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Jordan–Chevalley decomposition

The is a in the theory of finite-dimensional algebras over algebraically closed fields of characteristic zero, expressing every element x \in \mathfrak{g} as a unique sum x = s + n, where s is a semisimple element (meaning the adjoint \mathrm{ad}_s is diagonalizable), n is (\mathrm{ad}_n is ), and [s, n] = 0. This structure generalizes the classical of endomorphisms on vector spaces, where a linear splits into semisimple and commuting parts corresponding to its Jordan canonical form. In the setting of semisimple Lie algebras, the decomposition exists and is unique for all elements, with the semisimple part belonging to a and facilitating the study of root systems and . For general Lie algebras, the abstract Jordan–Chevalley decomposition—defined such that the parts s and n satisfy the property in every \pi: \mathfrak{g} \to \mathfrak{gl}(V)—exists for an element x if and only if x lies in the derived [\mathfrak{g}, \mathfrak{g}], with both s and n also in [\mathfrak{g}, \mathfrak{g}]. Moreover, every element of \mathfrak{g} admits such a decomposition precisely when \mathfrak{g} is perfect (i.e., [\mathfrak{g}, \mathfrak{g}] = \mathfrak{g}). Named after mathematicians Camille Jordan, who developed the matrix decomposition in the late , and , who extended it to in the mid-20th century, this theorem underpins key results in , including Cartan's criteria for solvability and semisimplicity, and the structure of reductive groups. It also ensures that images under Lie algebra homomorphisms preserve the semisimple and nilpotent components, aiding in the classification of representations and the analysis of algebraic groups.

Fundamentals

Definition and motivation

The Jordan–Chevalley decomposition is a fundamental tool in linear and that separates linear operators or elements into semisimple and components. In the context of linear , key prerequisite concepts include semisimple endomorphisms, which are diagonalizable over an (meaning their minimal has distinct roots), and nilpotent endomorphisms, for which some power is the zero operator (such as strictly upper-triangular matrices in a suitable basis). The decomposition requires the base field to be algebraically closed and of characteristic zero to ensure the existence of eigenvalues and the splitting of polynomials into linear factors. For a finite-dimensional V over such a , every x: V \to V admits a unique x = s + n, where s is semisimple, n is , and s and n commute (i.e., [s, n] = 0). This additive splitting captures the "regular" diagonalizable behavior in s and the "irregular" vanishing behavior in n, providing a way to analyze the structure of x. The motivation for this decomposition stems from the Jordan canonical form, which represents matrices as sums of diagonal and blocks, allowing the isolation of eigenvalues and generalized eigenspaces to study operator spectra and invariants. Chevalley's extension to Lie algebras further motivates its use in separating semisimple (diagonalizable action) and (vanishing powers) parts to examine representations, element conjugacy classes, and orbital structures in algebraic groups. In Lie algebras over an of characteristic zero, every element x decomposes uniquely as x = s + n, where s is semisimple (the \mathrm{ad}(s) is diagonalizable), n is (\mathrm{ad}(n) is ), and [s, n] = 0. This abstract version preserves the commuting property and extends the linear algebraic tool to infinite-dimensional settings when applicable, facilitating the study of derivations and structures.

