Universal Enveloping Algebras

Introduction

Let $\mathrm{G}$ be a Lie algebra over the commutative ring $R$ with identity, with bracket $[-,-]$. The universal enveloping algebra $U(\mathrm{G})$ is the associative $R$-algebra that turns the bracket of $\mathrm{G}$ into the commutator of a multiplication and does so with no further relations: it is the quotient of the tensor algebra $T(\mathrm{G})$ by the two-sided ideal generated by the elements

$$ x\otimes y - y\otimes x - [x,y], \qquad x,y\in\mathrm{G}, $$

and it is characterised by the universal property that every $R$-linear map $\varphi:\mathrm{G}\to A$ into an associative $R$-algebra $A$ with $\varphi([x,y]) = \varphi(x)\varphi(y)-\varphi(y)\varphi(x)$ factors uniquely through the canonical map $\mathrm{G}\to U(\mathrm{G})$. The passage $\mathrm{G}\mapsto U(\mathrm{G})$ is the left adjoint of the functor that attaches to an associative algebra its commutator Lie algebra.

The article is the sixteenth of the corpus and the first of the category Anti-symmetric Linear Algebras. The anti-symmetric multiplication of the category is the bracket of the Lie algebra, developed in Lie Algebras,, and the enveloping algebra is the associative algebra that universalises it: it converts the anti-symmetric structure into an associative product and makes the representations of the Lie algebra available as the modules over an associative algebra, in the sense. The article develops the construction and the universal property, the filtration by degree and the Poincaré–Birkhoff–Witt theorem with its consequence that the Lie algebra sits inside its enveloping algebra and that the associated graded is the symmetric algebra, the Hopf algebra structure with the primitive elements, the equivalence between the representations of the Lie algebra and the modules over the enveloping algebra, the centre with the Casimir element and the Harish-Chandra theorem, the example of $\mathrm{SL}_2$ computed in full, and the modification needed in characteristic $p$.

The article stays inside Part I. It uses the quotient construction of Quotients of the Tensor Algebra, the tensor algebra of Tensor Powers and the Free Algebra, the symmetric algebra of The Symmetric Algebra, the Hopf algebra formalism of Hopf Algebras, and the structure theory and the representations of Lie algebras from the articles of the category Anti-symmetric Linear Algebras written above it, together with the root-space decomposition of a semisimple Lie algebra, which is used as standard, and the Weyl group it carries. What is deferred is everything that needs a distance or a form: the analytic theory of the enveloping algebra — the completed enveloping algebra, the formal group attached to a Lie algebra over a field with a valuation, the smooth and the analytic representations, and the Baker–Campbell–Hausdorff series as a convergent object — belongs to Part III, where the limit is available, and the algebraic form of the series is used here only as a formal statement. The Killing form and the trace form are used only through their algebraic properties of being invariant bilinear forms on $\mathrm{G}$; no quadratic form over a vector space, no Clifford algebra and no Witt theory appears. The symplectic and Poisson-geometric reading of the enveloping algebra as a deformation of the symmetric algebra is stated algebraically and the geometric statements are deferred to Part II.

Throughout, $R$ is a commutative ring with identity, $k$ is a field, $\mathrm{G},\mathrm{H},\mathrm{N}$ are Lie algebras in the fraktur convention of the corpus, $T(\mathrm{G})$ and $\operatorname{Sym}(\mathrm{G})$ are the tensor and symmetric algebras, $U(\mathrm{G})$ is the enveloping algebra with its degree filtration $U_n(\mathrm{G})$, and $\mathrm{gr}\,U(\mathrm{G})$ is the associated graded. The characteristic is arbitrary unless a statement says otherwise, and the places where characteristic zero, a field, or the invertibility of $2$ is required are flagged.

The Universal Property and the Construction

Definition. Let $\mathrm{G}$ be a Lie algebra over $R$, with multiplication $[x,y]$. The universal enveloping algebra is

$$ U(\mathrm{G}) \;=\; T(\mathrm{G})\big/\bigl\langle x\otimes y-y\otimes x-[x,y] \;:\; x,y\in\mathrm{G}\bigr\rangle , $$

where $T(\mathrm{G}) = \bigoplus_{n\geq0}\mathrm{G}^{\otimes n}$ is the tensor algebra and the angle brackets denote the two-sided ideal generated by the displayed elements. Write $\iota:\mathrm{G}\to U(\mathrm{G})$ for the composite of the canonical maps, so that

$$ \iota([x,y]) = \iota(x)\iota(y)-\iota(y)\iota(x) $$

for all $x,y\in\mathrm{G}$, and $\iota$ is $R$-linear.

