This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| sp_{2n}(C) | |
|---|---|
| Name | sp_{2n}(C) |
| Type | Lie algebra |
| Dimension | n(2n+1) |
| Field | Complex numbers |
sp_{2n}(C)
sp_{2n}(C) is the complex symplectic Lie algebra of rank n and complex dimension n(2n+1), defined as the Lie algebra of 2n×2n complex matrices preserving a nondegenerate skew-symmetric bilinear form. It appears throughout representation theory, algebraic geometry, and mathematical physics, and relates to classical groups, root systems, and highest-weight classification.
sp_{2n}(C) is realized as the set of 2n×2n complex matrices X satisfying X^T J + J X = 0 for a fixed nonsingular skew form J. Standard choices of J connect to constructions used by Wilhelm Killing and Élie Cartan in the classification of complex simple Lie algebras, and to the matrix models seen in works by Hermann Weyl, Harish-Chandra, and Claude Chevalley. Conjugation by matrices in Gl(2n,C) preserves the defining relation, linking sp_{2n}(C) to the algebraic group actions studied by Alexander Grothendieck and Pierre Deligne. Explicit block forms decompose elements into symmetric and skew blocks, a viewpoint employed in computations by Bernard Kostant and Roger Howe.
As a simple Lie algebra of type C_n in Cartan's classification, sp_{2n}(C) is simple, centerless, and complex semisimple, properties articulated in the work of Élie Cartan and catalogued in tables used by Nathan Jacobson and Serge Lang. The Killing form is nondegenerate and determines the quadratic Casimir element studied by I. M. Gel'fand and Harish-Chandra. The derived algebra equals itself, and Levi decomposition considerations link to the representation-theoretic analyses of George Lusztig and David Vogan. Automorphisms include inner automorphisms from Sp(2n,C) and diagram automorphisms described in texts by Victor Kac and Robert Langlands.
A Cartan subalgebra h of sp_{2n}(C) consists of diagonal matrices with paired entries; the root system is type C_n with short and long roots as in classifications by Élie Cartan and H. S. M. Coxeter. Simple roots can be chosen to match the Dynkin diagram studied by Claude Chevalley and Bourbaki group expositions, yielding a Weyl group isomorphic to the hyperoctahedral group treated in work by Arthur Cayley and James Joseph Sylvester. The weight lattice, coroot lattice, and fundamental weights arise in the combinatorial frameworks developed by Weyl and George Mackey, and connections to moment map images appear in research by Michèle Vergne and Victor Guillemin.
Finite-dimensional irreducible representations of sp_{2n}(C) are classified by highest weights relative to a chosen Cartan and Borel, following the highest-weight theory of Hermann Weyl and Harish-Chandra. Dominant integral weights correspond to polynomial representations constructed by tensors and symmetrizations, methods refined by Fulton, Prasad, and William Fulton with Joe Harris in algebraic geometry contexts. Classical constructions include the defining representation, adjoint representation, and the symplectic traceless tensors used in branching laws investigated by Branching rules contributors such as Roger Howe and Sergei Khoroshkin. Character formulas use the Weyl character formula proven by Weyl and generalized in the Kazhdan–Lusztig theory developed by David Kazhdan and George Lusztig.
sp_{2n}(C) is the Lie algebra of the complex symplectic group Sp(2n,C), the connected algebraic group preserving J studied by Élie Cartan and later by Armand Borel and Alexander Grothendieck. The exponential map relates neighborhood structures as in works by John Milnor and Jean-Pierre Serre for complex groups. Homogeneous spaces such as the Lagrangian Grassmannian are quotients of Sp(2n,C) and feature in studies by Mihnea Popa and Claire Voisin in algebraic geometry; their Schubert calculus connects to enumerative geometry treated by William Fulton.
The ring of invariant polynomials on sp_{2n}(C) under the adjoint action is generated by even-degree traces, a result present in invariant theory expositions by David Hilbert and Emmy Noether. Chevalley invariants for type C_n yield algebra generators used in the construction of Casimir operators appearing in the work of I. M. Gel'fand and Marcel Riesz. Lie algebra cohomology for sp_{2n}(C) with coefficients in finite-dimensional modules has been computed in contexts by Bertram Kostant and relates to Hodge theory treatments by Pierre Deligne and Jean-Pierre Serre via spectral sequence techniques featured in studies by Alexander Beilinson.
sp_{2n}(C) appears in classical mechanics through linearized symplectic transformations encountered by Joseph-Louis Lagrange and William Rowan Hamilton and in quantum mechanics and representation-theoretic models used by Paul Dirac and Eugene Wigner. It underpins dualities in theoretical physics studied by Edward Witten and Nathan Seiberg, and features in the geometric representation theory of moduli spaces investigated by Simon Donaldson and Karen Uhlenbeck. Concrete examples include sp_{2}(C) ≅ sl_2(C) connections seen in early Lie theory by Sophus Lie and matrix realizations used in computational applications by John von Neumann.
Category:Complex simple Lie algebras