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| SL_n(C) | |
|---|---|
| Name | Special linear group |
| Notation | SL_n(C) |
| Type | Complex Lie group, Linear algebraic group |
| Dimension | n^2 − 1 |
| Field | Complex numbers |
| Rank | n − 1 |
SL_n(C)
SL_n(C) is the group of n×n complex matrices with determinant 1, forming a connected complex Lie group and a reductive linear algebraic group. It is a central example in Felix Kleinian geometry, Évariste Galois-inspired algebraic group theory, and the representation theory studied by Hermann Weyl, Claude Chevalley, and John T. Tate. SL_n(C) appears throughout modern mathematics and theoretical physics, including in work of Élie Cartan, Harish-Chandra, and applications in Yang–Mills theory and Quantum field theory.
SL_n(C) is defined as the set of invertible matrices A in GL_n(C) with det(A) = 1. As a subgroup of General linear group GL_n(C), it is Zariski-closed and an affine algebraic variety over C. SL_n(C) is simple for n ≥ 2 as an abstract algebraic group except for its center: the center consists of scalar matrices ζI with ζ^n = 1, linking to the cyclic group of roots of unity studied by Niels Henrik Abel and Carl Friedrich Gauss. The determinant map gives the exact sequence 1 → SL_n(C) → GL_n(C) → C^× → 1, reflecting its role in the classical groups treated by Wilhelm Killing and Élie Cartan.
Concrete matrix examples include elementary matrices, permutation matrices from Augustin-Louis Cauchy-related determinants, and companion matrices appearing in the theory of linear recurrences associated with Joseph-Louis Lagrange. For n = 2, SL_2(C) is generated by unipotent matrices and diagonal matrices of determinant 1, and it acts by Möbius transformations on the Riemann sphere studied by Bernhard Riemann and Henri Poincaré. For n = 3, matrices in SL_3(C) model projective transformations of the complex projective plane relevant to Giuseppe Veronese embeddings and Algebraic geometry associated with Federigo Enriques and Oscar Zariski. Elementary row operations correspond to left multiplication by elementary matrices, a viewpoint central to work of Arthur Cayley and James Joseph Sylvester.
As a Lie group, SL_n(C) is a complex Lie group of complex dimension n^2 − 1 and real dimension 2(n^2 − 1). Its Lie algebra sl_n(C) consists of traceless n×n complex matrices, studied by Élie Cartan in the classification of semisimple Lie algebras. The Killing form and root space decomposition connect to the Cartan subalgebras and Dynkin diagram of type A_{n−1}, topics developed by Weyl and Hermann Weyl's school. The exponential map exp: sl_n(C) → SL_n(C) is surjective onto a neighborhood of the identity, with global properties explored by John von Neumann and Shiing-Shen Chern in differential geometry.
Topologically, SL_n(C) is homotopy equivalent to its maximal compact subgroup SU(n), studied by Elie Cartan and Marston Morse. Thus π_1(SL_n(C)) ≅ Z/nZ for n ≥ 2 via the center and covering theory as in classics by H. Hopf and Jean-Pierre Serre. Higher homotopy groups relate to stable homotopy phenomena analyzed by J. H. C. Whitehead and Serre, and Bott periodicity results of Raoul Bott yield computations for π_k of classical groups important in topology and index theory of Atiyah–Singer.
Finite-dimensional irreducible representations of SL_n(C) are highest-weight modules classified by dominant integral weights corresponding to partitions and Young diagrams, a theory advanced by Weyl, James, and Schur. Tensor product decompositions use Littlewood–Richardson rules studied by Littlewood and Richardson, while characters are given by the Weyl character formula from Weyl's work. Infinite-dimensional representations, Harish-Chandra modules, and unitary dual questions were developed by Harish-Chandra, David Vogan, and Bernstein–Gelfand–Gelfand techniques, with links to automorphic representations in the program of Robert Langlands.
As an algebraic group, SL_n(C) contains standard subgroups: Borel subgroups of upper triangular matrices, maximal tori of diagonal matrices, and parabolic subgroups corresponding to flag varieties central to the works of Alexander Grothendieck and Jean-Pierre Serre. The Bruhat decomposition relates to the Weyl group isomorphic to the symmetric group S_n, topics traced to Francis Bruhat and André Weil. Important finite subgroups include images of Galois-theoretic and crystallographic constructions; central extensions and covers figure in studies by Steinberg and Tits.
SL_n(C) plays a role in moduli problems such as moduli of vector bundles on curves studied by Michael Atiyah and Nigel Hitchin, and in geometric invariant theory developed by David Mumford. In mathematical physics it underpins gauge groups in Yang–Mills theory and symmetry groups in string theory considered by Edward Witten. Related concepts include projective linear groups PGL_n(C), special orthogonal groups SO_n(C), symplectic groups Sp_{2n}(C), and quantum group deformations linked to Drinfeld and Vladimir Drinfeld's work. In number theory SL_n appears in Langlands reciprocity and automorphic forms examined by Langlands and Andrew Wiles.
Category:Linear algebraic groups