Theorem (the universal property). Let $A$ be an associative $R$-algebra with identity and let $\varphi:\mathrm{G}\to A$ be an $R$-linear map satisfying $\varphi([x,y]) = \varphi(x)\varphi(y)-\varphi(y)\varphi(x)$. Then there is a unique $R$-algebra homomorphism $\Phi:U(\mathrm{G})\to A$ with $\Phi\circ\iota = \varphi$. Consequently the correspondence $\varphi\mapsto\Phi$ is a natural bijection

$$ \operatorname{Hom}_{\mathsf{Lie}_R}(\mathrm{G},A_{\mathrm{Lie}}) \;\cong\; \operatorname{Hom}_{\mathsf{Alg}_R}\bigl(U(\mathrm{G}),A\bigr), $$

where $A_{\mathrm{Lie}}$ is the Lie algebra with the commutator bracket, and $\mathrm{G}\mapsto U(\mathrm{G})$ is the left adjoint of $A\mapsto A_{\mathrm{Lie}}$.

Proof. The map $\varphi$ extends to an algebra homomorphism $T(\mathrm{G})\to A$ by the universal property of the tensor algebra, since $T(\mathrm{G})$ is the free associative algebra on the $R$-module $\mathrm{G}$. The relation $\varphi([x,y]) = \varphi(x)\varphi(y)-\varphi(y)\varphi(x)$ says exactly that this homomorphism kills the generators $x\otimes y-y\otimes x-[x,y]$ of the ideal, hence factors through the quotient; uniqueness follows because $\mathrm{G}$ generates $U(\mathrm{G})$ as an algebra. The bijection is natural in $\mathrm{G}$ and $A$ by construction.

Corollary (functoriality). A Lie algebra homomorphism $f:\mathrm{G}\to\mathrm{H}$ induces a unique algebra homomorphism $U(f):U(\mathrm{G})\to U(\mathrm{H})$, and $U$ is a functor from Lie algebras to associative algebras; $U$ preserves direct sums, $U(\mathrm{G}\oplus\mathrm{H})\cong U(\mathrm{G})\otimes_R U(\mathrm{H})$ as algebras, and the abelian Lie algebra of an $R$-module $M$ has $U(M) = \operatorname{Sym}(M)$, the symmetric algebra of The Symmetric Algebra, since all brackets vanish.

Proof. The functoriality is the universal property applied to $\iota_{\mathrm{H}}\circ f$. For the direct sum, the two inclusions produce algebra maps into $U(\mathrm{G}\oplus\mathrm{H})$ whose images commute, giving $U(\mathrm{G})\otimes U(\mathrm{H})\to U(\mathrm{G}\oplus\mathrm{H})$, and the inverse is the map induced by $x+y\mapsto x\otimes1+1\otimes y$, which is a Lie algebra homomorphism because $[x,y] = 0$ whenever $x$ and $y$ lie in different summands, so that the commutator of the two images is the same as the image of the bracket. The abelian case is the identification of the quotient of $T(M)$ by the commutators with $\operatorname{Sym}(M)$.

Remark (the kernel of $\iota$). The map $\iota$ need not be injective for an arbitrary Lie algebra over an arbitrary ring: over a ring in which the module $\mathrm{G}$ has torsion the quotient can identify elements, and for a Lie algebra over a field in characteristic $p$ with $p$-th powers there are elements of $U(\mathrm{G})$ outside the image of $\mathrm{G}$. The injectivity of $\iota$ over a field is the first consequence of the theorem of the next section; the existence of primitive elements beyond $\mathrm{G}$ in characteristic $p$ is described in the last section.

The Filtration and the Poincaré–Birkhoff–Witt Theorem

Definition. The degree filtration of $U(\mathrm{G})$ is the increasing filtration $U_0\subseteq U_1\subseteq U_2\subseteq\cdots$ with $U_n$ the image of $\bigoplus_{m\leq n}\mathrm{G}^{\otimes m}$; it is exhaustive, $U(\mathrm{G}) = \bigcup_n U_n$, and the product satisfies $U_mU_n\subseteq U_{m+n}$. The associated graded algebra is $\mathrm{gr}\,U(\mathrm{G}) = \bigoplus_n U_n/U_{n-1}$ with the product induced by the multiplication, and it is generated by $\mathrm{gr}_1 = \mathrm{G}$ and commutative, because the defining relations of $U(\mathrm{G})$ have degree one in the graded sense: the leading terms of $x\otimes y-y\otimes x$ vanish. Hence there is a surjective algebra homomorphism

$$ \sigma:\operatorname{Sym}(\mathrm{G})\longrightarrow \mathrm{gr}\,U(\mathrm{G}), $$

the symbol map, induced by the identity on $\mathrm{G}$; it is an isomorphism if and only if the ordered monomials of the next theorem are linearly independent.

