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The Lie algebra of the general linear group

Statement

Assume the Axiom of Choice for the finite-type assertions inherited from the matrix-group supplier. Let $k$ be a field and $n \geq 1$ . Under the identification ${GL}_{n} = Spec k [x_{i j}, d^{- 1}]$ of The general linear group scheme and its coordinate ring, the tangent space at the identity is $Lie ({GL}_{n}) ≅ M_{n} (k) = {g l}_{n}$ , the isomorphism sending a matrix $X$ to the dual-number point $I_{n} + ε X$ ; in particular $Lie (G_{m}) ≅ k$ for $n = 1$ , generated by the functional dual to the cotangent class $[x_{11} - 1] = [t - 1]$ . For matrices $X, Y$ the commutator of the lifts $I + t X$ and $I + t^{'} Y$ in ${GL}_{n} (k [t, t^{'}] / (t^{2}, t^{' 2}))$ is $I + t t^{'} (X Y - Y X)$ , and the adjoint representation of The adjoint representation of an affine group scheme satisfies $Ad (A) X = A X A^{- 1}$ for all commutative $k$ -algebras $R$ , $A \in {GL}_{n} (R)$ and $X \in M_{n} (R)$ .

Facts & Assumptions

Given: The Axiom of Choice and a field $k$ , an integer $n \geq 1$ , the coordinate ring $A = k [x_{i j}, d^{- 1}]$ with $d = \det (x_{i j})$ , and matrices $X, Y \in M_{n} (k)$ .

[F1]

The general linear group scheme and its coordinate ring: ${GL}_{n} = Spec k [x_{i j}, d^{- 1}]$ is a group scheme of finite type over $k$ whose comultiplication is $Δ (x_{i j}) = \sum_{l} x_{i l} \otimes x_{l j}$ and whose $R$ -points are the invertible $n \times n$ matrices over $R$ ; for $n = 1$ this is $G_{m} = Spec k [t, t^{- 1}]$ with $t = x_{11}$ .

[F2]

The Lie algebra of a group scheme and Cotangent space at a rational point: at the $k$ -rational identity $e$ one has $Lie ({GL}_{n}) = {Hom}_{k} (m_{e} / m_{e}^{2}, k)$ , and the cotangent space is $Ω \otimes_{O_{G, e}} κ (e) ≅ m_{e} / m_{e}^{2}$ with $d x \otimes 1 \leftrightarrow [x - ε (x)]$ .

[F3]

The tangent space at the identity is a vector space, and Lie is a functor: for every commutative $k$ -algebra $R$ one has $Lie ({GL}_{n}) (R) ≅ Lie ({GL}_{n}) \otimes_{k} R$ and the elements $e^{ε X}$ correspond to $X$ ; the group law is addition, so the point $e^{ε X}$ with $ε$ -coefficient $X$ is the base-changed dual-number point.

[F4]

Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products, If $\det (A)$ is a unit, then $A^{- 1} = \det (A)^{- 1} adj (A)$ , For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix and The trace of a square matrix over a commutative ring: matrix multiplication is associative and distributive on both sides, $(I + ε X)^{- 1} = I - ε X$ when $ε^{2} = 0$ , and expanding $\det (I + ε X)$ by the Leibniz formula over permutations shows $\det (I + ε X) = 1 + ε tr X$ , a unit of $k [ε]$ .

[F5]

Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis: a basis is an independent spanning set; for a finite basis $v_{i j}$ of $V$ , the functionals $v_{i j}^{*}$ defined by $v_{i j}^{*} (v_{a b}) = δ_{i a} δ_{j b}$ form a basis of ${Hom}_{k} (V, k)$ , since a functional is determined by these finitely many values.

[F6]

The adjoint representation of an affine group scheme: $Ad : G \to {GL}_{g}$ satisfies $x e^{ε X} x^{- 1} = e^{ε Ad (x) X}$ for $x \in G (R)$ and $X \in g \otimes_{k} R$ .

