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Nilpotent Lie algebra

A nilpotent Lie algebra is a \mathfrak{g} over a whose lower central series terminates at the zero after finitely many steps, defined by \mathfrak{g}_1 = \mathfrak{g} and \mathfrak{g}_{m+1} = [\mathfrak{g}, \mathfrak{g}_m] for m \geq 1, with \mathfrak{g}_k = \{0\} for some finite k. This condition implies that iterated Lie brackets of length k vanish identically, and equivalently, the operators \mathrm{ad}_x for all x \in \mathfrak{g} are nilpotent endomorphisms. Nilpotent Lie algebras form an important subclass of solvable Lie algebras, as the lower central series terminating implies the derived series—defined by D_1(\mathfrak{g}) = [\mathfrak{g}, \mathfrak{g}] and D_{m+1}(\mathfrak{g}) = [D_m(\mathfrak{g}), D_m(\mathfrak{g})]—also terminates at zero. Key structural properties include a nontrivial Z(\mathfrak{g}) = \{ z \in \mathfrak{g} \mid [z, \mathfrak{g}] = \{0\} \}, which is itself an , ensuring that nontrivial nilpotent Lie algebras possess nonzero central elements. Finite-dimensional nilpotent Lie algebras over algebraically closed fields of characteristic zero admit faithful representations as subalgebras of strictly upper triangular matrices, reflecting their "triangular" structure in linear representations. Engel's theorem provides a foundational : a finite-dimensional of \mathfrak{[gl](/page/GL)}(V) over a of zero is if and only if every element acts nilpotently on V, allowing the construction of a flag of subspaces stabilized in a stepwise manner by the algebra. These algebras play a central role in the study of Lie group representations, algebraic geometry via nilpotent orbits, and the classification of low-dimensional Lie algebras, where examples like the Heisenberg algebra illustrate their nonabelian yet "mildly commutative" nature.

Definition and Characterizations

Definition

A Lie algebra \mathfrak{g} over a field K is nilpotent if its lower central series terminates at the zero subspace after finitely many steps. The lower central series is defined recursively by \mathfrak{g}^0 = \mathfrak{g} and \mathfrak{g}^{k+1} = [\mathfrak{g}, \mathfrak{g}^k] for k \geq 0, where [\mathfrak{g}, \mathfrak{g}^k] denotes the subspace spanned by all Lie brackets [x, y] with x \in \mathfrak{g} and y \in \mathfrak{g}^k. The first term \mathfrak{g}^1 = [\mathfrak{g}, \mathfrak{g}] is the derived algebra of \mathfrak{g}, and each subsequent term \mathfrak{g}^{k+1} is an ideal contained in \mathfrak{g}^k. Thus, \mathfrak{g} is nilpotent if there exists a positive integer n such that \mathfrak{g}^n = \{0\}, and this n is called the nilpotency class of \mathfrak{g}. The terms of the lower central series are often denoted by \gamma_k(\mathfrak{g}), where \gamma_0(\mathfrak{g}) = \mathfrak{g} and \gamma_{k+1}(\mathfrak{g}) = [\mathfrak{g}, \gamma_k(\mathfrak{g})] for k \geq 0, or alternatively starting with \gamma_1(\mathfrak{g}) = \mathfrak{g}. captures a form of higher-order commutativity in Lie algebras, where repeated applications of the Lie bracket eventually yield zero, analogous to the nilpotency condition in nilpotent groups via their lower central series.

