The Category of Finite-Dimensional Lie Algebras Enriched over Aff_k: Hom-schemes, automorphism group schemes, and derivations as infinitesimal symmetries

Lie Algebra Hom-Spaces as Affine Schemes1. Hom-schemes of finite-dimensional Lie algebras2. Composition is polynomial3. The endomorphism monoid scheme4. The automorphism group scheme5. The Lie algebra functor of an affine group scheme6. Example: the Lie algebra of $GL_n$7. Derivations as infinitesimal automorphisms8. The affine scheme of Lie algebra structures9. Change of basis and stabilizers

 

Lie Algebra Hom-Spaces as Affine Schemes

Let $k$$k$ be a field, and let

$${\mathbf{Lie}}_k^{\mathrm{fd}}$$

denote the category of finite-dimensional Lie algebras over $k$$k$.

The usual Hom-set

$${\mathrm{Hom}}_{{\mathbf{Lie}}_k}(\mathfrak{g},\mathfrak{h})$$

is the set of bracket-preserving linear maps from $\mathfrak{g}$$\mathfrak g$ to $\mathfrak{h}$$\mathfrak h$. In finite dimensions, this set naturally carries more structure: it is cut out by polynomial equations inside the affine space of all linear maps

$${\mathrm{Hom}}_k(\mathfrak{g},\mathfrak{h}).$$

Thus the Hom-sets of finite-dimensional Lie algebras can be upgraded to Hom-schemes. Composition is polynomial, so the category of finite-dimensional Lie algebras is naturally enriched over affine $k$$k$-schemes.

The same viewpoint also explains why endomorphisms form an affine monoid scheme, why automorphisms form an affine group scheme, and why derivations appear as infinitesimal automorphisms.

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1. Hom-schemes of finite-dimensional Lie algebras

Let $\mathfrak{g}$$\mathfrak g$ and $\mathfrak{h}$$\mathfrak h$ be finite-dimensional Lie algebras over $k$$k$.

Choose bases

$$e_1,\dots ,e_n$$

for $\mathfrak{g}$$\mathfrak g$, and

$$f_1,\dots ,f_m$$

for $\mathfrak{h}$$\mathfrak h$.

Write the Lie brackets in terms of structure constants:

$$[e_i,e_j{]}_{\mathfrak{g}}=\sum _r{c}_{ij}^r{e}_r,$$

and

$$[f_{\alpha },f_{\beta }{]}_{\mathfrak{h}}=\sum _{\gamma }{d}_{\alpha \beta }^{\gamma }{f}_{\gamma }.$$

A linear map

$$F:\mathfrak{g}\to \mathfrak{h}$$

is determined by matrix coordinates

$$F(e_i)=\sum _{\alpha }{x}_{\alpha i}{f}_{\alpha }.$$

The space of all linear maps is therefore the affine space

$${\mathrm{Hom}}_k(\mathfrak{g},\mathfrak{h})\cong {\mathbb{A}}_k^{mn}.$$

The condition that $F$$F$ is a Lie algebra homomorphism is

$$F([e_i,e_j{]}_{\mathfrak{g}})=[F(e_i),F(e_j){]}_{\mathfrak{h}}.$$

Expanding the left-hand side gives

$$F([e_i,e_j{]}_{\mathfrak{g}})=F\left(\sum _r{c}_{ij}^r{e}_r\right)=\sum _r{c}_{ij}^{r}F(e_r)=\sum _{r,\gamma }{c}_{ij}^r{x}_{\gamma r}{f}_{\gamma }.$$

Expanding the right-hand side gives

$$[F(e_i),F(e_j){]}_{\mathfrak{h}}=[\sum _{\alpha }{x}_{\alpha i}{f}_{\alpha },\sum _{\beta }{x}_{\beta j}{f}_{\beta }].$$

By bilinearity,

$$[F(e_i),F(e_j){]}_{\mathfrak{h}}=\sum _{\alpha ,\beta }{x}_{\alpha i}{x}_{\beta j}[f_{\alpha },f_{\beta }{]}_{\mathfrak{h}}.$$