Theorem (Poincaré–Birkhoff–Witt). Let $k$ be a field, let $\mathrm{G}$ be a Lie algebra over $k$ and let $(x_i)_{i\in I}$ be a basis of $\mathrm{G}$ ordered by a total order on $I$. Then the monomials

$$ x_{i_1}^{a_1}x_{i_2}^{a_2}\cdots x_{i_r}^{a_r}, \qquad i_1

together with the unit, form a $k$-basis of $U(\mathrm{G})$. Consequently the symbol map $\sigma:\operatorname{Sym}(\mathrm{G})\to\mathrm{gr}\,U(\mathrm{G})$ is an isomorphism of graded algebras, the canonical map $\iota:\mathrm{G}\to U(\mathrm{G})$ is injective, and $U(\mathrm{G})$ is a domain and, when $\mathrm{G}$ is finite-dimensional, a Noetherian ring.

Proof (outline). One shows that the ordered monomials span and are independent. For the spanning, one checks that the subspace spanned by them is closed under left multiplication by each $x_j$: the product $x_j\,x_{i_1}^{a_1}\cdots$ is brought to ordered form by the exchange move $x_jx_i = x_ix_j + [x_j,x_i]$ for $idiamond lemma of Bergman: the rewriting system consisting of the exchange moves is confluent because the two ways of resolving an overlap $x_kx_jx_i$ with $i

Corollary (the Poincaré series). Let $\mathrm{G}$ be finite-dimensional over $k$ of dimension $d$, and let $U_n$ be the $n$-th filtered piece. Then

$$ \dim U_n = \binom{n+d}{d}, \qquad \dim\mathrm{gr}_n U(\mathrm{G}) = \binom{n+d-1}{d-1}, $$

the first being the number of monomials of degree at most $n$ in $d$ indeterminates. In particular $U(\mathrm{G})$ and $\operatorname{Sym}(\mathrm{G})$ have the same Hilbert series, and the growth of the enveloping algebra is polynomial of degree $d-1$ in the graded pieces.

Proof. The ordered monomials of degree at most $n$ in the $d$ basis elements are in bijection with the multi-indices $(a_1,\dots,a_d)$ of total degree at most $n$, whose number is $\binom{n+d}{d}$ by the standard stars-and-bars count; the graded statement is the difference of consecutive values, $\binom{n+d}{d}-\binom{n+d-1}{d} = \binom{n+d-1}{d-1}$.

Corollary (the primitives in characteristic $p$). Over a field of characteristic $p>0$ the space of primitive elements of $U(\mathrm{G})$ is strictly larger than $\mathrm{G}$: it contains $x^p$ for every $x\in\mathrm{G}$, and it is closed under the $p$-operation $x\mapsto x^p$. When $\mathrm{G}$ is equipped with a $p$-map $[p]$ making it restricted, the elements $x^p-x^{[p]}$ are central, and the quotient by the ideal they generate is the restricted enveloping algebra of the last section.

Example. For the two-dimensional non-abelian Lie algebra with basis $x,y$ and $[x,y] = y$, the enveloping algebra is $k\langle x,y\rangle/(xy-yx-y)$ with the Poincaré–Birkhoff–Witt basis $x^my^n$, $m,n\geq0$, and its associated graded is the polynomial algebra on two generators. Sending $x$ to the operator $t\,\mathrm{d}/\mathrm{d}t$ and $y$ to multiplication by $t$ is a Lie algebra isomorphism onto a subalgebra of the first Weyl algebra, since $[t\,\mathrm{d}/\mathrm{d}t,t] = t$; the enveloping algebra of the two-dimensional non-abelian Lie algebra is therefore the universal associative algebra of the dilation and the translation of the line.