Equivalent conditions

A Lie algebra \mathfrak{g} over a k of characteristic zero is if and only if the \mathrm{ad}: \mathfrak{g} \to \mathrm{End}(\mathfrak{g}), defined by \mathrm{ad}_x(y) = [x, y] for x, y \in \mathfrak{g}, has the property that each \mathrm{ad}_x is a , meaning there exists some positive integer n (depending on x) such that (\mathrm{ad}_x)^n = 0. This condition holds for finite-dimensional Lie algebras over fields of characteristic zero. Engel's theorem provides a preliminary characterization in this context: for a finite-dimensional \mathfrak{g} over a of characteristic zero, every adjoint operator \mathrm{ad}_x is \mathfrak{g} is . The proof proceeds by on the of \mathfrak{g}, showing that if all \mathrm{ad}_x are , then the image of \mathrm{ad} (isomorphic to \mathfrak{g}/Z(\mathfrak{g}), where Z(\mathfrak{g}) is ) is , implying \mathfrak{g} itself is via properties of the lower central series. An equivalent definition uses the upper central series of \mathfrak{g}, defined recursively by Z_0(\mathfrak{g}) = \{0\} and Z_{k+1}(\mathfrak{g}) = \{ z \in \mathfrak{g} \mid [z, \mathfrak{g}] \subseteq Z_k(\mathfrak{g}) \} for k \geq 0, where each Z_k(\mathfrak{g}) is an ideal of \mathfrak{g}. The Lie algebra \mathfrak{g} is nilpotent if and only if there exists some positive integer m such that Z_m(\mathfrak{g}) = \mathfrak{g}. This equivalence to the termination of the lower central series (as in the definition) follows from the fact that the successive quotients Z_{k+1}(\mathfrak{g})/Z_k(\mathfrak{g}) are central in the quotient \mathfrak{g}/Z_k(\mathfrak{g}), mirroring the structure of the lower central series factors \mathfrak{g}_{k}/\mathfrak{g}_{k+1}, where \mathfrak{g}_k denotes the k-th term of the lower central series; thus, one series terminates if and only if the other does. The equivalence via the lower and upper central series is valid over arbitrary , while the nilpotency condition requires characteristic zero. Stronger forms (such as explicit bases or representation-theoretic implications) often require characteristic zero.

Examples

Strictly upper triangular matrices

A finite-dimensional example of a nilpotent is the \mathfrak{n}_k consisting of all k \times k strictly upper triangular matrices over a K of characteristic zero, equipped with the [A, B] = AB - BA. This structure arises as a of the general linear \mathfrak{gl}_k(K), where the matrices have zeros on and below the . The dimension of \mathfrak{n}_k is \binom{k}{2}, corresponding to the number of positions strictly above the diagonal. A consists of the matrix units E_{ij} for $1 \leq i < j \leq k, where E_{ij} is the matrix with a 1 in the (i,j)-entry and zeros elsewhere. The Lie bracket in this basis satisfies [E_{ij}, E_{pq}] = \delta_{jp} E_{iq} - \delta_{iq} E_{pj}, which is nonzero only when the indices form a "chain" connecting the rows and columns appropriately. To verify nilpotency, consider the lower central series defined by \mathfrak{n}_k^0 = \mathfrak{n}_k and \mathfrak{n}_k^{r+1} = [\mathfrak{n}_k, \mathfrak{n}_k^r] for r \geq 0. The term \mathfrak{n}_k^1 comprises matrices vanishing on the first superdiagonal (i.e., entries only where j - i \geq 2), while \mathfrak{n}_k^r consists of matrices vanishing on the first r superdiagonals (entries only where j - i > r). This series terminates with \mathfrak{n}_k^{k-1} = \{0\}, confirming that \mathfrak{n}_k is nilpotent of class at most k-1. In the \mathrm{ad}: \mathfrak{n}_k \to \mathfrak{gl}(\mathfrak{n}_k), the action of \mathrm{ad}_A for A \in \mathfrak{n}_k on basis elements E_{pq} effectively shifts the "steps" in the indices upward: repeated applications move entries beyond the matrix boundaries, rendering \mathrm{ad}_A^m = 0 for sufficiently large m depending on A. This ad-nilpotency of all elements characterizes the nilpotency of \mathfrak{n}_k. Algebraically, \mathfrak{n}_k serves as the of the unipotent group U_k(K) of k \times k upper triangular matrices with 1's on the diagonal.