Using the structure constants of $\mathfrak{h}$$\mathfrak h$, this becomes

$$\sum _{\alpha ,\beta ,\gamma }{d}_{\alpha \beta }^{\gamma }{x}_{\alpha i}{x}_{\beta j}{f}_{\gamma }.$$

Therefore the condition

$$F([e_i,e_j])=[F(e_i),F(e_j)]$$

is equivalent to the system of polynomial equations

$$\sum _r{c}_{ij}^r{x}_{\gamma r}-\sum _{\alpha ,\beta }{d}_{\alpha \beta }^{\gamma }{x}_{\alpha i}{x}_{\beta j}=0.$$

These are quadratic equations in the matrix coordinates of $F$$F$.

Hence the Lie homomorphism space is represented by the affine scheme

$${\underset{―}{\mathrm{Hom}}}_{\mathrm{Lie}}(\mathfrak{g},\mathfrak{h})=\mathrm{Spec}\frac{k[x_{\alpha i}]}{(\sum _r{c}_{ij}^r{x}_{\gamma r}-\sum _{\alpha ,\beta }{d}_{\alpha \beta }^{\gamma }{x}_{\alpha i}{x}_{\beta j})}.$$

The coordinate presentation depends on the chosen bases, but the affine scheme itself does not. The basis only gives coordinates.

Equivalently, the intrinsic definition is that

$${\underset{―}{\mathrm{Hom}}}_{\mathrm{Lie}}(\mathfrak{g},\mathfrak{h})$$

represents the functor

$$R⟼{\mathrm{Hom}}_{{\mathrm{Lie}}_R}(\mathfrak{g}{\otimes }_{k}R,\mathfrak{h}{\otimes }_{k}R),$$

where $R$$R$ ranges over commutative $k$$k$-algebras.

Thus the Hom-scheme is the affine scheme of Lie algebra homomorphisms after arbitrary extension of scalars.

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2. Composition is polynomial

Let

$$\mathfrak{g},\mathfrak{h},\mathfrak{l}$$

be finite-dimensional Lie algebras over $k$$k$.

Given Lie homomorphisms

$$F:\mathfrak{g}\to \mathfrak{h},G:\mathfrak{h}\to \mathfrak{l},$$

their composition

$$G\circ F:\mathfrak{g}\to \mathfrak{l}$$

is again a Lie homomorphism.

In matrix coordinates, composition is just matrix multiplication. If

$$F(e_i)=\sum _{\alpha }{x}_{\alpha i}{f}_{\alpha }$$

and

$$G(f_{\alpha })=\sum _{\beta }{y}_{\beta \alpha }{u}_{\beta },$$

then

$$(G\circ F)(e_i)=G\left(\sum _{\alpha }{x}_{\alpha i}{f}_{\alpha }\right)=\sum _{\alpha ,\beta }{y}_{\beta \alpha }{x}_{\alpha i}{u}_{\beta }.$$

Thus the coordinate functions of $G\circ F$$G\circ F$ are

$$(G\circ F{)}_{\beta i}=\sum _{\alpha }{y}_{\beta \alpha }{x}_{\alpha i}.$$

This is polynomial in the coordinates of $F$$F$ and $G$$G$.

Therefore composition gives a morphism of affine schemes

$${\underset{―}{\mathrm{Hom}}}_{\mathrm{Lie}}(\mathfrak{h},\mathfrak{l}){\times }_k{\underset{―}{\mathrm{Hom}}}_{\mathrm{Lie}}(\mathfrak{g},\mathfrak{h})\to {\underset{―}{\mathrm{Hom}}}_{\mathrm{Lie}}(\mathfrak{g},\mathfrak{l}).$$

The identity map on $\mathfrak{g}$$\mathfrak g$ gives a $k$$k$-point

$$\mathrm{Spec}k\to {\underset{―}{\mathrm{Hom}}}_{\mathrm{Lie}}(\mathfrak{g},\mathfrak{g}).$$

The associativity and identity axioms follow from the ordinary associativity and identity laws for composition of maps.