The Hopf Algebra Structure

Definition. Let $\mathrm{G}$ be a Lie algebra over $R$. Define maps on the generators by

$$ \Delta(x) = x\otimes1+1\otimes x, \qquad \varepsilon(x) = 0, \qquad S(x) = -x, $$

extended to algebra homomorphisms $\Delta:U(\mathrm{G})\to U(\mathrm{G})\otimes U(\mathrm{G})$ and $\varepsilon:U(\mathrm{G})\to R$, and to the anti-homomorphism $S:U(\mathrm{G})\to U(\mathrm{G})$. The elements $x\in\mathrm{G}$ are primitive, $\Delta(x) = x\otimes1+1\otimes x$, and the conditions $(\varepsilon\otimes\mathrm{id})\Delta = \mathrm{id} = (\mathrm{id}\otimes\varepsilon)\Delta$ and $m(S\otimes\mathrm{id})\Delta = \eta\varepsilon = m(\mathrm{id}\otimes S)\Delta$ hold, where $m$ is the multiplication and $\eta$ the unit.

Theorem. With these maps $U(\mathrm{G})$ is a Hopf algebra in the sense of Hopf Algebras, cocommutative, with augmentation ideal $\mathrm{G}U(\mathrm{G}) = U(\mathrm{G})\mathrm{G} = \ker\varepsilon$, and the coproduct is filtered: $\Delta(U_n)\subseteq\sum_{i+j=n}U_i\otimes U_j$. The algebra $U(\mathrm{G})$ is generated as an algebra by its primitive elements, and the set of primitive elements is $\mathrm{G}$.

Proof. The maps are well defined because they respect the relations: $\Delta$ takes $xy-yx-[x,y]$ to the commutator of the images minus $\Delta([x,y])$, which vanishes by the computation $\Delta(x)\Delta(y)-\Delta(y)\Delta(x) = [x,y]\otimes1+1\otimes[x,y]$; the coassociativity and the counit and antipode axioms are checked on generators, where they reduce to the corresponding statements for the tensor algebra. The filtration statement is the homogeneity of $\Delta$. For the primitives: if $u\in U(\mathrm{G})$ is primitive, write $u = u_0+u_1+\cdots$ with $u_i$ of filtration degree $i$; the equality $\Delta(u) = u\otimes1+1\otimes u$ in the filtration gives in degree one $u_1\in\mathrm{G}$ and in higher degrees forces $u_i = 0$ when the characteristic is zero, since $\Delta$ multiplies the degree while the right-hand side has the degree-one part concentrated in the two terms; hence the primitives are exactly $\mathrm{G}$ in characteristic zero.

Corollary. The category of $\mathrm{G}$-modules is the category of modules over the cocommutative Hopf algebra $U(\mathrm{G})$, and the tensor product of two $\mathrm{G}$-modules is again a $\mathrm{G}$-module by the coproduct, the action being $x\cdot(v\otimes w) = xv\otimes w+v\otimes xw$; the dual of a finite-dimensional module is a module via the antipode, and the isomorphism $U(\mathrm{G}\oplus\mathrm{H})\cong U(\mathrm{G})\otimes U(\mathrm{H})$ is an isomorphism of Hopf algebras.

Pro. The module structure transported through the coproduct is the standard construction of Hopf Algebras: the axioms of a module over a Hopf algebra reproduce the derivation property $x\cdot(v\otimes w) = (xv)\otimes w+v\otimes(xw)$ that makes the tensor product of two representations a representation. The Hopf-algebra statement for the direct sum follows since $\Delta$ is the tensor product of the two coproducts.

Remark (the characteristic-$p$ correction). In characteristic $p>0$ the space of primitives of $U(\mathrm{G})$ is larger than $\mathrm{G}$: it is the restricted Lie algebra generated by $\mathrm{G}$ with the $p$-operation $x\mapsto x^p$, that is the primitive elements of degree $p$ are $x^p$ and its linear combinations, and the augmentation filtration of $U(\mathrm{G})$ has a divided-power structure. The statement that a cocommutative Hopf algebra over a field of characteristic zero is the enveloping algebra of its primitives is the theorem of Milnor–Moore; in characteristic $p$ it holds for the restricted or the divided-power version, and the corresponding algebra is the restricted enveloping algebra $u(\mathrm{G}) = U(\mathrm{G})/(x^p-x^{[p]})$.

Representations and the Equivalence of Categories

Definition. A representation of the Lie algebra $\mathrm{G}$ on an $R$-module $M$ is an $R$-linear map $\rho:\mathrm{G}\to\operatorname{End}_R(M)$ with $\rho([x,y]) = [\rho(x),\rho(y)]$. A $\mathrm{G}$-module is such a pair $(M,\rho)$; a morphism is an $R$-linear map commuting with the actions.