Heisenberg algebras

The three-dimensional Heisenberg algebra \mathfrak{h}_3 over a field K of characteristic zero is the Lie algebra with basis \{x, y, z\} and Lie bracket relations [x, y] = z, [x, z] = [y, z] = 0. This structure makes \mathfrak{h}_3 a prototypical example of a non-abelian nilpotent Lie algebra. This algebra generalizes to higher dimensions as the (2n+1)-dimensional Heisenberg algebra \mathfrak{h}_{2n+1} over K, with basis \{x_1, \dots, x_n, y_1, \dots, y_n, z\} and nonzero Lie brackets given by [x_i, y_j] = \delta_{ij} z for i, j = 1, \dots, n. All other brackets vanish, reflecting the central extension of the abelian Lie algebra K^{2n} by the one-dimensional ideal \langle z \rangle. The Heisenberg algebras are uniformly 2-step nilpotent, with lower central series \mathfrak{h}^1 = [\mathfrak{h}, \mathfrak{h}] = \langle z \rangle and \mathfrak{h}^2 = [\mathfrak{h}, \mathfrak{h}^1] = 0. Their center coincides with the derived algebra: Z(\mathfrak{h}) = \langle z \rangle = [\mathfrak{h}, \mathfrak{h}]. A concrete realization of \mathfrak{h}_3 is the Lie subalgebra of $3 \times 3 strictly upper triangular matrices over K, generated by the basis elements x = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad y = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \quad z = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, satisfying the defining relations via matrix commutators. In applications, the Heisenberg algebra models the structure of the creation and annihilation operators in the quantum harmonic oscillator, capturing the canonical commutation relations [q, p] = i\hbar in a Lie-theoretic framework.

Cartan subalgebras

In a semisimple Lie algebra \mathfrak{g} over an algebraically closed field of characteristic zero, a Cartan subalgebra \mathfrak{h} is defined as a nilpotent subalgebra that is self-normalizing, meaning its normalizer N_{\mathfrak{g}}(\mathfrak{h}) = \{ x \in \mathfrak{g} \mid [x, \mathfrak{h}] \subseteq \mathfrak{h} \} coincides with \mathfrak{h} itself. This property ensures that \mathfrak{h} is maximal among nilpotent subalgebras with this normalization condition. Such s are abelian, with the Lie bracket [\mathfrak{h}, \mathfrak{h}] = 0, making them of 1. In the context of semisimple Lie algebras, this abelian structure facilitates the decomposition of \mathfrak{g} relative to \mathfrak{h}. The presence of a \mathfrak{h} induces the decomposition of \mathfrak{g}: \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta} \mathfrak{g}_\alpha, where \Delta is the and each \mathfrak{g}_\alpha is the corresponding to the \alpha: \mathfrak{h} \to K. The Lie bracket satisfies [h, x_\alpha] = \alpha(h) x_\alpha for h \in \mathfrak{h} and x_\alpha \in \mathfrak{g}_\alpha, and [\mathfrak{g}_\alpha, \mathfrak{g}_\beta] \subseteq \mathfrak{g}_{\alpha + \beta}, highlighting how the nilpotency of \mathfrak{h} interacts with the semisimple structure of \mathfrak{g}. A concrete example occurs in the special linear Lie algebra \mathfrak{sl}(n, K), where the subalgebra of trace-zero diagonal matrices forms a Cartan subalgebra; in contrast, the strictly upper triangular matrices constitute the nilradical of the Borel subalgebra containing this Cartan. The dimension of any Cartan subalgebra equals the rank of \mathfrak{g}, which is the dimension of the abelian factor in the Levi decomposition or the number of simple roots in the root system. In characteristic zero, Cartan subalgebras are toral, meaning every element h \in \mathfrak{h} is ad-semisimple (i.e., \mathrm{ad}_h is diagonalizable). This toral property aligns with the self-normalizing condition and underpins the existence and conjugacy of Cartan subalgebras in semisimple algebras.

