Lie Algebra Actions as Internal Algebraic Structures

Lie Algebra Actions as Internal Algebraic Structures

Derivations as the one-dimensional case

What does it mean for a Lie algebra to act on an algebraic structure?

For an associative algebra, we expect an action by derivations. For a Lie algebra, the action should preserve the bracket infinitesimally. For a Hopf algebra, it should also preserve the coproduct. These requirements have a common interpretation:

An algebraic structure with a compatible Lie algebra action is an algebraic structure internal to the corresponding representation category.

We will formulate this precisely for PROP models. Derivations will be defined through algebraic automorphism groups, and the usual Leibniz identities will follow from equivariance. A single derivation will then appear by specializing the acting Lie algebra to dimension one.

Throughout, $k$$k$ is an algebraically closed field of characteristic zero, and tensor products are over $k$$k$. We write ${\mathrm{Rep}}_k(\mathfrak{l})$$\operatorname{Rep}_k(\mathfrak l)$ for all representations of a Lie algebra $\mathfrak{l}$$\mathfrak l$, and use the superscript $\mathrm{fd}$$\mathrm{fd}$ for finite-dimensional representations.

The symmetric monoidal category of representations

The universal enveloping algebra $U(\mathfrak{l})$$U(\mathfrak l)$ has a Hopf algebra structure determined by

$$\mathrm{\Delta }(\xi )=\xi \otimes 1+1\otimes \xi ,ϵ(\xi )=0,S(\xi )=-\xi (\xi \in \mathfrak{l}).$$

The coproduct and counit extend as algebra homomorphisms, and the antipode extends as an algebra antihomomorphism. This Hopf algebra is cocommutative, so its module category is symmetric monoidal.

The equivalence between Lie algebra representations and modules over the enveloping algebra is a strong symmetric monoidal equivalence:

$${\mathrm{Rep}}_k(\mathfrak{l})\simeq U(\mathfrak{l})\text{-}\mathrm{Mod}.$$

Explicitly, the tensor product action is

$${\rho }_{V\otimes W}(\xi )={\rho }_V(\xi )\otimes 1+1\otimes {\rho }_W(\xi ).$$

The tensor unit is $k$$k$ with the zero Lie algebra action, and the symmetry is the usual flip $v\otimes w\mapsto w\otimes v$$v\otimes w\mapsto w\otimes v$. These structures preserve finite dimensionality.

The tensor product here is ${\otimes }_k$$\otimes_k$ with its diagonal action. In particular, for a polynomial enveloping algebra it is not the relative tensor product over that algebra.

The forgetful functor to vector spaces is also strong symmetric monoidal. Consequently, an algebraic structure internal to ${\mathrm{Rep}}_k(\mathfrak{l})$$\operatorname{Rep}_k(\mathfrak l)$ has an underlying algebraic structure in vector spaces, and all its structural maps must be $\mathfrak{l}$$\mathfrak l$-equivariant.

PROP models and their infinitesimal symmetries

Let $\mathcal{P}$$\mathcal P$ be a one-coloured $k$$k$-linear PROP. Its objects are the natural numbers, with tensor product given by addition. A model in a symmetric monoidal $k$$k$-linear category $\mathcal{C}$$\mathcal C$ is a $k$$k$-linear strong symmetric monoidal functor

$$A:\mathcal{P}⟶\mathcal{C}.$$

Morphisms of models are monoidal natural transformations.

For a model in vector spaces, write $V=A(1)$$V=A(1)$. The monoidal structure identifies $A(n)$$A(n)$ with $V^{\otimes n}$$V^{\otimes n}$, so each operation $p:m\to n$$p:m\to n$ gives a linear map

$$A(p):V^{\otimes m}⟶V^{\otimes n}.$$

A model morphism $f:A\to B$$f:A\to B$ is determined by a linear map $f:V\to W$$f:V\to W$ satisfying

$$f^{\otimes n}A(p)=B(p)f^{\otimes m}$$

for every $p:m\to n$$p:m\to n$. We use $V^{\otimes 0}=k$$V^{\otimes0}=k$ and $f^{\otimes 0}={\mathrm{id}}_k$$f^{\otimes0}=\operatorname{id}_k$.

