By Stefano Ossicini, Lorenzo Pavesi, Francesco Priolo, L. Pavesi, F. Priolo

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A( p, q). This group consists of matrices that are the sum of an identity matrix and the upper right-hand off-diagonal block of a ( p, q) blocked matrix. 4 Bilinear and quadratic constraints 39 elements satisfy Ai, j Aα,β Aα, j Ai,β = δi, j 1 ≤ i, j ≤ p = δα,β p + 1 ≤ α, β ≤ p + q =0 = arbitrary This group is abelian or commutative: AB = B A for all elements (matrices) in this group. 9): x → x = x + a. 4 Bilinear and quadratic constraints In (8)–(11) we treat groups that preserve a metric, represented by a matrix G.

1). Solvable groups are strictly upper triangular. 11) These matrices have the same structure as the group generated by exponentials of the photon number operator (nˆ = a † a), the creation (a † ) and annihilation (a) operators, and their commutator (I = aa † − a † a = [a, a † ]). We will use this identification between operator and matrix groups to develop some powerful operator disentangling theorems. 6. N il(n). Nilpotent groups are subgroups of Sol(n) whose diagonal matrix elements are all +1.

The scale factor can always be chosen so that y is in the unit sphere in R n+1 : y ∈ S n ⊂ R n+1 . n+1 2 1/2 In fact, two values of λ can be chosen: λ = ±1/( i=1 xi ) . In R 3 the straight line containing (x, y, z) can be represented by homogeneous coordinates (X, Y ) = (x/z, y/z) if z = 0. Straight lines through the origin of R 3 are mapped to straight lines in R 3 by x → x = M x, M ∈ S L(3; R). 6 Problems 33 representing the two lines containing x and x are related by the linear fractional transformation X Y → X Y m 11 m 21 = m 12 m 22 X Y + m 13 m 23 [ m 31 m 32 ] X Y + m 33 Generalize for linear fractional transformations R P n → R P n .