Other examples

Filiform Lie algebras provide a family of Lie algebras achieving the maximal possible nilpotency . For a finite-dimensional Lie algebra \mathfrak{g} over a of characteristic zero, the nilpotency is at most \dim \mathfrak{g} - 1, and filiform algebras are precisely those attaining this bound. They admit a basis \{e_1, e_2, \dots, e_n\} such that the Lie brackets generate a flag of ideals \mathfrak{g} = \langle e_1, \dots, e_n \rangle \supset \langle e_2, \dots, e_n \rangle \supset \cdots \supset \langle e_n \rangle \supset \{0\}, with [e_1, e_i] = e_{i+1} for $1 \leq i < n and all other brackets vanishing in the model case. Every nilpotent Lie algebra \mathfrak{g} gives rise to a naturally graded Lie algebra via its lower central series \mathfrak{g} = \mathfrak{g}^1 \supset \mathfrak{g}^2 \supset \cdots \supset \mathfrak{g}^k = \{0\}, where \mathfrak{g}^{i+1} = [\mathfrak{g}, \mathfrak{g}^i]. The associated graded Lie algebra is \mathrm{gr}(\mathfrak{g}) = \bigoplus_{i=1}^k \mathfrak{g}^i / \mathfrak{g}^{i+1}, with the Lie bracket induced by the commutator in \mathfrak{g} and zero across different grading components unless adjacent. This graded structure captures the "steps" of nilpotency and is itself nilpotent, often used to study deformations and cohomology of \mathfrak{g}. Classifications in low dimensions highlight the scarcity and structure of nilpotent Lie algebras. In dimension 2, the only nilpotent Lie algebra is the abelian one. Dimension 3 yields two isomorphism classes: the abelian algebra and the with basis \{x, y, z\} and [x, y] = z, all other brackets zero. In dimension 4, there are five non-isomorphic nilpotent Lie algebras over \mathbb{R} or \mathbb{C}, including the direct sum \mathfrak{h}_3 \oplus K (where K is 1-dimensional abelian), the abelian algebra, and others like the 4-dimensional or filiform models. These classifications rely on solving structure equations and are complete up to dimension 6. Contact Lie algebras extend the Heisenberg example to higher-step nilpotency in the context of contact geometry. While the standard Heisenberg algebra is 2-step nilpotent and underlies contact structures on odd-dimensional manifolds, higher-step analogs include multi-step nilpotent Lie algebras preserving contact forms of higher codimension, such as 3-step nilpotent algebras in dimensions 5 or 7 that admit contact gradings where the center is complemented by derived ideals. These structures generalize the symplectic leaves of the Heisenberg case to more complex coadjoint orbits and appear in classifications of nilpotent algebras with non-degenerate invariant forms.

Properties

Relation to solvability

A Lie algebra \mathfrak{g} over a field is called solvable if its derived series terminates at the zero subspace. The derived series is defined by setting \mathfrak{g}^{(0)} = \mathfrak{g} and \mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}] for k \geq 0, so solvability means there exists some integer s such that \mathfrak{g}^{(s)} = \{0\}. Nilpotent Lie algebras are always solvable. To see this, recall that the lower central series of \mathfrak{g} is given by \gamma_0(\mathfrak{g}) = \mathfrak{g} and \gamma_{i+1}(\mathfrak{g}) = [\mathfrak{g}, \gamma_i(\mathfrak{g})] for i \geq 0, with nilpotency meaning \gamma_c(\mathfrak{g}) = \{0\} for some class c. The derived series is refined by the lower central series in the sense that \mathfrak{g}^{(k)} \subseteq \gamma_{2^k}(\mathfrak{g}) for all k \geq 0. Thus, if \mathfrak{g} is nilpotent with class c, then \mathfrak{g}^{(k)} = \{0\} for all k such that $2^k > c, implying the derived series terminates and \mathfrak{g} is solvable. The converse does not hold: there exist solvable Lie algebras that are not nilpotent. A standard example is the Borel subalgebra \mathfrak{b} of \mathfrak{sl}(2, \mathbb{C}), consisting of trace-zero upper triangular $2 \times 2 matrices. Here, \mathfrak{b}^{(1)} = [\mathfrak{b}, \mathfrak{b}] is the one-dimensional span of the strictly upper triangular matrix, and \mathfrak{b}^{(2)} = \{0\}, so \mathfrak{b} is solvable of length 2. However, the lower central series stabilizes at \gamma_2(\mathfrak{b}) = [\mathfrak{b}, \mathfrak{b}] since [\mathfrak{b}, \gamma_2(\mathfrak{b})] = \gamma_2(\mathfrak{b}) \neq \{0\}, so \mathfrak{b} is not nilpotent. The nilpotency class provides a bound on the solvability length: if a Lie algebra is nilpotent of class c, its derived series terminates in at most \lceil \log_2 (c+1) \rceil steps due to the refinement relation above. Another counterexample of a solvable but non-nilpotent Lie algebra is the two-dimensional affine Lie algebra over \mathbb{R} or \mathbb{C}, with basis \{X, Y\} and bracket [X, Y] = Y. The derived series is \mathfrak{g}^{(1)} = \operatorname{span}\{Y\} and \mathfrak{g}^{(2)} = \{0\}, confirming solvability, but the lower central series has \gamma_2(\mathfrak{g}) = \operatorname{span}\{Y\} and \gamma_3(\mathfrak{g}) = [\mathfrak{g}, \gamma_2(\mathfrak{g})] = \operatorname{span}\{Y\} \neq \{0\}, so it is not nilpotent. Over fields of characteristic zero, such as \mathbb{R} or \mathbb{C}, nilpotency imposes stricter conditions on representations than mere solvability. While solvable Lie algebras admit finite-dimensional representations that are simultaneously triangularizable by Lie's theorem, nilpotent ones admit representations that are simultaneously strictly upper triangularizable, meaning the images consist of unipotent matrices (with all eigenvalues 1 and nilpotent Jordan blocks). This follows from the combination of Lie's theorem and the nilpotency of the .