Suppose now that $V$$V$ is finite-dimensional. For a commutative $k$$k$-algebra $R$$R$, let $A_R$$A_R$ be the scalar extension of the model and define

$$G_A(R)={\mathrm{Aut}}_{{\mathcal{P}}_R}(A_R).$$

Equivalently,

$$G_A(R)=\{g\in {\mathrm{GL}}_R(V_R)|g^{\otimes n}{A}_R(p)=A_R(p)g^{\otimes m}\text{ for every }p:m\to n\}.$$

These are polynomial equations in the entries of $g$$g$, so $G_A$$G_A$ is a closed subgroup scheme of $\mathrm{GL}(V)$$\operatorname{GL}(V)$. In characteristic zero it is a linear algebraic group, which we denote by ${\underset{―}{\mathrm{Aut}}}_{\mathcal{P}}(A)$$\underline{\operatorname{Aut}}_{\mathcal P}(A)$.

Definition. The Lie algebra of derivations of $A$$A$ is

$${\mathrm{Der}}_{\mathcal{P}}(A):=\mathrm{Lie}{\underset{―}{\mathrm{Aut}}}_{\mathcal{P}}(A).$$

A compatible infinitesimal action of $\mathfrak{l}$$\mathfrak l$ on $A$$A$ is a Lie algebra homomorphism

$$\rho :\mathfrak{l}⟶{\mathrm{Der}}_{\mathcal{P}}(A).$$

No integration to an algebraic group action is assumed.

To unpack the definition, use the dual numbers $R_{\epsilon }=k[\epsilon ]/({\epsilon }^2)$$R_\varepsilon=k[\varepsilon]/(\varepsilon^2)$. The Lie algebra of $G_A$$G_A$ is its tangent space at the identity:

$$\mathrm{Lie}(G_A)=\ker (G_A(R_{\epsilon })⟶G_A(k)).$$

Its elements have the form $\mathrm{id}+\epsilon d$$\operatorname{id}+\varepsilon d$, with inverse $\mathrm{id}-\epsilon d$$\operatorname{id}-\varepsilon d$. This identifies ${\mathrm{Der}}_{\mathcal{P}}(A)$$\operatorname{Der}_{\mathcal P}(A)$ with a Lie subalgebra of ${\mathrm{End}}_k(V)$$\operatorname{End}_k(V)$, using the commutator bracket.

For $r\ge 1$$r\geq1$, put

$$d_{(r)}=\sum _{i=1}^r{1}^{\otimes (i-1)}\otimes d\otimes 1^{\otimes (r-i)},d_{(0)}=0.$$

Proposition. An endomorphism $d$$d$ is a $\mathcal{P}$$\mathcal P$-derivation if and only if

$$\begin{array}{}\text{(1)}& \boxed{{\displaystyle d_{(n)}A(p)=A(p)d_{(m)}}}\end{array}$$

for every operation $p:m\to n$$p:m\to n$.

Proof. Over the dual numbers,

$$(\mathrm{id}+\epsilon d{)}^{\otimes r}=\mathrm{id}+\epsilon d_{(r)}.$$

Thus preservation of $p$$p$ by $\mathrm{id}+\epsilon d$$\operatorname{id}+\varepsilon d$ is equivalent to

$$(\mathrm{id}+\epsilon d_{(n)})A(p)=A(p)(\mathrm{id}+\epsilon d_{(m)}).$$

Comparing the coefficients of $\epsilon$$\varepsilon$ gives (1). Conversely, these equations make $\mathrm{id}+\epsilon d$$\operatorname{id}+\varepsilon d$ an automorphism of the entire model. $◻$$\square$

The Leibniz rule is therefore an infinitesimal equivariance condition: differentiating all output factors equals differentiating all input factors.

Actions are internal models

Let ${\mathrm{Act}}_{\mathfrak{l}}(\mathcal{P}{)}^{\mathrm{fd}}$$\operatorname{Act}_{\mathfrak l}(\mathcal P)^{\mathrm{fd}}$ be the category of pairs

$$(A,\rho ),\rho :\mathfrak{l}⟶{\mathrm{Der}}_{\mathcal{P}}(A),$$

where $A(1)$$A(1)$ is finite-dimensional. A morphism $(A,\rho )\to (B,\sigma )$$(A,\rho)\to(B,\sigma)$ is a model morphism $f:A\to B$$f:A\to B$ such that

$$f\rho (\xi )=\sigma (\xi )f(\xi \in \mathfrak{l}).$$

Theorem. There are equivalences of categories

$$\begin{array}{c}\boxed{{\displaystyle \begin{array}{rl}{\mathrm{Mod}}_{\mathcal{P}}({\mathrm{Rep}}_k^{\mathrm{fd}}(\mathfrak{l}))& \simeq {\mathrm{Act}}_{\mathfrak{l}}(\mathcal{P}{)}^{\mathrm{fd}}\\ & \simeq {\mathrm{Mod}}_{\mathcal{P}}(U(\mathfrak{l})\text{-}{\mathrm{Mod}}^{\mathrm{fd}}).\end{array}}}\end{array}$$

Proof. An internal model assigns a representation $(V,\rho )$$(V,\rho)$ to $1$$1$. On $V^{\otimes r}$$V^{\otimes r}$, the element $\xi \in \mathfrak{l}$$\xi\in\mathfrak l$ acts as $\rho (\xi {)}_{(r)}$$\rho(\xi)_{(r)}$.

