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| Adjoint representation | |
|---|---|
| Name | Adjoint representation |
| Type | Mathematical concept |
| Field | Sophus Lie, Wilhelm Killing, Élie Cartan |
| Related | Lie algebra, Lie group, representation theory, linear algebra |
Adjoint representation is a construction in Lie algebra and Lie group theory that associates to each element of a Lie algebra or Lie group a linear transformation on the algebra itself; it plays a central role in the study of Élie Cartan’s classification, Wilhelm Killing’s form, and Sophus Lie’s original work on continuous symmetries. The adjoint connects structural invariants such as the Killing form, Cartan subalgebra, and root system with concrete linear operators, and it underpins many aspects of representation theory and applications in Albert Einstein-inspired physics, Hermann Weyl’s harmonic analysis, and modern Pierre Deligne-influenced algebraic geometry.
For a Lie algebra g over a field (often the real numbers or complex numbers) the adjoint map ad assigns to each x in g the endomorphism ad(x): g → g given by ad(x)(y) = [x,y], where [ , ] denotes the Lie bracket introduced by Sophus Lie and developed by Élie Cartan and Wilhelm Killing. For a Lie group G with Lie algebra g the adjoint action Ad is a group homomorphism Ad: G → GL(g) defined by conjugation at the group level: Ad(g)(X) = (d/dt)|_{t=0} g exp(tX) g^{-1}, relating constructions of Hermann Weyl, Élie Cartan, and Claude Chevalley.
Given a finite-dimensional Lie algebra g, the adjoint representation is the linear representation ad: g → gl(g) sending x to ad(x). The map ad is a Lie algebra homomorphism satisfying ad([x,y]) = [ad(x), ad(y)] in gl(g), a fact exploited in work by Nathan Jacobson and Harish-Chandra to analyze ideals, solvable and semisimple structure via Levi decomposition and the Killing form introduced by Wilhelm Killing. The kernel of ad is the center Z(g) of g, a concept studied by Élie Cartan and Hermann Weyl in classification, and the image ad(g) identifies g/ Z(g) with a subalgebra of gl(g), a perspective central to Ado's theorem and results by Jean-Pierre Serre.
For a Lie group G with exponential map exp: g → G the adjoint representation Ad: G → GL(g) is defined by Ad(g) = d( Inn_g )_{e} where Inn_g: G → G is conjugation by g. The differential at the identity yields Ad(g)(X) = gXg^{-1} in matrix groups like GL(n, R), SL(n, C), SO(n), and Sp(n). The map Ad is a continuous homomorphism studied in the contexts of Élie Cartan’s theory of symmetric spaces, Hermann Weyl’s character theory, and Claude Chevalley’s study of algebraic groups; its image, the adjoint group Ad(G), features in classification results by Armand Borel and Jean-Pierre Serre.
The adjoint action preserves structural decompositions such as the root system decomposition relative to a Cartan subalgebra h, a theme present in the work of Élie Cartan and Killing. Eigenvalues of ad(H) for H in h yield roots α ∈ h*, and the corresponding root spaces g_α underpin the classification of semisimple algebras by Dynkin diagrams as developed by Eugene Dynkin. The adjoint representation is faithful precisely when g is centerless, a condition in semisimple Lie algebra theory; its trace pairing produces the Killing form, which is nondegenerate for semisimple g, a criterion used by Cartan and Weyl to distinguish simple components. The representation interacts with universal constructions like the universal enveloping algebra U(g) studied by Benoit Mandelbrot—note: Mandelbrot worked in fractals, but I. M. Gelfand and Joseph Bernstein contributed to enveloping algebra theory—and satisfies identities derived from the Jacobi identity.
Classical matrix Lie algebras provide concrete adjoint representations: for g = sl(n, C) the adjoint action is conjugation by traceless matrices inside GL(n, C), an approach used by Hermann Weyl and Élie Cartan in representation computations; for so(n) and sp(n) the adjoint coincides with commutator action inside O(n), SO(n), and Sp(n, C). In low dimensions, ad for g = sl(2, C) realizes the three-dimensional irreducible representation linked historically to work by Sophus Lie and Élie Cartan and later analyzed by Eugene Dynkin and Harish-Chandra. Exceptional Lie algebras like g2, f4, e6, e7, e8 have adjoint representations of dimensions 14, 52, 78, 133, 248 respectively, classified through contributions by Wilhelm Killing, Élie Cartan, and later by Bourbaki and Armand Borel.
The adjoint representation is central in the study of symmetry in mathematical physics, where Paul Dirac, Richard Feynman, and Albert Einstein’s frameworks utilize Lie symmetries, and in gauge theories formulated by Yang Chen Ning and Robert Mills where structure constants from ad appear in field strength expressions. In pure mathematics it underlies representation theory of semisimple groups, influences the formulation of the Langlands program by Robert Langlands, and plays a role in geometric constructions by Alexandre Grothendieck and Pierre Deligne in algebraic group theory. The adjoint action also informs the study of cohomology of groups by Jean-Pierre Serre and deformation theory as seen in work by Max Karoubi and Gerald Hochschild.
Category:Lie theory