Subalgebras, ideals, and quotients

A key structural property of nilpotent Lie algebras is their closure under certain operations that preserve nilpotency. Specifically, , ideals, and quotients inherit the nilpotent structure from the ambient algebra, ensuring that these substructures maintain the defining central series termination. Consider Lie first. If \mathfrak{h} is a Lie subalgebra of a nilpotent Lie algebra \mathfrak{g}, then \mathfrak{h} is itself nilpotent. Moreover, the nilpotency class of \mathfrak{h}—the smallest integer k such that the k-th term of the lower central series of \mathfrak{h} vanishes—is at most that of \mathfrak{g}. This follows from the inclusion of the lower central series terms: the k-th term D_k(\mathfrak{h}) satisfies D_k(\mathfrak{h}) \subseteq D_k(\mathfrak{g}), so if D_m(\mathfrak{g}) = 0, then D_m(\mathfrak{h}) = 0. Ideals provide a related preservation. Any ideal \mathfrak{i} of a Lie algebra \mathfrak{g} that is nilpotent remains nilpotent, as ideals are special cases of subalgebras. In particular, if \mathfrak{g} is nilpotent, all its ideals are nilpotent subalgebras. Furthermore, every finite-dimensional Lie algebra possesses a unique maximal nilpotent ideal, known as the nilradical, which is the sum of all nilpotent ideals. For quotients, if \mathfrak{g} is nilpotent and \mathfrak{i} is a nilpotent ideal of \mathfrak{g}, the quotient Lie algebra \mathfrak{g}/\mathfrak{i} is nilpotent. The lower central series of the quotient corresponds to the images of the series terms in \mathfrak{g}, which terminate since those of \mathfrak{g} do; specifically, the map \mathfrak{g} \to \mathfrak{g}/\mathfrak{i} induces isomorphisms D_k(\mathfrak{g})/ (D_k(\mathfrak{g}) \cap \mathfrak{i}) \cong D_k(\mathfrak{g}/\mathfrak{i}), preserving the descent to zero. This extends to homomorphic images: any homomorphic image of a nilpotent Lie algebra is nilpotent, as it is isomorphic to a quotient by the kernel ideal. In the special case where \mathfrak{g} is abelian (nilpotency class 1), these properties simplify: subalgebras reduce to vector subspaces, which are automatically abelian as Lie algebras, and quotients by ideals (also subspaces) remain abelian.