Every operation $A(p):V^{\otimes m}\to V^{\otimes n}$$A(p):V^{\otimes m}\to V^{\otimes n}$ must be equivariant, meaning

$$\rho (\xi {)}_{(n)}A(p)=A(p)\rho (\xi {)}_{(m)}.$$

By (1), this says precisely that $\rho (\xi )$$\rho(\xi)$ belongs to ${\mathrm{Der}}_{\mathcal{P}}(A)$$\operatorname{Der}_{\mathcal P}(A)$. The representation identity

$$\rho ([\xi ,\eta ])=[\rho (\xi ),\rho (\eta )]$$

then makes $\rho$$\rho$ a Lie algebra homomorphism into that derivation Lie algebra.

Conversely, such a homomorphism equips every tensor power with an action making all operations equivariant. The existing PROP relations remain valid, so the model lifts to the representation category. The descriptions of morphisms agree.

The equivalence with models in $U(\mathfrak{l})$$U(\mathfrak l)$-modules follows by transporting models along the strong symmetric monoidal equivalence above. $◻$$\square$

Thus we may begin with a representation and equip it with an internal algebraic structure, or begin with an algebraic structure and equip it with a compatible infinitesimal action.

Associative, Lie, and Hopf algebra objects

Associative algebras. Equivariance of multiplication $\mu :A\otimes A\to A$$\mu:A\otimes A\to A$ means

$$\rho (\xi )(ab)=\rho (\xi )(a)b+a\rho (\xi )(b).$$

Equivariance of the unit means $\rho (\xi )(1_A)=0$$\rho(\xi)(1_A)=0$. Hence monoid objects in ${\mathrm{Rep}}_k(\mathfrak{l})$$\operatorname{Rep}_k(\mathfrak l)$ are associative unital algebras on which $\mathfrak{l}$$\mathfrak l$ acts by algebra derivations.

Lie algebras. An internal Lie algebra is a Lie algebra $\mathfrak{a}$$\mathfrak a$ with an action satisfying

$$\rho (\xi )[a,b]=[\rho (\xi )a,b]+[a,\rho (\xi )b].$$

It is therefore exactly a Lie algebra equipped with a homomorphism

$$\mathfrak{l}⟶{\mathrm{Der}}_{\mathrm{Lie}}(\mathfrak{a}).$$

The acting Lie algebra $\mathfrak{l}$$\mathfrak l$ and the internal Lie algebra $\mathfrak{a}$$\mathfrak a$ play different roles.

For example, take $\mathfrak{a}=\mathfrak{l}$$\mathfrak a=\mathfrak l$ with its adjoint action. The required equivariance becomes

$$[\xi ,[a,b]]=[[\xi ,a],b]+[a,[\xi ,b]],$$

which is the Jacobi identity. Thus a finite-dimensional Lie algebra, with its adjoint action, is a Lie algebra object in its own finite-dimensional representation category.

Hopf algebras. For a Hopf algebra $B$$B$, write $d_{\xi }=\rho (\xi )$$d_\xi=\rho(\xi)$. Equivariance of its structural maps gives

$$\begin{array}{rlrl}{d}_{\xi }\mu & =\mu (d_{\xi }\otimes 1+1\otimes d_{\xi }),& d_{\xi }\eta & =0,\\ {\mathrm{\Delta }}_B{d}_{\xi }& =(d_{\xi }\otimes 1+1\otimes d_{\xi }){\mathrm{\Delta }}_B,& {ϵ}_B{d}_{\xi }& =0,\\ d_{\xi }{S}_B& =S_B{d}_{\xi }.\end{array}$$

Every $d_{\xi }$$d_\xi$ is both a derivation and a coderivation. Compatibility with the antipode follows from the other identities: $\mathrm{id}+\epsilon d_{\xi }$$\operatorname{id}+\varepsilon d_\xi$ is then a bialgebra automorphism, which necessarily commutes with the unique antipode.

The Hopf algebra $B$$B$ need not be cocommutative. The cocommutativity that makes the ambient category symmetric belongs to $U(\mathfrak{l})$$U(\mathfrak l)$.

