Then there exists a unique mapping $τ : 𝔘 𝔤 \to A$ such that $τ (1) = 1_{A}$ and $α = τ \circ α_{0}$ , 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 $(x_{i})_{i \in Λ}$ . Then the elements $x_{i_{1}} x_{i_{2}} \dots x_{i_{n}}$ of the enveloping algebra $𝔘 𝔤$ , where $(i_{1} \leq i_{2} \leq \dots \leq i_{n})$ is an arbitrary increasing finite seqeuce of elements of $Λ$ , form a basis of $𝔘 𝔤$ .

1.4 It follows from the Poincaré-Birkoff-Witt theorem that the canonical map of $𝔤$ into $𝔘 𝔤$ is injective. Thus we can view $𝔤$ as a subspace of $𝔘 𝔤$ . Also it is natural to give $𝔘 𝔤$ a filtraition by defining $𝔘_{n} = span - \{x_{i_{1}} x_{i_{2}} \dots x_{i_{k}} ∣ k \leq n, x_{i_{j}} \in 𝔤\} .$ Then $𝔘 𝔤 = \sum 𝔘_{m}, 𝔘_{m} \subseteq 𝔘_{n}, if m \leq n, and 𝔘_{m} 𝔘_{n} \subseteq 𝔘_{m + n} .$

1.5 The enveloping algebra of $𝔤$ is made into a Hopf algebra by defining a comultiplication $Δ : 𝔘 𝔤 \to 𝔘 𝔤 \otimes 𝔘 𝔤$ by $Δ (x) = 1 \otimes x + x \otimes 1, for all x \in 𝔤,$ a counit $ϵ : 𝔘 𝔤 \to k$ by $ϵ (x) = 0, for all x \in 𝔤,$ and an antipode $S : 𝔘 𝔤 \to 𝔘 𝔤$ by $S (x) = - x, for all x \in 𝔤 .$

1.6 The algebra $𝔘 𝔤 \otimes 𝔘 𝔤$ also has a filtration given by $(𝔘 𝔤 \otimes 𝔘 𝔤)_{n} = \sum_{r + s = n} 𝔘_{r} \otimes 𝔘_{s} .$ If $y \in 𝔘_{m}$ then $Δ (y) \in (𝔘 𝔤 \otimes 𝔘 𝔤)_{m}$ since if $x \in 𝔤$ then $Δ (x) = 1 \otimes x + x \otimes 1 \in \in (𝔘 𝔤 \otimes 𝔘 𝔤)_{1}$ .

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

If $char (k) = 0$ then the subspace $𝔤$ of $𝔘 𝔤$ is the set of primitive elements of $𝔘 𝔤$ .

		Proof:
	Suppose that $x_{1}, x_{2}, \dots$ is a basis of $𝔤$ . Then the monomials $x_{1}^{k_{1}} x_{1}^{k_{2}} \dots x_{1}^{k_{m}}$ form a basis of $𝔘 𝔤$ and the tensors $x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m}} \otimes x_{1}^{l_{1}} x_{2}^{l_{2}} \dots x_{n}^{l_{n}}$ form a basis of $𝔘 𝔤 \otimes 𝔘 𝔤$ . Then $\begin{array}{rcl} Δ (x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m}}) & = & {(x_{1} \otimes 1 \otimes 1 \otimes x_{1})}^{k_{1}} \dots {(x_{m} \otimes 1 \otimes 1 \otimes x_{m})}^{k_{m}} \\ = & x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m}} \otimes 1 \\ + k_{1} x_{1}^{k_{1} - 1} x_{2}^{k_{2}} \dots x_{m}^{k_{m}} \otimes x_{1} + k_{2} x_{1}^{k_{1}} x_{2}^{k_{2} - 1} \dots x_{m}^{k_{m}} \otimes x_{2} + \dots + k_{m} x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m} - 1} \otimes x_{m} \\ + \dots + 1 \otimes x_{1}^{k_{1}} x_{1}^{k_{2}} \dots x_{1}^{k_{m}} . \end{array}$ Note that The term $x_{1}^{k_{1} - 1} x_{2}^{k_{2}} \dots x_{m}^{k_{m}} \otimes x_{1}$ cannot appear with non-xero coefficeint in $Δ (x_{1}^{l_{1}} x_{2}^{l_{2}} \dots x_{n}^{l_{n}})$ for any basis element $x_{1}^{l_{1}} x_{2}^{l_{2}} \dots x_{n}^{l_{n}}$ different from $\otimes x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m}}$ The coproduct $Δ (x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m}})$ is a linear combination of terms of the form $x_{1}^{j_{1}} x_{2}^{j_{2}} \dots x_{m}^{j_{m}} \otimes 1$ and $1 \otimes x_{1}^{l_{1}} x_{2}^{l_{2}} \dots x_{m}^{l_{m}}$ only if all the $k_{i}$ are $0$ except for one, and this one is $1$ , i.e., only if $x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m}} = x_{i} \in 𝔤$ for some $i$ . It follows from 1) that if $a \in 𝔘 𝔤$ and the expansion of $Δ (a)$ contains only terms of the form $x_{1}^{j_{1}} x_{2}^{j_{2}} \dots x_{m}^{j_{m}} \otimes 1$ and $1 \otimes x_{1}^{l_{1}} x_{2}^{l_{2}} \dots x_{m}^{l_{m}}$ then $a$ is a linear combination of terms $x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m}}$ such that $Δ (x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{m}^{k_{m}})$ is a linear combination of terms of the form $x_{1}^{j_{1}} x_{2}^{j_{2}} \dots x_{m}^{j_{m}} \otimes 1$ and $1 \otimes x_{1}^{l_{1}} x_{2}^{l_{2}} \dots x_{m}^{l_{m}}$ . 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 $𝔘 𝔤$ , then $a \in 𝔤$ . $□$

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.

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