Center and central series

The center of a Lie algebra \mathfrak{g}, denoted Z(\mathfrak{g}), is the subalgebra consisting of all elements z \in \mathfrak{g} such that [z, x] = 0 for every x \in \mathfrak{g}. This center is always an ideal of \mathfrak{g}, as the Lie bracket with elements outside the center vanishes. For any non-abelian nilpotent Lie algebra, the center is non-trivial, meaning Z(\mathfrak{g}) \neq 0. The upper central series of \mathfrak{g} is the ascending sequence of ideals defined by Z_0(\mathfrak{g}) = \{0\} and Z_{k+1}(\mathfrak{g}) = \{z \in \mathfrak{g} \mid [z, x] \in Z_k(\mathfrak{g}) \text{ for all } x \in \mathfrak{g}\}, so that Z_1(\mathfrak{g}) = Z(\mathfrak{g}). Each successive factor Z_{k+1}(\mathfrak{g}) / Z_k(\mathfrak{g}) lies in the center of the quotient \mathfrak{g} / Z_k(\mathfrak{g}), making these factors abelian Lie algebras. A Lie algebra \mathfrak{g} is nilpotent if and only if this series reaches \mathfrak{g} after finitely many steps, i.e., Z_s(\mathfrak{g}) = \mathfrak{g} for some positive s. The quotient \mathfrak{g} / Z(\mathfrak{g}) inherits nilpotency from \mathfrak{g}, with the nilpotency class of the quotient being at most one less than that of \mathfrak{g}. Iterating this process along the upper central series yields a chain of quotients that are successively central and thus abelian. In particular, for a nilpotent Lie algebra of class 2, the derived algebra satisfies [\mathfrak{g}, \mathfrak{g}] \subseteq Z(\mathfrak{g}). In the Heisenberg algebra, a canonical example of a class-2 nilpotent Lie algebra, the center coincides with the derived algebra, both one-dimensional and consisting of scalar multiples of the central basis element.

Engel's theorem

Engel's theorem characterizes nilpotent subalgebras of \mathfrak{gl}(V) over fields of characteristic zero. Specifically, over an algebraically closed field k of characteristic zero, a finite-dimensional Lie subalgebra \mathfrak{h} \subseteq \mathfrak{gl}(V) consisting entirely of nilpotent endomorphisms is simultaneously strictly upper triangularizable: there exists a basis of V such that all matrices in \mathfrak{h} are strictly upper triangular (zeros on and below the diagonal). The proof uses on \dim V. For \dim V = 1, trivial. For larger, a key shows that such a \mathfrak{h} has a common zero eigenvector v \in V (i.e., \mathfrak{h} v = 0), found by considering maximal proper subalgebras and nilpotency of actions. The spanned by orbits or the kernel complement allows , building the . This refines Lie's for solvable subalgebras (upper triangular), specializing to zero diagonal due to nilpotency. A direct is that every finite-dimensional nilpotent \mathfrak{g} over such a admits a faithful into the of strictly upper triangular matrices. This follows by via Ado's or the adjoint (adjusted for center) into \mathfrak{gl}(\mathfrak{g}), then applying Engel's to triangularize strictly, preserving faithfulness. This triangularization relates to Jordan form: nilpotent operators have only zero eigenvalue, with Engel ensuring a common flag refining individual structures. In positive p > 0, Engel's theorem fails: there exist finite-dimensional representations of Lie algebras that are not simultaneously triangularizable. For instance, the Heisenberg algebra over a field of p possesses irreducible representations of dimension p, lacking a common eigenvector. Instead, the Engel condition—that [\dots [x, y], x], \dots , x] = 0 after n brackets for some fixed n and all x, y—characterizes p-nilpotency or local nilpotency in this setting, with bounded Engel conditions implying by results of Zelmanov. Applications include the of irreducible representations: over algebraically closed fields of characteristic zero, all irreducible modules over a nilpotent Lie algebra are one-dimensional, as any higher-dimensional irreducible would contradict the existence of a common eigenvector (proper ) by Lie's theorem.

