Group unimodular

It is known as the unitary unimodular group, or the special unitary group denoted by SU(2). Because of the extra condition on the determinant, SU(2) is a three-parameter group. 

Formula (5.29) is the special case (odd k values) of the Casimir operator of the more general so-called unitary unimodular group SU21+1, ie. 

Rotations in a vector space of three orbitals are described by the group SO(3) of orthogonal 3x3 matrices with determinant +1. To embed the octahedral rotation group in this covering group one needs a matrix representation of O which also consists of orthogonal and unimodular 3x3 matrices. Such a matrix representation is sometimes called the fundamental vector representation of the point group. In the case of O the fundamental vector representation is Ti and not T2. Indeed the 7] matrices are unimodular, i.e. have determinant +1, while the determinants of the T2 matrices are equal to the characters of the one-dimensional representation A2. 

The set of 2 x 2 unimodular unitary matrices constitutes the special unitary group SU(2). Such matrices can be parametrized by... 

To establish the connection between the spinor and the vector, we now need to verify how transformations in the spinor are manifested as transformations in the vector. Consider a finite unitary transformation of the spinor. The transformation belongs to the unitary group, U(2), and, as we have seen, the determinant of this matrix is unimodular. We consider the special case, however, where the determinant is +1. Such matrices form the special unitary group, SU 2). The most general form of an SU(2) matrix involves two complex parameters, say a and b, subject to the condition that their squared norm, a + b, equals unity. These parameters are also known as the Cayley-Klein parameters. (Cf. Problem 2.1.) One has... 

The group into which / maps will be denoted by N, We have / 2gZ — 2ql = N, We have already set a basis in N, Let us numerate it with the same letter— Cg,...,e29). If we change the basis in the group iV, the matrix of the homomorphism will change too. Suppose that the gluing has already been made. Note that if for some t there holds the equality 7 = 0, then from the integervaluedness and unimodularity condition on the matrix it follows = 1. 

The product function defined above clearly violates this requirement, and although acceptable as a solution of (1.2.4) it is not acceptable as a wavefunction defining the state of two indistinguishable particles. The functions P(r2, fi) and W ri, rz) should clearly differ at most by a unimodular complex number e, and it can be shown frcan group theory that the only essentially distinct possibilities that need be considered are... 

A further reduction occurs when we use only real unimodular matrices we then obtain the real orthogonal group R(m), in which U = U, and the distinction between co- and contravariant transformations disappears. 

The group R(3), for example, describes all possible rotations in 3-dimensional space. To include reflections, the unimodular condition must be relaxed and the group becomes R (3), which includes the improper rotations with detU = -1. It is because all these groups are infinite and contain matrices with elements that are continuously variable that the results obtained for finite groups (Appendix 3) are not always immediately applicable. 

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