Erlangen Program at Large—0: Starting with the Group Sl2(r)

The simplest objects with non-commutative multiplication may be 2 × 2 matrices with real entries. Such matrices of determinant one form a closed set under multiplication (since det(AB) = detA · detB), the identity matrix is among them and any such matrix has an inverse (since detA 6= 0). In other words those matrices form a group, the SL2(R) group [8]— one of the two most important Lie groups in analysis. The other group is the Heisenberg group [3]. By contrast the “ax + b”-group, which is often used to build wavelets, is only a subgroup of SL2(R), see the numerator in (1). The simplest non-linear transforms of the real line—linear-fractional or Möbius maps—may also be associated with 2×2matrices [1, Ch. 13]: (1)