Killing form and automorphisms

The Killing form of a finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero is the symmetric bilinear form defined by B(X, Y) = \trace(\ad_X \circ \ad_Y) for X, Y \in \mathfrak{g}, where \ad_X: \mathfrak{g} \to \mathfrak{g} is the adjoint map given by \ad_X(Z) = [X, Z]. For a nilpotent Lie algebra \mathfrak{g}, each \ad_X is a nilpotent endomorphism, so \ad_X \circ \ad_Y is also nilpotent and thus has trace zero; hence, B vanishes identically on \mathfrak{g}. This degeneracy of the Killing form implies that nilpotent Lie algebras are not semisimple, as semisimplicity requires the Killing form to be non-degenerate. Moreover, the vanishing form means that nilpotent Lie algebras admit no non-degenerate invariant symmetric bilinear form, distinguishing them from reductive cases where such forms exist. The automorphism group \Aut(\mathfrak{g}) of a nilpotent algebra \mathfrak{g} consists of Lie algebra automorphisms, which are invertible linear maps preserving the bracket. Inner automorphisms arise from the adjoint action of the associated connected simply connected , forming the image of the \Ad: G \to \GL(\mathfrak{g}); however, for nilpotent \mathfrak{g}, this image is unipotent and often a proper of \Aut(\mathfrak{g}). Outer automorphisms, not inner, exist in many nilpotent cases, contributing to the structure of \Aut(\mathfrak{g}) beyond the adjoint group. For the Heisenberg algebra \mathfrak{h}_3, the three-dimensional nilpotent over \mathbb{R} or \mathbb{C} with basis \{X, Y, Z\} and relations [X, Y] = Z, [X, Z] = [Y, Z] = 0, the includes inner automorphisms from the unipotent group exponentials and outer ones such as s that act non-trivially on \langle Z \rangle, for example, maps sending Z \mapsto \lambda Z for \lambda \neq 0 while adjusting X, Y to preserve the bracket. In the \mathfrak{n} of n \times n strictly upper triangular matrices over a of characteristic zero, automorphisms include inner conjugations by unipotent matrices preserving the nilpotent structure, as well as outer ones like automorphisms reordering basis elements compatibly with the of ideals and central () automorphisms on graded components. The derivation algebra \Der(\mathfrak{g}) of a nilpotent Lie algebra \mathfrak{g} over a of characteristic zero consists of all linear endomorphisms D: \mathfrak{g} \to \mathfrak{g} satisfying D([X, Y]) = [D(X), Y] + [X, D(Y)]; inner derivations are those of the form \ad_X for X \in \mathfrak{g}, and outer derivations form the \Der(\mathfrak{g}) / \Inn(\mathfrak{g}). Every non-zero nilpotent Lie algebra admits outer derivations, as established by results showing the existence of non-inner derivations acting nilpotently on the central series. In characteristic zero, the Killing form remains zero overall due to nilpotency.

Derived algebra in solvable Lie algebras

A fundamental result in the theory of Lie algebras over fields of characteristic zero states that if \mathfrak{g} is a finite-dimensional , then its derived algebra \mathfrak{g}' = [\mathfrak{g}, \mathfrak{g}] is . This theorem provides a partial converse to the inclusion that every Lie algebra is solvable. The proof relies on Lie's theorem, which asserts that for a over \mathbb{C}, there exists a basis in which every operator \mathrm{ad}_x (for x \in \mathfrak{g}) is represented by an . Consequently, for any x, y \in \mathfrak{g}, the operator \mathrm{ad}_{[x,y]} = [\mathrm{ad}_x, \mathrm{ad}_y] is strictly upper triangular, hence . Since every element of \mathfrak{g}' is ad-nilpotent, Engel's theorem implies that \mathfrak{g}' itself is . This result has several implications for the structure of solvable Lie algebras. The derived series \mathfrak{g}^{(k)} strictly decreases in dimension at each step until reaching zero, with \dim \mathfrak{g}' < \dim \mathfrak{g} unless \mathfrak{g} is abelian. Moreover, the nilpotency class of \mathfrak{g}' is bounded by the dimension of \mathfrak{g}, providing control over the complexity of iterated commutators. A example arises in the Borel \mathfrak{b} of \mathfrak{[sl](/page/SL)}(n, \mathbb{C}), consisting of trace-zero upper triangular matrices, which is solvable. Its derived is the Lie of strictly upper triangular matrices, which is of at most n-1. In positive , the does not hold without additional conditions; for instance, there exist solvable Lie algebras whose derived algebras are not . Such cases often require hypotheses like restricted solvability to ensure nilpotency of the derived . This property aids in classifying solvable Lie algebras by highlighting their nilpotent ideals, particularly the nilradical, which contains \mathfrak{g}' and facilitates decompositions into nilpotent components.