Historical development

The Jordan–Chevalley decomposition originated in the context of linear algebra with Camille Jordan's seminal 1870 treatise Traité des substitutions et des équations algébriques, where he established the for over algebraically closed fields, enabling the unique additive decomposition of an into a diagonalizable (semisimple) part and a part. This work laid the groundwork for understanding structure beyond mere eigenvalues, influencing subsequent developments in by providing a tool to separate commuting components with distinct spectral properties. This matrix decomposition was later connected to Lie theory through Wilhelm Killing's classification of complex semisimple Lie algebras in the 1880s. In the early , the decomposition gained broader significance through connections to , particularly via Élie Cartan's refinement of Killing's of semisimple Lie algebras over the complex numbers in his 1894 doctoral thesis and his of those over the reals in 1914, which emphasized the role of semisimple elements in root decompositions. Hermann Weyl's contemporaneous contributions in 1925–1926 further bridged linear algebra and Lie groups by developing the of compact semisimple Lie groups, where semisimple and operators naturally arise in highest weight modules and Cartan subalgebras. These advancements highlighted the decomposition's utility in and , setting the stage for its abstraction beyond matrices. Claude Chevalley generalized the decomposition to arbitrary s over fields of characteristic zero in his 1946 book Theory of Lie Groups, introducing the semisimple-nilpotent split for elements in the of an algebraic group, which preserves the ad-action and commutation properties. This formulation unified Jordan's matrix-level insight with the structural theory of , proving existence and uniqueness under the . In the 1950s and 1960s, and others recognized its centrality in the theory of linear algebraic groups, integrating it into the study of reductive groups and their Borel subgroups over arbitrary fields, as detailed in foundational texts on algebraic group structure. Post-2000 refinements have extended the decomposition to positive characteristic, addressing limitations in characteristic zero by adapting it to modular Lie algebras, where nilpotency criteria are modified via restricted enveloping algebras. Recent applications in the include its role in quantum groups, where the decomposition aids in analyzing Drinfeld-Jimbo quantizations and q-deformations of semisimple elements, facilitating computations in braided categories and representations. Similarly, in modular Lie algebras, it supports the decomposition of irreducible modules, enhancing classifications in positive characteristic settings relevant to representations.

Decomposition for endomorphisms

Uniqueness theorem

The Jordan–Chevalley decomposition theorem asserts that for an x on a finite-dimensional V over an of characteristic zero, if x = s + n = s' + n' where s and s' are semisimple endomorphisms, n and n' are endomorphisms, [s, n] = 0, and [s', n'] = 0, then s = s' and n = n'. This uniqueness ensures that the decomposition is , providing a well-defined separation of the semisimple and nilpotent components of x. The proof of uniqueness relies on the primary decomposition theorem for endomorphisms. The minimal polynomial m_x(t) of x factors uniquely as m_x(t) = m_s(t) m_n(t), where m_s(t) is the product of distinct linear factors corresponding to the eigenvalues (reflecting the semisimple structure) and m_n(t) is the product of the contributions (higher powers of those linear factors). By the primary decomposition theorem, V decomposes uniquely into generalized eigenspaces V = \bigoplus_{\lambda} \ker((x - \lambda I)^{k_\lambda}), where each summand is under x. On each such space, the semisimple part s acts as multiplication by \lambda, while the part n accounts for the Jordan blocks; any alternative decomposition s' + n' must respect this decomposition and thus coincide with s + n. This uniqueness arises necessarily from the structure imposed by the , as the coprime factorization of the minimal guarantees that the projections onto the semisimple and components are uniquely determined. In particular, the Fitting decomposition of the over the k induced by x, which separates the torsion-free and torsion parts, further supports this by implying that the component is confined to the Fitting submodule, uniquely identifying both parts. An alternative argument considers s - s' = n' - n; since both s and s' are polynomials in x, their difference is semisimple and commutes with x, while n' - n is , forcing the difference to be zero as the only operator that is both semisimple and .