Hence

$${\mathbf{Lie}}_k^{\mathrm{fd}}$$

is naturally enriched over

$$({\mathbf{Aff}}_k,{\times }_k).$$

In short:

$$\boxed{{\displaystyle \text{finite-dimensional Lie algebra Hom-sets are affine schemes, and composition is polynomial.}}}$$

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3. The endomorphism monoid scheme

Taking

$$\mathfrak{h}=\mathfrak{g},$$

we obtain the affine scheme

$${\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g})={\underset{―}{\mathrm{Hom}}}_{\mathrm{Lie}}(\mathfrak{g},\mathfrak{g}).$$

It carries a composition morphism

$${\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g}){\times }_k{\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g})\to {\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g}),$$

and a unit point corresponding to

$${\mathrm{id}}_{\mathfrak{g}}.$$

Thus

$${\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g})$$

is an affine monoid scheme.

On coordinate rings, the monoid structure gives a comultiplication. If $x_{ij}$$x_{ij}$ are the matrix-coordinate functions, then composition of matrices gives

$$\mathrm{\Delta }(x_{ij})=\sum _r{x}_{ir}\otimes x_{rj}.$$

The identity endomorphism gives the counit

$$\epsilon (x_{ij})={\delta }_{ij}.$$

Therefore

$$\mathcal{O}({\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g}))$$

is a commutative bialgebra.

This is just the algebraic dual of the fact that endomorphisms compose.

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4. The automorphism group scheme

The automorphism group is the subfunctor of endomorphisms whose underlying linear map is invertible.

After choosing a basis, this is the determinant condition

$$\det \left(X\right)\ne 0.$$

Therefore

$${\underset{―}{\mathrm{Aut}}}_{\mathrm{Lie}}(\mathfrak{g})=D(\det )\subseteq {\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g}).$$

If

$$A=\mathcal{O}({\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g})),$$

then

$$\mathcal{O}({\underset{―}{\mathrm{Aut}}}_{\mathrm{Lie}}(\mathfrak{g}))=A[{\det }^{-1}].$$

Thus the automorphism group scheme is obtained by localizing the coordinate ring of the endomorphism monoid scheme at the determinant.

Equivalently,

$${\underset{―}{\mathrm{Aut}}}_{\mathrm{Lie}}(\mathfrak{g})={\underset{―}{\mathrm{End}}}_{\mathrm{Lie}}(\mathfrak{g})\cap GL(\mathfrak{g}).$$

The inverse map is regular on this open subscheme. Hence

$${\underset{―}{\mathrm{Aut}}}_{\mathrm{Lie}}(\mathfrak{g})$$

is an affine group scheme.

Over an algebraically closed field in the classical reduced setting, one often calls it an affine algebraic group. In general, the more precise phrase is affine group scheme.

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5. The Lie algebra functor of an affine group scheme

Let $G$$G$ be an affine group scheme over $k$$k$.

For every commutative $k$$k$-algebra $R$$R$, define

$$R[\epsilon ]=R[X]/(X^2).$$

Thus

$$R[\epsilon ]=R\oplus R\epsilon$$

as an $R$$R$-module, with

$${\epsilon }^2=0.$$

There are natural $k$$k$-algebra homomorphisms

$$R\stackrel{i}{\to }R[\epsilon ]\stackrel{\pi }{\to }R,$$

where

$$i(a)=a+\epsilon 0$$

and

$$\pi (a+\epsilon b)=a.$$

They satisfy

$$\pi \circ i={\mathrm{id}}_R.$$

Applying the group functor $G$$G$, we obtain group homomorphisms

$$G(R)\stackrel{i}{\to }G(R[\epsilon ])\stackrel{\pi }{\to }G(R),$$

again satisfying

$$\pi \circ i={\mathrm{id}}_{G(R)}.$$

The Lie algebra functor of $G$$G$ is defined by

$$\mathrm{Lie}(G)(R)=\ker (G(R[\epsilon ])\stackrel{\pi }{\to }G(R)).$$

Thus

$$\mathrm{Lie}(G)(R)$$

consists of those $R[\epsilon ]$$R[\varepsilon]$-points of $G$$G$ which reduce to the identity element over $R$$R$.

