sl2

SL2 refers to the special linear group of degree 2, denoted SL(2,F), which consists of all 2x2 matrices with determinant equal to 1 over a field F. When F is the real numbers, SL(2,R) is a real Lie group of dimension 3; when F is the complex numbers, SL(2,C) is a complex Lie group of dimension 3 (real dimension 6). The discrete subgroup SL(2,Z) occupies a central place in number theory and the theory of modular forms.

Its Lie algebra, sl2(F), consists of 2x2 matrices with trace zero. A standard basis is H = [ [1,0], [0,-1] ], E = [ [0,1], [0,0] ], F = [ [0,0], [1,0] ], with commutation relations [H,E] = 2E, [H,F] = -2F, [E,F] = H. The algebra is simple of type A1.

For F = R, SL(2,R) acts by linear fractional transformations on the projective line and on the upper half-plane; factoring out its center {±I} yields PSL(2,R), the orientation-preserving isometry group of the hyperbolic plane. SL(2,C) acts on hyperbolic 3-space, and PSL(2,C) is its isometry group; the Lie algebras satisfy sl2(C) ≅ so(3,1), so SL(2,C) is a double cover of PSL(2,C).

SL(2,Z) consists of 2x2 matrices with integer entries and determinant 1, acting on the upper half-plane by Möbius transformations. It underpins the theory of modular forms and is generated by S = [ [0,-1], [1,0] ] and T = [ [1,1], [0,1] ], with S^2 = (ST)^3 = -I in SL(2,Z).

In representation theory, SL(2) provides a fundamental example: finite-dimensional representations are classified by highest weight. In number theory, automorphic forms relate to SL(2,Z); in physics, SL(2,C) features as the double cover of the Lorentz group, relevant to spinor representations.