associated Lie algebras and exponential maps

to their Lie Groups.

This appendix summarizes some material and sets conventions for the noncompact Lie Group SU(1, 1) and its algebra su(1, 1). For every irreducible unitary finite dimensional representation of SU(2), there being one for every finite n, we can analytically continue, the representation to a finite dimensional, necessarily non-unitary representation of SU(1, 1). Considering the the lowest dimensional, IRREP, of su(2), which is the defining representation, by the Pauli spin matrices:

|0 1| sigma_1 = (1/2) | | |1 0| |0 -i| sigma_2 = (1/2) | | |i 0| |1 0| sigma_3 = (1/2) | | |0 -1| so [sigma_1, sigma_2] = i sigma_3 [sigma_2, sigma_3] = i sigma_1 [sigma_3, sigma_1] = i sigma_2 and exponentiating | cos(theta_1/2) i sin(theta_1/2) | exp( i theta_1 sigma_1 ) = | | | i sin(theta_1/2) cos(theta_1/2) | | cos(theta_2/2) sin(theta_2/2) | exp( i theta_2 sigma_2 ) = | | | -sin(theta_2/2) cos(theta_2/2) | | exp(i theta_3/2) 0 | exp( i theta_3 sigma_3 ) = | | | 0 exp(-i theta_3/2) | Then an arbitrary element of the group SU(2) can be represented as a product of the three exponentials. U(phi_1, theta, phi_2) = exp( i phi_3 sigma_2 ) exp( i theta sigma_1 ) exp( i phi_2 sigma_3 ) or in 2x2 matrix form: | exp(-i xi/2) cos( theta ) exp(+i eta/2) sin( theta ) | | | | -exp(-i eta/2) sin( theta ) exp(+i xi/2) cos( theta ) | where an element of U(2) has, in addition, a factor exp(i phi/2), so the determininant then becomes exp(i phi). The normalized invariant Haar measure on the group manifold is d(mu) = (1/16 pi^2) sin( theta ) d(phi_1) d(theta) d(phi_2) with coordinate ranges 0 <= phi_1 < 4 pi, 0 <= theta <= pi, 0 <= phi_2 < 2 pi

The coordinates are those of a solid sphere in R^3 of radius (4 pi), to which the group manifold is homeomorphic. The group manifold is then clearly, compact, connected, and simply connected.

The exhaustive list of su(2) IRREPS, one for each value of n is given by construction from the fundamental B(n) and B!(n) in [Section XIV].

The prescription for the analytic continuation is to multiply any one element or any two elements by "i". In exponentiating the algebra elements to one parameter subgroups, this is equivalent to continuing the parameter by a rotation of pi/2. Thus, (i times Miller's choice) [Miller 1968].

|0 i| j_1 = (1/2) | | |i 0| |0 -1| j_2 = (1/2) | | |1 0| |1 0| j_3 = (1/2) | | |0 -1| Then [j_1, j_2] = +i j_3, [j_3, j_1] = -i j_2, [j_2, j_3] = +i j_1 and exponentiating | cosh(theta_1/2) sinh(theta_1/2) | exp( i theta_1 j_1 ) = | | | sinh(theta_1/2) cosh(theta_1/2) | | cosh(theta_2/2) i sinh(theta_2/2) | exp( i theta_2 j_2 ) = | | | -i sinh(theta_2/2) cosh(theta_2/2) | | exp(+i theta_3/2) 0 | exp( i theta_3 j_3 ) = | | | 0 exp(-i theta_3/2) | Then again, an arbitrary element of the group SU(1, 1) can be represented as a product of the three exponentials. G(phi_1, theta, phi_2) = exp( i phi_1 J_3 ) exp( i theta J_1 ) exp( i phi_2 J_3 ) or in 2x2 matrix form: | exp(-i xi/2) cosh( theta ) exp(+i eta/2) sinh( theta ) | | | | exp(-i eta/2) sinh( theta ) exp(+i xi/2) cosh( theta ) | where an element of U(1, 1)) has, in addition, a factor exp(i phi/2), so the determininant then becomes exp(i phi).