This definition is often the cleanest way to understand tangent vectors at the identity.

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6. Example: the Lie algebra of $G{L}_n$$GL_n$

Let

$$G=G{L}_n.$$

Then

$$G(R)=G{L}_n(R).$$

An element of

$$\mathrm{Lie}(G{L}_n)(R)$$

is an invertible matrix over $R[\epsilon ]$$R[\varepsilon]$ which reduces to the identity matrix modulo $\epsilon$$\varepsilon$.

Hence it must have the form

$$I_n+\epsilon A$$

for some

$$A\in M_n(R).$$

Conversely, every matrix of the form

$$I_n+\epsilon A$$

is invertible, with inverse

$$I_n-\epsilon A.$$

Indeed,

$$(I_n+\epsilon A)(I_n-\epsilon A)=I_n-{\epsilon }^2{A}^2=I_n.$$

Therefore

$$\mathrm{Lie}(G{L}_n)(R)=M_n(R).$$

In particular,

$$\mathrm{Lie}(G{L}_n)(k)=M_n(k).$$

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7. Derivations as infinitesimal automorphisms

Now take

$$G={\underset{―}{\mathrm{Aut}}}_{\mathrm{Lie}}(\mathfrak{g}),$$

where $\mathfrak{g}$$\mathfrak g$ is a finite-dimensional Lie algebra over $k$$k$.

By definition,

$$G(R)={\mathrm{Aut}}_{{\mathrm{Lie}}_R}({\mathfrak{g}}_R),$$

where

$${\mathfrak{g}}_R=\mathfrak{g}{\otimes }_{k}R.$$

Therefore

$$\mathrm{Lie}(G)(R)=\ker ({\mathrm{Aut}}_{{\mathrm{Lie}}_{R[\epsilon ]}}({\mathfrak{g}}_{R[\epsilon ]})\to {\mathrm{Aut}}_{{\mathrm{Lie}}_R}({\mathfrak{g}}_R)).$$

An element of this kernel is a Lie algebra automorphism of

$${\mathfrak{g}}_{R[\epsilon ]}=\mathfrak{g}{\otimes }_{k}R[\epsilon ]$$

which reduces to the identity modulo $\epsilon$$\varepsilon$.

Such an automorphism has the form

$$F_{\epsilon }=\mathrm{id}+\epsilon D,$$

where

$$D\in {\mathrm{End}}_R({\mathfrak{g}}_R).$$

As an $R[\epsilon ]$$R[\varepsilon]$-linear map, it is automatically invertible, with inverse

$$F_{\epsilon }^{-1}=\mathrm{id}-\epsilon D.$$

The nontrivial condition is that $F_{\epsilon }$$F_\varepsilon$ must be a Lie algebra automorphism. Hence it must preserve the bracket:

$$F_{\epsilon }([x,y])=[F_{\epsilon }(x),F_{\epsilon }(y)]$$

for all

$$x,y\in {\mathfrak{g}}_R.$$

Substituting

$$F_{\epsilon }=\mathrm{id}+\epsilon D,$$

the left-hand side is

$$F_{\epsilon }([x,y])=[x,y]+\epsilon D([x,y]).$$

The right-hand side is

$$[F_{\epsilon }(x),F_{\epsilon }(y)]=[x+\epsilon D(x),y+\epsilon D(y)].$$

By bilinearity of the bracket,

$$[x+\epsilon D(x),y+\epsilon D(y)]=[x,y]+\epsilon [D(x),y]+\epsilon [x,D(y)]+{\epsilon }^2[D(x),D(y)].$$

