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Y = 1 is precisely the condition that the vector (a, b, c, d) lie on the unit sphere in 4-space. Thus again we obtain the homomorphism of SU(2) onto S0(3, R) as before. Diagram 1 summarizes the interrelations between the various groups discussed here. Here we have denoted the linear fractional transformation group, or projective special linear group, by PSL(2, C). 19 TENSOR PRODUCT OF VECTOR SPACES The study of the local structure of Lie groups, based on the theory of Lie algebras, makes heavy use of linear algebra and vector space concepts.

The sum and intersection of ideals of a Lie algebra are again ideals, and the ideals of a Lie algebra form a lattice under these two operations. In addition, it follows from the Jacobi identity that the Lie product [7ls I2] of two ideals is again an ideal. The situation regarding subalgebras is a little bit different since the sum of two subalgebras need not be a subalgebra, although the intersection of any set of subalgebras is still a subalgebra. Nevertheless, the subalgebras of a Lie algebra still form a lattice.

For example, the real Lie algebra s/(n, C) of the special linear group consists of the matrices with trace zero. The real Lie algebra o(n, C) of the complex orthogonal group consists of all complex antisymmetric matrices of order n, while the Lie algebra o(n, R) consists of the real ones. Similarly, the real Lie algebra u(ri) of the unitary group consists of the anti-Hermitian matrices. The tangent space to the intersection of two manifolds at a point is the intersection of their tangent spaces.