The finite dimensional IRREPS of su(1, 1) obtained by continuing those of su(2) are, of course, nonunitary; but su(1, 1) also has infinite dimensional IRREPS. An infinite dimensional representation unitary of su(1, 1) can be given by:

| 0 1 0 0 0 ... 0 ... | | 1 0 2 0 0 ... 0 ... | | 0 2 0 3 0 ... 0 ... | k_1 = (1/2) | 0 0 3 0 4 ... 0 ... | | 0 0 0 4 0 ... 0 ... | | 0 0 0 0 5 ... 0 ... | | ... ... 0 ... | = (1/2)( SHA N + N SHA! ) | 0 +i 0 0 0 ... 0 ... | | -i 0 +i2 0 0 ... 0 ... | | 0 -i2 0 +i3 0 ... 0 ... | k_2 = (1/2) | 0 0 -i3 0 +i4 ... 0 ... | | 0 0 0 -i4 0 ... 0 ... | | 0 0 0 0 -i5 ... 0 ... | | ... ... 0 ... | = (i/2)( SHA N - N SHA! ) | 1 0 0 0 0 ... 0 ... | | 0 3 0 0 0 ... 0 ... | | 0 0 5 0 0 ... 0 ... | k_3 = (1/2) | 0 0 0 7 0 ... 0 ... | | 0 0 0 0 9 ... 0 ... | | 0 0 0 0 0 ... 0 ... | | ... ... 0 ... | = (1/2)( 2N + I ) where N is the standard number operator and SHA is the shift operator [Equation (2.31)] with commutation relations: [k_1, k_2] = -i k_3, [k_3, k_1] = +i k_2, [k_2, k_3] = -i k_1

From the point of view of the Lie algebras, su(2) and su(1, 1) have the same complexification, and su(2) is said to be the compact form of su(1, 1). The complexification of either algebra is the Lie algebra sl(2, C) whose associated group is the covering group of the Lorentz group. From the point of view of the groups, the finite dimensional, and therefore nonunitary IRREPS of su(1, 1) can be obtained by analytically continuing the group parameters in an IRREP of su(2). The correspondence consistent with the above realizations are:

SU(1, 1) SU(2)theta_1 itheta_1 theta_2 -itheta_2 theta_3 theta_3

With the indicated parameter substitutions, any IRREP of SU(2) can be analytically continued to a non-unitary IRREP of SU(1, 1). The procedure generalizes to higher dimensional IRREPS of SU(2), to obtain the other non-unitary IRREPS of SU(1, 1). Both SU(2) and SU(1, 1) are Lie groups, of rank 1, and so possess one invariant (Casimir) operator whose eigenvalues label the IRREPS. For SU(2) it is given by equation (14.6). For SU(1, 1),

J_1^2(n) + J_2^2(n) - J_3^2(n) = -(n^2 - 1)/2 I(n)

The extension to IRREPS of U(2) and U(1, 1) consist of including in the product of the one parameter subgroups, and additional factor

exp( i theta_0 sigma_0 )where sigma_0 is the identity of the Pauli algebra. The algebra complexification of u(2) and u(1, 1) is the algebra of the general linear group gl(2, C).

With regard to classical Lie algebras expressed in terms of 2x2 matrices, the following algebra isomorphisms corresponding to local group isomorphisms, hold:

su(2) <-> so(3) <-> sp(1) sl(2, R) <-> su(1, 1) <-> so(2, 1) <-> sp(1, R) sp(1, C) <-> sl(2, C) <-> so(3, 1) where generally sp(n) = Sp(n, C) < u(2n) For L in SU(1, 1) | 1 0 | | 1 0 | L | | L! = | | | 0 -1 | | 0 -1 | and for M in Sp(1, R) | 0 -1 | | 0 -1 | M | | M! = | | | 1 0 | | 1 0 | Then the map M -> L = exp( -i (pi/2) sigma_1 ) M exp( +i (pi/2) sigma_1 )maps Sp(1, R) onto su(1, 1), and by complexification of the respective algebras also maps Sp(1, C) onto sl(2, C).

The full eight dimensional algebra of 2x2 matrices gl(2, C) is isomorphic to the algebra of complex quaternions and to the Clifford algebras [Appendix B],

C(2, C) %10:%00 C(3, %10R%00)the bivectors of which are the generators of rotations in the space with which the Clifford algebra is associated.

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Created: August 1997

Last Updated: January 15, 2001

Last Updated: July 21, 2002