Theorem (the equivalence). The universal property induces an equivalence of categories

$$ \mathsf{Rep}_R(\mathrm{G}) \;\simeq\; \mathsf{Mod}_{U(\mathrm{G})}, $$

between the representations of the Lie algebra and the left modules over its enveloping algebra; the equivalence is the identity on the underlying $R$-modules and sends a representation $\rho$ to the module structure $u\cdot m = \Phi_\rho(u)m$ obtained from the universal property.

Proof. Given a representation, the universal property applied to $\rho:\mathrm{G}\to\operatorname{End}_R(M)$ gives an algebra homomorphism $U(\mathrm{G})\to\operatorname{End}_R(M)$, that is a module structure; conversely a module structure restricts to a representation, and the two constructions are mutually inverse because $\mathrm{G}$ generates $U(\mathrm{G})$. Naturality is immediate.

Corollary. Subrepresentations are submodules, irreducible representations are simple modules, the tensor product, the direct sum, the dual of a finite-dimensional representation and the Hom functor are those ; and a Lie algebra representation of $\mathrm{G}$ on $M$ of finite rank over a field extends to the enveloping algebra, so that every statement about the representation theory of $\mathrm{G}$ is a statement about the module category of $U(\mathrm{G})$.

Example (the adjoint representation). The bracket $\mathrm{ad}:\mathrm{G}\to\operatorname{End}(\mathrm{G})$ is a representation, since $\mathrm{ad}([x,y]) = [\mathrm{ad}_x,\mathrm{ad}_y]$ is the Jacobi identity; the corresponding $U(\mathrm{G})$-module is $\mathrm{G}$ with $u\cdot x = \mathrm{ad}_u(x)$, and the invariants of this module are the centre $\mathrm{Z}(\mathrm{G})$ of the Lie algebra. The symmetric and the exterior powers of the adjoint representation are the modules that carry the trace forms, and the Casimir element of the next section is the invariant element of $U(\mathrm{G})$ of degree two acting on them.

The Centre and the Casimir Element

Definition. Let $\mathrm{G}$ be finite-dimensional and semisimple over a field $k$ of characteristic zero, and let $\langle-,-\rangle$ be a non-degenerate invariant symmetric bilinear form on $\mathrm{G}$ — the Killing form $\langle x,y\rangle = \operatorname{tr}(\mathrm{ad}_x\mathrm{ad}_y)$, or the trace form of any faithful finite-dimensional representation. Let $(x_i)$ and $(y_i)$ be dual bases, $\langle x_i,y_j\rangle = \delta_{ij}$. The Casimir element is

$$ \Omega \;=\; \sum_i x_iy_i \;\in\; U(\mathrm{G}). $$

Theorem. The Casimir element is independent of the choice of the dual bases, it lies in the centre of $U(\mathrm{G})$, and it acts on an irreducible $\mathrm{G}$-module of highest weight $\lambda$ by the scalar $\langle\lambda,\lambda+2\rho\rangle$, where $\rho$ is the half-sum of the positive roots.

Proof. For the independence and the centrality: if $z\in\mathrm{G}$, then $[z,x_i] = \sum_j\langle[z,x_i],y_j\rangle x_j$ and $[z,y_i] = \sum_j\langle[z,y_i],x_j\rangle y_j$ by the expansion of an element in the dual basis, so

$$ [z,\Omega] = \sum_i[z,x_i]y_i+\sum_ix_i[z,y_i] = \sum_{i,j}\Bigl(\langle[z,x_i],y_j\rangle+\langle x_i,[z,y_j]\rangle\Bigr)x_jy_i = 0, $$

the bracket in the second factor having been rewritten by the invariance $\langle[z,x_i],y_j\rangle+\langle x_i,[z,y_j]\rangle = 0$ of the form. The eigenvalue statement is computed on a highest weight vector $v_\lambda$, which is killed by the root vectors attached to the positive roots: with $\Omega$ written as the sum of the terms $x_\alpha y_{-\alpha}$ over the positive roots together with the terms of a basis of the Cartan, the first group contributes the shift by $2\rho$ through the commutation relations and the second contributes the value of the weight, so that the total is $\langle\lambda,\lambda+2\rho\rangle$; this is the standard computation of the eigenvalue of the Casimir operator, recorded in the representation theory of semisimple Lie algebras.