Existence proofs

The existence of the Jordan–Chevalley decomposition for a linear A of a finite-dimensional V over a k follows from constructive proofs that explicitly build the semisimple part s and nilpotent part n = A - s. An elementary proof proceeds by on \dim V. For \dim V = 1, A is by a scalar, which is semisimple with n = 0. For higher dimensions and k algebraically closed, let \lambda be an eigenvalue of A. The space V decomposes as a of the generalized eigenspace W = \ker (A - \lambda I)^m (where m is the algebraic multiplicity) and a complementary A- U. By , the restriction of A|_W - \lambda I is nilpotent, so A|_W = \lambda I_W + n_W with n_W ; the restriction A|_U decomposes as s_U + n_U. Defining s = \lambda \mathrm{proj}_W + s_U and n = n_W + n_U yields the decomposition, as s is semisimple (diagonalizable on W and by on U) and n is (on W and by on U), with s and n commuting since they preserve the decomposition. A related elementary method uses the theorem, applicable over any where the minimal polynomial m(t) of A factors into distinct irreducible factors p_i(t)^{e_i}. Then V = \bigoplus_i \ker p_i(A)^{e_i}, and on each primary component V_i = \ker p_i(A)^{e_i}, the restriction A|_{V_i} has primary minimal polynomial p_i(t)^{e_i}. If p_i(t) is linear (over algebraically closed k), A|_{V_i} = \lambda_i I_{V_i} + n_i with n_i nilpotent by the above induction or Fitting decomposition (splitting V_i into nilpotent and invertible parts relative to A - \lambda_i I). The semisimple part is s = \sum_i \lambda_i \mathrm{proj}_{V_i} (extended by zero elsewhere), which is semisimple as a direct sum of scalars, and n = A - s is nilpotent as a direct sum. For general irreducible p_i, the semisimple part on V_i is the scalar multiple corresponding to the root in a splitting . These constructions yield an explicit polynomial p(t) \in k such that s = p(A), confirming s is a polynomial in A (hence commutes with n). Specifically, using the Chinese Remainder Theorem on the factors of m(t), one finds p(t) satisfying p(\lambda) = \lambda on each root \lambda of m(t) (modulo higher powers) while ensuring nilpotency of A - p(A). For the case of distinct eigenvalues \lambda_1, \dots, \lambda_r (as in the minimal polynomial of s), this specializes to Lagrange interpolation: s = \sum_{\lambda} \lambda \prod_{\mu \neq \lambda} \frac{A - \mu I}{\lambda - \mu}, where the product is over distinct eigenvalues \mu; on the generalized eigenspace for \nu \neq \lambda, the Lagrange basis polynomial vanishes, while on the eigenspace for \lambda it acts as the identity. This formula projects onto eigenspaces, yielding the semisimple part directly, and is computable via computer algebra systems for concrete matrices by factoring the characteristic polynomial. To establish existence over arbitrary fields k (not necessarily algebraically closed), embed k in a separable closure k_s and base-change V \otimes_k k_s, where the decomposition exists by the elementary methods above: \overline{A} = \overline{s} + \overline{n}. Uniqueness (established separately) implies the \mathrm{Gal}(k_s/k) acts on the decomposition, preserving the semisimple and parts up to conjugation. The semisimple part \overline{s} descends to a k-defined s via Galois : specifically, s is the unique element fixed by the action such that its base-change is \overline{s}, obtained by averaging over the group or via invariant polynomials in A with coefficients in k. The part n = A - s then follows, ensuring both are defined over k. This works for perfect fields and extends generally using separability.

Examples and counterexamples

A canonical example of the Jordan–Chevalley decomposition arises with nilpotent endomorphisms. Consider the 3×3 nilpotent Jordan block matrix N = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} over an of , such as \mathbb{C}. Here, the semisimple part is the s = 0, and the nilpotent part is n = N itself, since N satisfies N^3 = 0 but N^2 \neq 0, and it is not diagonalizable. For a non-trivial decomposition, take the 2×2 Jordan block A = \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix} over \mathbb{C}, where \lambda \neq 0. The semisimple part is the scalar multiple of the s = \lambda I = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix}, which is diagonalizable, and the nilpotent part is n = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, satisfying n^2 = 0. This illustrates how the decomposition separates the eigenvalue contribution from the non-diagonalizable defect. A to the existence of the over non-algebraically closed fields occurs with in \mathbb{[R](/page/R)}^2. The 90-degree R = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} has no real eigenvalues, as its x^2 + 1 = 0 is irreducible over \mathbb{[R](/page/R)}. Thus, R admits no real semisimple part that is diagonalizable over \mathbb{[R](/page/R)}, though over \mathbb{C} it decomposes as semisimple with eigenvalues i and -i, and nilpotent part zero. In positive characteristic, the decomposition may fail over non-perfect fields. Over the function field \mathbb{F}_2(t) in characteristic 2, the A = \begin{pmatrix} 0 & t \\ 1 & 0 \end{pmatrix} of the irreducible inseparable x^2 + t lacks a Jordan–Chevalley decomposition because the field is imperfect, preventing the separation into commuting semisimple and parts. Geometrically, the Jordan–Chevalley decomposition for endomorphisms of a can be interpreted via the associated action on the flag , where the semisimple part corresponds to actions preserving flags, while the part generates unipotent orbits within the , linking to the structure of nilpotent cones as normal varieties in .