One-dimensional symmetry is a derivation

Now specialize to the one-dimensional Lie algebra $\mathfrak{l}=kx$$\mathfrak l=kx$, with chosen basis $x$$x$ and zero bracket.

A representation is determined by a single endomorphism

$$d=\rho (x).$$

Conversely, any endomorphism defines such a representation. Likewise, a homomorphism

$$kx⟶{\mathrm{Der}}_{\mathcal{P}}(A)$$

is determined by a single $\mathcal{P}$$\mathcal P$-derivation $d$$d$.

Let ${\mathrm{Diff}}_k$$\operatorname{Diff}_k$ be the category of pairs $(V,d)$$(V,d)$, with morphisms $f:(V,d)\to (W,e)$$f:(V,d)\to(W,e)$ satisfying $fd=ef$$fd=ef$. Its tensor product and unit are

$$(V,d)\otimes (W,e)=(V\otimes W,d\otimes 1+1\otimes e),\mathbf{1}=(k,0).$$

There is no condition that $d^2=0$$d^2=0$.

Since $U(kx)=k[x]$$U(kx)=k[x]$, we obtain strong symmetric monoidal equivalences

$$\boxed{{\displaystyle {\mathrm{Rep}}_k(kx)\simeq k[x]\text{-}\mathrm{Mod}\simeq {\mathrm{Diff}}_k,}}$$

where the polynomial Hopf algebra has

$$\mathrm{\Delta }(x)=x\otimes 1+1\otimes x,ϵ(x)=0,S(x)=-x.$$

The action of a polynomial is $p(x)\cdot v=p(d)(v)$$p(x)\cdot v=p(d)(v)$.

Writing ${\mathrm{DiffMod}}_{\mathcal{P}}^{\mathrm{fd}}$$\operatorname{DiffMod}_{\mathcal P}^{\mathrm{fd}}$ for finite-dimensional models with one derivation and model morphisms commuting with it, the theorem specializes to

$$\boxed{{\displaystyle {\mathrm{Mod}}_{\mathcal{P}}({\mathrm{Diff}}_k^{\mathrm{fd}})\simeq {\mathrm{DiffMod}}_{\mathcal{P}}^{\mathrm{fd}}.}}$$

Here are three concrete instances.

An associative algebra. Let

$$A=k[t]/(t^3),d=t\frac{d}{dt}.$$

The operator preserves $(t^3)$$(t^3)$, so it descends to the quotient:

$$d(1)=0,d(t)=t,d(t^2)=2t^2.$$

For example,

$$d(t\cdot t)=2t^2=d(t)t+td(t).$$

Thus $(A,d)$$(A,d)$ is a monoid object in ${\mathrm{Diff}}_k$$\operatorname{Diff}_k$. Its infinitesimal automorphism is obtained from the algebra automorphisms ${\varphi }_a(t)=at$$\phi_a(t)=at$: over the dual numbers,

$${\varphi }_{1+\epsilon }=\mathrm{id}+\epsilon d.$$

A Hopf algebra. Take Sweedler's four-dimensional Hopf algebra

$$B=k⟨g,u⟩/(g^2-1,u^2,gu+ug),$$

with basis $\{1,g,u,gu\}$$\{1,g,u,gu\}$ and structure

$$\begin{array}{rlrl}{\mathrm{\Delta }}_B(g)& =g\otimes g,& {\mathrm{\Delta }}_B(u)& =u\otimes 1+g\otimes u,\\ {ϵ}_B(g)& =1,& {ϵ}_B(u)& =0,\\ S_B(g)& =g,& S_B(u)& =-gu.\end{array}$$

For $a\in k^{\times }$$a\in k^\times$, the assignments

$${\varphi }_a(g)=g,{\varphi }_a(u)=au$$

preserve all the Hopf operations and define Hopf automorphisms. Setting $a=1+\epsilon$$a=1+\varepsilon$ gives the infinitesimal action

$$d(1)=d(g)=0,d(u)=u,d(gu)=gu.$$

For instance, its compatibility with the coproduct on $u$$u$ is

$$\begin{array}{rl}{\mathrm{\Delta }}_{B}d(u)& =u\otimes 1+g\otimes u\\ & =(d\otimes 1+1\otimes d){\mathrm{\Delta }}_B(u).\end{array}$$

Also $d{S}_B(u)=-gu=S_{B}d(u)$$dS_B(u)=-gu=S_Bd(u)$. Consequently, $(B,d)$$(B,d)$ is a Hopf algebra object in ${\mathrm{Diff}}_k$$\operatorname{Diff}_k$.