Theorem (Harish-Chandra, standard). Let $\mathrm{G}$ be semisimple over a field of characteristic zero, let $\mathrm{H}\subseteq\mathrm{G}$ be a Cartan subalgebra and let $W$ be the Weyl group of the corresponding root system. Then the Harish-Chandra homomorphism restricts central elements to the Cartan and gives an isomorphism

$$ Z(U(\mathrm{G})) \;\cong\; \operatorname{Sym}(\mathrm{H})^W $$

onto the Weyl-group invariants, with the twist by $\rho$ understood: the isomorphism is the composite of the projection $Z(U(\mathrm{G}))\to U(\mathrm{H}) = \operatorname{Sym}(\mathrm{H})$ with the automorphism of $\operatorname{Sym}(\mathrm{H})$ induced by the translation by $\rho$. Consequently $Z(U(\mathrm{G}))$ is a polynomial algebra on $\operatorname{rank}\mathrm{G}$ generators, and for $\mathrm{G} = \mathrm{SL}_2$ it is the polynomial algebra on the Casimir element alone.

Proof (outline). The projection of a central element to the Cartan subalgebra is well defined because the decomposition of $U(\mathrm{G})$ into the subalgebra generated by the negative root vectors, the Cartan and the subalgebra generated by the positive root vectors is a triangular decomposition, and a central element lies in the intersection of the two opposite Borel parts, which is the Cartan part; the image lies in the $W$-invariants because the action of the Weyl group on the Cartan is realised by conjugation by elements of $U(\mathrm{G})$ up to the twist by $\rho$. The map is injective because a central element with zero image has zero action on the Verma modules of generic highest weight, and it is surjective because the symmetrised monomials in a basis of $\mathrm{H}$ produce central elements realising any $W$-invariant polynomial. The polynomial-algebra statement is the theorem of Chevalley on the invariants of a finite reflection group, recorded.

Proposition. Let $\mathrm{G}$ be semisimple of characteristic zero with Casimir element $\Omega$ normalised by the trace form of a faithful representation of highest weight $\theta$. Then $\Omega$ acts on the adjoint representation by the scalar $2h^\vee$, the dual Coxeter number, and for $\mathrm{SL}_2$ by $4$; on the defining representation of $\mathrm{SL}_2$ it acts by $\frac32$, and on the irreducible module of highest weight $m$ by $\frac{m(m+2)}{2}$.

Proof. The adjoint statement is the eigenvalue formula with $\lambda = \theta$ and $\langle\theta,\theta+2\rho\rangle = 2h^\vee$, the standard evaluation for the highest root. For $\mathrm{SL}_2$ with the standard basis $e,f,h$ and $\Omega = ef+fe+\frac12h^2$: on a highest weight vector $v$ of weight $m$, so that $ev = 0$ and $hv = mv$, one has $fev = 0$ and $efv = (fe+h)v = 0+mv = mv$, using $[e,f] = ef-fe = h$; hence $\Omega v = (0+m+\frac12m^2)v$ and the eigenvalue is $\frac{m(m+2)}{2}$. For the defining representation $m = 1$ this is $\frac32$, computed directly from the matrices, $ef+fe+\frac12h^2 = \frac32\,\mathrm{id}$.

The Example of $\mathrm{SL}_2$

Example. Let $\mathrm{G} = \mathrm{SL}_2(k)$ with the basis $e,f,h$ and the relations

$$ [h,e] = 2e, \qquad [h,f] = -2f, \qquad [e,f] = h . $$

By the Poincaré–Birkhoff–Witt theorem the monomials $e^af^bh^c$ and, equally, the ordered monomials in any total order, form a basis of $U(\mathrm{SL}_2)$; the algebra is a domain, is Noetherian, has Gelfand–Kirillov dimension $3$, and its centre is $k[\Omega]$ with $\Omega = ef+fe+\frac12h^2$. The Casimir relations

$$ [\Omega,e] = [\Omega,f] = [\Omega,h] = 0 $$

are verified by direct computation from the relations: for instance

$$ [\Omega,e] = [ef+fe,e]+\tfrac12[h^2,e] = -(eh+he)+(he+eh) = 0, $$

using $[ef,e] = -eh$, $[fe,e] = -he$ and $[h^2,e] = 2(he+eh)$.

Proposition. For $\mathrm{SL}_2$ over a field of characteristic zero, the irreducible modules are the modules $L(m)$, $m\geq0$, of dimension $m+1$, with highest weight vector $v$ satisfying $ev = 0$, $hv = mv$ and $L(m) = \bigoplus_{i=0}^{m}k\,f^iv$; the Casimir acts on $L(m)$ by $\frac{m(m+2)}{2}$; the tensor product decomposes as $L(m)\otimes L(n)\cong\bigoplus_{j}L(m+n-2j)$, the sum over $j$ with $0\leq j\leq\min(m,n)$, the Clebsch–Gordan rule; and the universal highest weight object is the Verma module $M(m) = U(\mathrm{SL}_2)\otimes_{U(\mathrm{B})}k_m$, where $\mathrm{B} = \mathrm{H}\oplus ke$ is the Borel subalgebra and $k_m$ is the one-dimensional $\mathrm{B}$-module on which $h$ acts by $m$ and $e$ by zero.

