SL2R

sl2R, or sl(2,R), is the real Lie algebra of all 2x2 real matrices with trace zero. It is a real form of the complex Lie algebra sl2(C), and it is a three-dimensional, simple Lie algebra. As the Lie algebra of the real Lie group SL(2,R), it plays a central role in the study of linear actions and hyperbolic geometry.

A common real basis is given by

H = [ [1, 0], [0, -1] ], E = [ [0, 1], [0, 0] ], F = [ [0, 0], [1, 0] ].

These satisfy the commutation relations [H, E] = 2E, [H, F] = -2F, and [E, F] = H. The adjoint representation of sl2R has dimension 3, and the Killing form is non-degenerate with signature (2,1), reflecting its non-compact nature. Sl2R is isomorphic to so(2,1) as real Lie algebras, and it is the split real form of sl2(C).

Relation to the group SL(2,R) is given by exponentiation: sl2R is the tangent space at the identity to SL(2,R), the group of 2x2 real matrices with determinant 1. The action of SL(2,R) by conjugation induces the adjoint representation on sl2R.

Representations: finite-dimensional representations of sl2R correspond to the well-known sl2C highest-weight representations and are completely reducible over R. Irreducibles have dimension n+1 for nonnegative integers n. Because SL(2,R) is non-compact, it has no nontrivial finite-dimensional unitary representations; its unitary representation theory is infinite-dimensional, including the principal and discrete series.

See also: Lie algebra, Lie group SL(2,R), so(2,1), representation theory of sl2.