Since

$${\epsilon }^2=0,$$

the last term vanishes. Therefore the right-hand side becomes

$$[x,y]+\epsilon ([D(x),y]+[x,D(y)]).$$

Thus the bracket-preservation condition is equivalent to

$$[x,y]+\epsilon D([x,y])=[x,y]+\epsilon ([D(x),y]+[x,D(y)]).$$

Comparing the coefficients of $\epsilon$$\varepsilon$, we obtain

$$D([x,y])=[D(x),y]+[x,D(y)].$$

This is exactly the derivation condition.

Therefore

$$\mathrm{Lie}({\underset{―}{\mathrm{Aut}}}_{\mathrm{Lie}}(\mathfrak{g}))(R)={\mathrm{Der}}_R({\mathfrak{g}}_R).$$

In particular,

$$\mathrm{Lie}({\underset{―}{\mathrm{Aut}}}_{\mathrm{Lie}}(\mathfrak{g}))(k)={\mathrm{Der}}_k(\mathfrak{g}).$$

This is the precise sense in which derivations are infinitesimal automorphisms.

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8. The affine scheme of Lie algebra structures

There is a parallel geometric picture for Lie algebra structures themselves.

Fix a finite-dimensional vector space $V$$V$ over $k$$k$.

A Lie algebra structure on $V$$V$ is an alternating bilinear map

$$\mu :{\wedge }^{2}V\to V$$

satisfying the Jacobi identity

$$\mu (x,\mu (y,z))+\mu (y,\mu (z,x))+\mu (z,\mu (x,y))=0.$$

The space of all alternating bilinear maps is the affine space

$${\mathrm{Hom}}_k({\wedge }^{2}V,V).$$

The Jacobi identity is a system of quadratic equations. Hence the set of Lie algebra structures on $V$$V$ forms an affine scheme

$$\mathcal{L}(V)\subseteq {\mathrm{Hom}}_k({\wedge }^{2}V,V).$$

Choosing a basis

$$e_1,\dots ,e_n$$

and writing

$$\mu (e_i,e_j)=\sum _r{c}_{ij}^r{e}_r,$$

the Jacobi identity becomes

$$\sum _s(c_{jk}^s{c}_{is}^r+c_{ki}^s{c}_{js}^r+c_{ij}^s{c}_{ks}^r)=0.$$

Thus $\mathcal{L}(V)$$\mathcal L(V)$ is explicitly cut out by quadratic equations in the structure constants.

A point of $\mathcal{L}(V)$$\mathcal L(V)$ is a Lie bracket on the fixed underlying vector space $V$$V$.

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9. Change of basis and stabilizers

The group

$$GL(V)$$

acts on

$$\mathcal{L}(V)$$

by transport of structure:

$$(g\cdot \mu )(x,y)=g\mu (g^{-1}x,g^{-1}y).$$

This action records change of basis.

The orbit

$$GL(V)\cdot \mu$$

consists of all Lie brackets on $V$$V$ which are isomorphic to $\mu$$\mu$.

The stabilizer of $\mu$$\mu$ is

$${\mathrm{Stab}}_{GL(V)}(\mu )=\{g\in GL(V)\mid g\cdot \mu =\mu \}.$$

Expanding the condition

$$g\cdot \mu =\mu$$

gives

$$g\mu (x,y)=\mu (gx,gy).$$

In bracket notation, this is

$$g([x,y])=[gx,gy].$$

This is precisely the condition that $g$$g$ is a Lie algebra automorphism of $(V,\mu )$$(V,\mu)$.

Therefore

$${\mathrm{Stab}}_{GL(V)}(\mu )={\underset{―}{\mathrm{Aut}}}_{\mathrm{Lie}}(V,\mu ).$$

So the automorphism group scheme of a Lie algebra is the stabilizer of its corresponding point in the affine scheme of Lie algebra structures.