Extension to Lie algebras

Statement in Lie algebras

In the context of Lie algebras, the Jordan–Chevalley decomposition provides an additive splitting of elements that respects the Lie bracket structure. For a finite-dimensional semisimple Lie algebra \mathfrak{g} over an algebraically closed field k of characteristic zero, every element x \in \mathfrak{g} admits a unique decomposition x = s + n, where s, n \in \mathfrak{g}, the adjoint endomorphism \mathrm{ad}(s): \mathfrak{g} \to \mathfrak{g} is diagonalizable (i.e., semisimple), \mathrm{ad}(n) is nilpotent, and [s, n] = 0. This decomposition arises from the \mathrm{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}), where \mathrm{ad}_x = \mathrm{ad}_s + \mathrm{ad}_n with [\mathrm{ad}_s, \mathrm{ad}_n] = 0 forms the decomposition of the \mathrm{ad}_x into commuting semisimple and parts. Unlike the classical decomposition, which applies to arbitrary endomorphisms in \mathfrak{gl}(V) and yields parts within the endomorphism , the Jordan–Chevalley version ensures that the semisimple and nilpotent components s and n lie intrinsically within \mathfrak{g} itself, compatible with the via the commutativity condition. The condition [s, n] = 0 underscores the intrinsic nature of the decomposition to the structure, as it implies that the subalgebra generated by s and n is a of the semisimple and parts under the action.

Proof of existence

The existence of the Jordan–Chevalley decomposition in the context of finite-dimensional s over an of characteristic zero follows from adapting the decomposition to the Lie structure via the . For a \mathfrak{L}, the map \mathrm{ad}: \mathfrak{L} \to \mathfrak{gl}(\mathfrak{L}) is an injective , embedding \mathfrak{L} as a semisimple of endomorphisms on itself. Given any x \in \mathfrak{L}, the \mathrm{ad}_x decomposes as \mathrm{ad}_x = e + n, where e is semisimple and n is in \mathfrak{gl}(\mathfrak{L}). Since \mathfrak{L} is semisimple, the images under \mathrm{ad} of its elements are closed under taking semisimple and parts in this representation, yielding unique s, n \in \mathfrak{L} such that \mathrm{ad}_x = \mathrm{ad}_s + \mathrm{ad}_n, with \mathrm{ad}_s semisimple and \mathrm{ad}_n . Thus, x = s + n, where s is semisimple (its has eigenvalues corresponding to evaluated on a containing s) and n is (satisfying (\mathrm{ad}_n)^k = 0 for some k \leq \dim \mathfrak{L}). The key equation is \mathrm{ad}_s(y) = [s, y] for all y \in \mathfrak{L}, ensuring the decomposition respects the Lie bracket. An elementary approach for semisimple Lie algebras leverages the structure theorem, where \mathfrak{L} is a direct sum of simple ideals, each isomorphic to a subalgebra of \mathfrak{sl}(n) for some n. The matrix Jordan decomposition in \mathfrak{sl}(n) directly provides the semisimple and nilpotent parts, which lift to \mathfrak{L} via the isomorphism. For the general finite-dimensional case over characteristic zero, the abstract exists for an element x x \in [\mathfrak{L}, \mathfrak{L}], in which case the unique parts s and n also lie in [\mathfrak{L}, \mathfrak{L}]. This is established using , ensuring that the semisimple and nilpotent parts in any representation of \mathfrak{L} coincide with those from the adjoint action when defined. Ado's theorem provides faithful representations into \mathfrak{gl}(V), where the decomposition aligns with the abstract one. An advanced proof employs the universal enveloping U(\mathfrak{L}) and the Poincaré–Birkhoff–Witt (PBW) theorem to separate the semisimple and components . The adjoint action extends to U(\mathfrak{L}), and PBW provides a basis of ordered monomials, allowing the semisimple part of x to be expressed as a p(x) that commutes with a and diagonalizes thereon, while the remainder satisfies higher powers vanishing in the adjoint action. This method highlights the in the enveloping , ensuring the is compatible with the . Extensions to restricted Lie algebras in positive , beyond the characteristic zero assumption, were established in the using p-polynomials and minimal p-equations to construct the , provided the algebra satisfies certain solvability or conditions.