Pro. The classification of the irreducible modules is the standard highest-weight theory of $\mathrm{SL}_2$, recorded, and the module $L(m)$ is the unique irreducible quotient of $M(m)$; the Clebsch–Gordan rule is the classical decomposition, proved by the standard computation of the highest weights occurring in the tensor product; the Casimir eigenvalue is the proposition of the previous section. The Verma module is a free $U(\mathrm{N}^-)$-module on one generator, with $\mathrm{N}^- = kf$, and its Jordan–Hölder factors are the $L(m-2i)$, a statement used in the cohomology.

Remark (the filtration and the deformation). The associated graded of $U(\mathrm{SL}_2)$ is the polynomial algebra $k[e,f,h]$, and the passage from the symmetric algebra to the enveloping algebra is a flat deformation of Poisson algebras: the Lie–Poisson structure on $\operatorname{Sym}(\mathrm{G}) = k[e,f,h]$ defined by $\{a,b\} = $ the leading term of $[\tilde a,\tilde b]$ makes $\mathrm{gr}\,U(\mathrm{G})$ a Poisson algebra, and the enveloping algebra is the deformation whose semi-classical limit is that Poisson structure. The algebraic statement is recorded here; the geometric form of the statement — that the symmetric algebra of the dual of a Lie algebra is the algebra of functions on a Poisson manifold — is the subject of Deformation Quantization, where the manifold is available, and is not used.

Characteristic $p$ and Restricted Enveloping Algebras

Definition. Let $k$ be a field of characteristic $p>0$. A restricted Lie algebra is a Lie algebra $\mathrm{G}$ together with a map $x\mapsto x^{[p]}$ satisfying the axioms $(\lambda x)^{[p]} = \lambda^p x^{[p]}$, $[x^{[p]},y] = [x,[x,\dots,[x,y]\cdots]]$ with $p$ brackets, and $(x+y)^{[p]} = x^{[p]}+y^{[p]}+\sum_{i=1}^{p-1}s_i(x,y)$ with $s_i$ the standard polynomials. The restricted enveloping algebra is

$$ u(\mathrm{G}) = U(\mathrm{G})\big/\bigl\langle x^p-x^{[p]} : x\in\mathrm{G}\bigr\rangle . $$

Theorem. Let $\mathrm{G}$ be a finite-dimensional restricted Lie algebra over $k$ of characteristic $p$, of dimension $d$. Then $u(\mathrm{G})$ has dimension $p^d$, with basis the monomials $x_1^{a_1}\cdots x_d^{a_d}$, $0\leq a_i

Proof (outline). The relations $x^p = x^{[p]}$ make every monomial reducible to one of the displayed form, giving the spanning statement and the bound $p^d$; the independence is the restricted Poincaré–Birkhoff–Witt theorem, proved by the same rewriting argument as in characteristic zero with the additional reductions by the $p$-relations. The centrality of $x^p-x^{[p]}$ is the computation $[y,x^p-x^{[p]}] = \mathrm{ad}_y(x)^p-[y,x^{[p]}] = 0$, valid in characteristic $p$ because $\mathrm{ad}_y$ is a derivation and the $p$-th power of a derivation in characteristic $p$ is again a derivation, and equal to the bracket with $x^{[p]}$ by the defining axiom of the restricted structure applied to the inner derivation $\mathrm{ad}_x$; the simple-module classification for $\mathrm{SL}_2$ is the standard theory of the restricted representations of the three-dimensional algebra, in which the Steinberg module $L(p-1)$ of dimension $p$ plays the role of the projective generator.

Remark. The restricted theory is needed for the representation theory of semisimple Lie algebras in characteristic $p$ and for the structure of the finite groups of Lie type, where the restricted enveloping algebra is the small algebra of the theory and the Frobenius twist is the operation distinguishing the modules of the same dimension. The unrestricted representation theory in characteristic $p$, with its infinite-dimensional simple modules, is the subject that requires the analysis of the "centre" of $U(\mathrm{G})$ by the methods, and the arithmetic statements about the finite groups of Lie type belong to Part III, where the finite group and its characters are developed.

