2.2 Derivations and automorphisms of Lie algebras

Derivations and automorphisms are key tools for understanding Lie algebra structure. Derivations are linear maps that follow the Leibniz rule, while automorphisms are bijective linear maps preserving the Lie bracket.

These concepts help us analyze Lie algebras more deeply. We'll learn how to identify and compute derivations and automorphisms, and explore their relationships and properties within Lie algebra theory.

Derivations and Automorphisms of Lie Algebras

Definition of Derivations

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• A derivation of a Lie algebra $L$ is a linear map $D: L \to L$ satisfying the Leibniz rule: $D([x, y]) = [D(x), y] + [x, D(y)]$ for all $x, y \in L$
• The set of derivations of a Lie algebra $L$, denoted $Der(L)$, forms a Lie algebra under the commutator bracket: $[D_1, D_2] = D_1 \circ D_2 - D_2 \circ D_1$
• Inner derivations are derivations of the form $ad_x: L \to L$, where $ad_x(y) = [x, y]$ for some fixed $x \in L$
• The set of inner derivations, denoted $Inn(L)$, is an ideal of $Der(L)$

Definition of Automorphisms

• An automorphism of a Lie algebra $L$ is a bijective linear map $\phi: L \to L$ that preserves the Lie bracket: $\phi([x, y]) = [\phi(x), \phi(y)]$ for all $x, y \in L$
• The set of automorphisms of a Lie algebra $L$, denoted $Aut(L)$, forms a group under composition of maps
• The identity map is the identity element of $Aut(L)$, and the inverse of an automorphism is its inverse map

Identifying Derivations and Automorphisms

Checking if a Linear Map is a Derivation

• To check if a linear map $D: L \to L$ is a derivation, verify that it satisfies the Leibniz rule: $D([x, y]) = [D(x), y] + [x, D(y)]$ for all basis elements $x, y \in L$
• Example: Let $L$ be the Lie algebra of $2 \times 2$ matrices with basis $\{e_1, e_2, e_3\}$. Define $D: L \to L$ by $D(e_1) = e_2, D(e_2) = 0, D(e_3) = e_1$. Check if $D$ is a derivation by verifying the Leibniz rule for all basis elements
• If the Leibniz rule holds for all basis elements, then $D$ is a derivation due to the linearity of $D$ and the bilinearity of the Lie bracket

Checking if a Linear Map is an Automorphism

• To check if a linear map $\phi: L \to L$ is an automorphism, verify that it is bijective and preserves the Lie bracket: $\phi([x, y]) = [\phi(x), \phi(y)]$ for all basis elements $x, y \in L$
• A linear map $\phi: L \to L$ is an automorphism if and only if its matrix representation with respect to a basis of $L$ is invertible and satisfies the condition: $[\phi]_B [x, y]_B = [[\phi]_B x_B, [\phi]_B y_B]$ for all basis elements $x, y \in L$, where $[\cdot]_B$ denotes the matrix representation with respect to the basis $B$
• Example: Let $L$ be the Lie algebra of $2 \times 2$ matrices with basis $\{e_1, e_2, e_3\}$. Define $\phi: L \to L$ by $\phi(e_1) = e_2, \phi(e_2) = e_1, \phi(e_3) = -e_3$. Check if $\phi$ is an automorphism by verifying that it is bijective and preserves the Lie bracket for all basis elements

Computing Derivation Algebras and Automorphism Groups

Computing the Derivation Algebra

• To compute the derivation algebra $Der(L)$ of a Lie algebra $L$, find all linear maps $D: L \to L$ that satisfy the Leibniz rule: $D([x, y]) = [D(x), y] + [x, D(y)]$ for all basis elements $x, y \in L$
• The derivation algebra $Der(L)$ is a Lie algebra under the commutator bracket: $[D_1, D_2] = D_1 \circ D_2 - D_2 \circ D_1$
• Example: Compute the derivation algebra of the Lie algebra of $2 \times 2$ upper triangular matrices by finding all linear maps that satisfy the Leibniz rule for the basis elements

Computing the Automorphism Group

• To compute the automorphism group $Aut(L)$ of a Lie algebra $L$, find all bijective linear maps $\phi: L \to L$ that preserve the Lie bracket: $\phi([x, y]) = [\phi(x), \phi(y)]$ for all basis elements $x, y \in L$
• The automorphism group $Aut(L)$ is a group under composition of maps, with the identity map as the identity element and the inverse of an automorphism being its inverse map
• Example: Compute the automorphism group of the Lie algebra of $2 \times 2$ diagonal matrices by finding all bijective linear maps that preserve the Lie bracket for the basis elements

Derivations vs Automorphisms

Relationship between Derivations and Automorphisms

• Every derivation $D \in Der(L)$ generates a one-parameter subgroup of automorphisms $\{\exp(tD) \mid t \in \mathbb{R}\} \subseteq Aut(L)$ via the exponential map
• The exponential map $\exp: Der(L) \to Aut(L)$ is defined by $\exp(D) = \sum_{n=0}^\infty (D^n / n!)$, where $D^n$ denotes the $n$-fold composition of $D$ with itself
• The exponential map is a homomorphism from the Lie algebra $Der(L)$ to the Lie group $Aut(L)$, i.e., $\exp([D_1, D_2]) = \exp(D_1) \circ \exp(D_2) \circ \exp(-D_1) \circ \exp(-D_2)$

Inner Automorphism Group and Nilpotent Derivations

• The image of the exponential map, $\exp(Der(L))$, generates a connected subgroup of $Aut(L)$ called the inner automorphism group, denoted $Inn(L)$
• The kernel of the exponential map, $\ker(\exp) = \{D \in Der(L) \mid \exp(D) = id_L\}$, consists of nilpotent derivations
• The kernel of the exponential map is related to the center of the Lie algebra $L$, which consists of elements $x \in L$ such that $[x, y] = 0$ for all $y \in L$
• Example: For the Lie algebra of $n \times n$ matrices, the inner derivations are of the form $ad_A(B) = [A, B] = AB - BA$ for some fixed matrix $A$, and the inner automorphism group consists of conjugation maps $\phi_A(B) = ABA^{-1}$ for invertible matrices $A$