Key properties

The Jordan–Chevalley in algebras exhibits several intrinsic properties that underscore its utility in . A fundamental property is its preservation under : if \phi is an automorphism of the \mathfrak{g}, and x = s + n is the decomposition of x \in \mathfrak{g}, then \phi(x) = \phi(s) + \phi(n) is the decomposition of \phi(x). This compatibility follows from the fact that the is a , ensuring the semisimple and parts transform accordingly. Another important feature concerns centralizers. The centralizer C_{\mathfrak{g}}(x) of x \in \mathfrak{g} coincides with the intersection C_{\mathfrak{g}}(s) \cap C_{\mathfrak{g}}(n). This property arises because elements commuting with x must commute with both components, given that [s, n] = 0 and the uniqueness of the decomposition. In the context of a \mathfrak{h}, the centralizer C_{\mathfrak{g}}(\mathfrak{h}) = \mathfrak{h}, and for semisimple x \in \mathfrak{h}, this reinforces the self-centralizing nature of toral subalgebras. The component n satisfies a clear : n is if and only if all eigenvalues of \mathrm{ad}_n are zero, reflecting the fact that over an of characteristic zero, nilpotency equates to the absence of nonzero eigenvalues in the . For real semisimple algebras, additional invariants like the Killing form can provide necessary conditions such as B(n, n) = 0, but full characterization involves the real Jordan form of \mathrm{ad}_n. Regarding semisimplicity, the semisimple component s is semisimple if and only if it lies in some or, equivalently, generates a toral subalgebra (an abelian consisting entirely of semisimple elements). Maximal toral s are precisely the Cartan subalgebras in semisimple Lie algebras, providing a structural test for the semisimple part via its centralizer and abelian nature. In extensions to Kac–Moody algebras, which are infinite-dimensional generalizations of finite-dimensional semisimple Lie algebras, the Jordan–Chevalley decomposition persists under suitable conditions, such as when elements induce locally finite actions. For symmetrizable Kac–Moody algebras, the decomposition aligns with the root space structure and triangular s, facilitating the study of orbits and representations, though uniqueness may require additional assumptions like cleftness in locally finite settings.