Summary

Let $\mathrm{G}$ be a Lie algebra over a commutative ring $R$. The universal enveloping algebra $U(\mathrm{G}) = T(\mathrm{G})/\langle x\otimes y-y\otimes x-[x,y]\rangle$ is the associative algebra universal for the property that the bracket becomes the commutator; it is the left adjoint of the commutator functor, it is functorial, and it preserves direct sums. It carries the degree filtration, whose associated graded receives the symbol map $\sigma:\operatorname{Sym}(\mathrm{G})\to\mathrm{gr}\,U(\mathrm{G})$, which the Poincaré–Birkhoff–Witt theorem shows to be an isomorphism: over a field, the ordered monomials in a basis of $\mathrm{G}$ form a basis of $U(\mathrm{G})$, the map $\mathrm{G}\to U(\mathrm{G})$ is injective, and $U(\mathrm{G})$ is a domain, Noetherian when $\mathrm{G}$ is finite-dimensional, with $\dim U_n = \binom{n+d}{d}$ and $\dim\mathrm{gr}_n U = \binom{n+d-1}{d-1}$ for $d = \dim\mathrm{G}$. With the coproduct $\Delta(x) = x\otimes1+1\otimes x$, the counit $\varepsilon(x) = 0$ and the antipode $S(x) = -x$ it is a cocommutative Hopf algebra, whose primitive elements are exactly $\mathrm{G}$ in characteristic zero, the tensor product of representations being given by the coproduct. The representations of $\mathrm{G}$ are the modules over $U(\mathrm{G})$, an equivalence of categories. For $\mathrm{G}$ semisimple of characteristic zero, the Casimir element $\Omega = \sum x_iy_i$ built from dual bases for an invariant form lies in the centre and acts on the irreducible module of highest weight $\lambda$ by $\langle\lambda,\lambda+2\rho\rangle$, and the Harish-Chandra isomorphism identifies the centre with the Weyl-group invariants $\operatorname{Sym}(\mathrm{H})^W$, a polynomial algebra on $\operatorname{rank}\mathrm{G}$ generators. For $\mathrm{SL}_2$ the centre is $k[\Omega]$, the irreducible modules are $L(m)$ of dimension $m+1$ with Casimir eigenvalue $\frac{m(m+2)}{2}$, the tensor products obey Clebsch–Gordan, and the Verma modules $M(m)$ are free of rank one over $U(kf)$. In characteristic $p$ the enveloping algebra is modified by the $p$-operation to the restricted enveloping algebra $u(\mathrm{G})$ of dimension $p^{\dim\mathrm{G}}$, whose modules are the restricted representations; the analytic completion, the formal group and the geometric reading of the deformation are deferred to Parts III and II respectively.

Summary of Notation

Symbol	Meaning
$\mathrm{G},\mathrm{H},\mathrm{B},\mathrm{N}$	Lie algebras; Borel and nilpotent subalgebras
$T(\mathrm{G})$	tensor algebra on $\mathrm{G}$
$\operatorname{Sym}(\mathrm{G})$	symmetric algebra on $\mathrm{G}$
$U(\mathrm{G})$	universal enveloping algebra, $T(\mathrm{G})/\langle x\otimes y-y\otimes x-[x,y]\rangle$
$\iota:\mathrm{G}\to U(\mathrm{G})$	canonical map, injective over a field
$U_n(\mathrm{G})$, $\mathrm{gr}\,U(\mathrm{G})$	degree filtration, associated graded
$\sigma$	symbol map $\operatorname{Sym}(\mathrm{G})\to\mathrm{gr}\,U(\mathrm{G})$
$\Delta,\varepsilon,S$	coproduct, counit, antipode of the Hopf algebra
$A_{\mathrm{Lie}}$	commutator Lie algebra of an associative algebra $A$
$\langle-,-\rangle$	invariant symmetric form on $\mathrm{G}$ (Killing or trace form)
$\Omega$	Casimir element, $\Omega = \sum_ix_iy_i$
$Z(U(\mathrm{G}))$	centre of the enveloping algebra
$\rho$, $W$, $\mathrm{H}$	half-sum of positive roots, Weyl group, Cartan subalgebra
$M(\lambda)$, $L(\lambda)$	Verma module, irreducible highest weight module
$x^{[p]}$, $u(\mathrm{G})$	$p$-operation, restricted enveloping algebra in characteristic $p$
$\mathrm{ad}_x(y) = [x,y]$	adjoint map