Proof

1.1F1F2F3F4F5algebra

The cotangent space and dual-number points. Set $y_{i j} = x_{i j} - δ_{i j}$ and $J = (y_{i j}) \subseteq A$ . Then $A = k [y_{i j}] [D^{- 1}]$ with $D = \det (δ_{i j} + y_{i j}) \equiv 1 m o d J$ , so $A / J^{2} ≅ k [y_{i j}] / (y_{i j})^{2}$ has basis $1, [y_{i j}]$ . Here $J$ is the augmentation ideal in $A$ , and $m_{e} = J A_{J}$ is the maximal ideal in $O_{G, e} = A_{J}$ . Every element of $A ∖ J$ has nonzero constant term and is a unit modulo $J^{2}$ , with inverse given by the square-zero formula; thus localization does not change this quotient and $m_{e} / m_{e}^{2}$ has basis $[y_{i j}]$ . Its dual is $M_{n} (k)$ via $θ_{X} ([y_{i j}]) = X_{i j}$ by [F2] and [F5]. The coefficient correspondence sends $θ_{X}$ to the algebra map $A \to k [ε]$ , $y_{i j} \mapsto ε X_{i j}$ , since $\det (I + ε X) = 1 + ε tr X$ is a unit by [F4]; hence $e^{ε X} = I + ε X$ . By [F3] this identification extends naturally to $M_{n} (R)$ for every $R$ . For $n = 1$ the Lie-algebra generator is the functional taking $[t - 1]$ to $1$ , dual to the cotangent generator.

1.2F4algebra

The commutator of the two lifts. In $B = k [t, t^{'}] / (t^{2}, t^{' 2})$ one has $t^{2} = t^{' 2} = 0$ , so $(I + t X)^{- 1} = I - t X$ and $(I + t^{'} Y)^{- 1} = I - t^{'} Y$ by [F4], and using associativity and distributivity one computes $(I + t X) (I + t^{'} Y) (I - t X) (I - t^{'} Y) = I + t t^{'} (X Y - Y X)$ : the terms linear in $t$ or $t^{'}$ cancel in pairs and the only surviving second-order term is $t t^{'} X Y - t t^{'} Y X$ , the other products $t^{2}, t^{' 2}$ vanishing.

2.1F4F6step 1.1algebra

The adjoint action. Let $R$ be a commutative $k$ -algebra, $A \in {GL}_{n} (R)$ and $X \in M_{n} (R)$ . By step 1.1 applied over $R$ , the element $e^{ε X} \in Lie ({GL}_{n}) (R)$ is the point $I + ε X$ of ${GL}_{n} (R [ε])$ , and $A$ is invertible in $M_{n} (R) \subseteq M_{n} (R [ε])$ ; conjugating with $A$ and using the matrix arithmetic of [F4] gives $A (I + ε X) A^{- 1} = I + ε (A X A^{- 1})$ . Comparing with the defining identity $A e^{ε X} A^{- 1} = e^{ε Ad (A) X}$ of [F6] identifies the $ε$ -coefficient, so $Ad (A) X = A X A^{- 1}$ .

3.1step 1.1step 1.2step 2.1∎

Conclusion. Step 1.1 identifies the tangent space at the identity with $M_{n} (k)$ via $X \mapsto I + ε X$ and gives the $n = 1$ case; step 1.2 computes the commutator of the two independent lifts as $I + t t^{'} (X Y - Y X)$ ; and step 2.1 computes the adjoint representation as conjugation. This proves all the displayed claims.

Remarks

The commutator of step 1.2 is the computational heart of the bracket $[X, Y] = X Y - Y X$ ; making it the definition of a functorial bracket on $Lie (G)$ for arbitrary affine $G$ is the content of the following theorem.

Depends on

Used by

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
SGA 3, Expose II (M. Demazure), Fibres tangents - Algebres de Lie, corrected 14 October 2024 edition (standard reference, not scraped)