A Lie algebra. Let $\mathfrak{a}={\mathfrak{sl}}_2(k)$$\mathfrak a=\mathfrak{sl}_2(k)$, with

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

Restrict its adjoint action along the Lie algebra homomorphism $kx\to {\mathfrak{sl}}_2(k)$$kx\to\mathfrak{sl}_2(k)$ sending $x$$x$ to $h$$h$. The resulting operator is

$$d=\mathrm{ad}h,d(e)=2e,d(h)=0,d(f)=-2f.$$

For example,

$$d([e,f])=0=[2e,f]+[e,-2f].$$

Thus $({\mathfrak{sl}}_2,\mathrm{ad}h)$$(\mathfrak{sl}_2,\operatorname{ad}h)$ is a Lie algebra object in ${\mathrm{Diff}}_k$$\operatorname{Diff}_k$.

The corresponding infinitesimal automorphism can be seen by conjugation with $\mathrm{diag}(a,a^{-1})$$\operatorname{diag}(a,a^{-1})$:

$${\varphi }_a(e)=a^{2}e,{\varphi }_a(h)=h,{\varphi }_a(f)=a^{-2}f.$$

Over the dual numbers, ${\varphi }_{1+\epsilon }=\mathrm{id}+\epsilon \mathrm{ad}h$$\phi_{1+\varepsilon}=\operatorname{id}+\varepsilon\operatorname{ad}h$.

Several derivations and their relations

The same construction also describes several derivations. Their commutator relations are recorded by the acting Lie algebra.

Required operators	Acting Lie algebra	Enveloping Hopf algebra
One derivation	$kx$$kx$	$k[x]$$k[x]$
$r$$r$ commuting derivations	The abelian Lie algebra $k^r$$k^r$	$k[x_1,\dots ,x_r]$$k[x_1,\ldots,x_r]$
$r$$r$ derivations with no prescribed relations	The free Lie algebra on $x_1,\dots ,x_r$$x_1,\ldots,x_r$	$k⟨x_1,\dots ,x_r⟩$$k\langle x_1,\ldots,x_r\rangle$
Derivations satisfying specified Lie relations	The Lie algebra $\mathfrak{l}$$\mathfrak l$ with those generators and relations	$U(\mathfrak{l})$$U(\mathfrak l)$

In every row, the chosen Lie generators are primitive. Even the free associative algebra in the third row has a cocommutative Hopf structure with this coproduct.

Iterating the differential-object construction gives commuting operators: an endomorphism $d_2$$d_2$ of $(V,d_1)$$(V,d_1)$ in ${\mathrm{Diff}}_k$$\operatorname{Diff}_k$ must satisfy $d_2{d}_1=d_1{d}_2$$d_2d_1=d_1d_2$.

Allowing infinite-dimensional models

Finite dimensionality was used to realize the automorphism functor as an algebraic group of finite type. The infinitesimal calculation itself works in any dimension.

For an arbitrary model, retain

$$G_A(R)={\mathrm{Aut}}_{{\mathcal{P}}_R}(A_R)$$

and define

$${\mathrm{Der}}_{\mathcal{P}}(A)=\{d\in {\mathrm{End}}_k(A(1))|\mathrm{id}+\epsilon d\in G_A(k[\epsilon ]/({\epsilon }^2))\}.$$

This is the tangent Lie algebra of the automorphism group functor; no representability by a finite-type algebraic group is asserted. Equation (1) and the proof of the theorem are unchanged. Thus the theorem extends to

$${\mathrm{Mod}}_{\mathcal{P}}({\mathrm{Rep}}_k(\mathfrak{l}))\simeq {\mathrm{Act}}_{\mathfrak{l}}(\mathcal{P}).$$

For example, on $A=k[t]$$A=k[t]$, the derivations

$$D=\frac{d}{dt},E=t\frac{d}{dt}$$

satisfy

$$[E,D](f)=t{f}^{″}-(f^{\prime }+t{f}^{″})=-f^{\prime },[E,D]=-D.$$

Consequently, $k[t]$$k[t]$ is an associative algebra object in the representation category of the two-dimensional Lie algebra

$$\mathfrak{l}=ke\oplus k\delta ,[e,\delta ]=-\delta ,$$

where $e$$e$ acts as $E$$E$ and $\delta$$\delta$ as $D$$D$. Restricting along the one-dimensional subalgebra $k\delta$$k\delta$ recovers the familiar differential algebra $(k[t],d/dt)$$(k[t],d/dt)$.