Starting with the Group SL2(R)
نویسنده
چکیده
T he simplest objects with noncommutativemultiplicationmay be 2×2matrices 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 ≠ 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 a subgroup of SL2(R), see the numerator in (1). The simplest nonlinear transformations of the real line—the linear-fractional or Möbius maps— may also be associated with 2 × 2 matrices [1, Ch. 13]: (1)
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