Algebraic and group generalizations

Multiplicative decomposition

In a G over a of characteristic zero, the multiplicative Jordan–Chevalley provides an analog of the additive for elements of the group. Specifically, for any g \in G(k) where k is the base , there exist unique elements g_s, g_u \in G(k) such that g = g_s g_u = g_u g_s, with g_s semisimple (diagonalizable over an of k) and g_u unipotent (all eigenvalues equal to 1). In the case of a connected group over an , the semisimple part g_s lies in a of G. For the general linear group GL_n(k), the semisimple part g_s is diagonalizable over an , with eigenvalues that are roots of unity or elements of k, while the unipotent part g_u takes the form I + N where N is a . For instance, a such as \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} decomposes as g_s = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} (semisimple) and g_u = \begin{pmatrix} 1 & 1/2 \\ 0 & 1 \end{pmatrix} (unipotent), satisfying the commutation and product conditions. The existence of this decomposition follows from embedding G into GL(V) for a faithful representation V, applying the additive Jordan–Chevalley decomposition to the image of g, and reconstructing g_s and g_u using the polynomial equations defining G. A key step involves taking the Lie algebra logarithm of g (possible in characteristic zero for elements in the image of the exponential map), decomposing the resulting element X = \log(g) additively as X = s + n with s semisimple and n nilpotent in the Lie algebra \mathfrak{g}, and then exponentiating back to obtain g = \exp(s + n) = \exp(s) \exp(n). Uniqueness inherits from the uniqueness of the additive Jordan decomposition in the endomorphism algebra. The equality \exp(s + n) = \exp(s) \exp(n) holds because s and n commute, allowing the to simplify: since [s, n] = 0, the series terminates, and the product formula applies directly in the case. This extends to applications in , particularly in the study of Springer fibers, which parametrize the geometry of flags stabilized by elements in the ; developments in the 2000s have linked it to computations and actions on these fibers.

Applications to Lie groups

In linear algebraic groups over an of characteristic zero, the Jordan–Chevalley decomposition extends multiplicatively to classify group elements as products g = s u, where s is semisimple and u is unipotent, with both commuting and the decomposition unique. This classification distinguishes purely semisimple elements (diagonalizable over the ), unipotent elements (all eigenvalues 1), and mixed types, facilitating the study of structures. Specifically, Borel subgroups, which are maximal solvable connected s, decompose as a of a (semisimple) and its unipotent radical, leveraging the decomposition to analyze parabolic inductions and flag varieties. For real semisimple Lie groups such as \mathrm{SL}(n, \mathbb{R}) or \mathrm{SO}(n), the decomposition adapts to the real setting via the Cartan \theta, which splits the \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p} into compact (\mathfrak{k}) and noncompact (\mathfrak{p}) parts. An element X \in \mathfrak{g} decomposes as X = E + H + N, where E is elliptic (semisimple in \mathfrak{k}), H is (semisimple in \mathfrak{p}), and N is , all commuting; at the group level, g = e h u with e elliptic, h hyperbolic, and u unipotent. The unipotent component aligns with the G = K A N, where N is the nilpotent part and A captures the hyperbolic direction. A concrete example arises in \mathrm{SL}(2, \mathbb{R}), where elements classify as (|\mathrm{trace}| > 2, real eigenvalues; conjugate to \begin{pmatrix} \cosh t & \sinh t \\ \sinh t & \cosh t \end{pmatrix} for t > 0 if trace > 2, or to \begin{pmatrix} -\cosh t & \sinh t \\ \sinh t & -\cosh t \end{pmatrix} for trace < -2), parabolic (|\mathrm{trace}| = 2, single Jordan block; unipotent if eigenvalues 1, i.e., trace = 2), or elliptic (|\mathrm{trace}| < 2, eigenvalues on the unit , rotations). This mirrors dynamics on the hyperbolic plane, with elements acting as translations along geodesics, parabolic as horocyclic translations, and elliptic as rotations around fixed points. The plays a central role in classifying conjugacy classes in semisimple groups, as semisimple and unipotent parts determine stable and rigid classes, essential for orbital integrals in . In , nilpotent orbits—parametrized by weighted Dynkin diagrams via the Bala– —underlie the structure of coadjoint orbits and Whittaker models, with integrals over these orbits computing characters of unitary representations. Recent applications in automorphic forms, post-2010, utilize the to expand coefficients along orbits, linking endoscopic transfers and Langlands correspondences for groups like \mathrm{PGL}(n).