By Alexandru Buium
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Extra resources for Differential Function Fields and Moduli of Algebraic Varieties
Example text
Some Important Concepts and Examples in Lie Algebras 27 group by the translations is the rotation group. In fact, ideals of Lie algebras always correspond to normal subgroups of the corresponding Lie group. (G) The set of commutators [g, g] is an ideal of 9 (prove it), called g(1). Similarly g(2) = [g(1), g(l)] is an ideal of g(1). We define g(n+1) = [g(n), g(n)]. If this sequence terminates in zero, we say 9 is solvable. For example e(3)(1) = e(3), so e(3)(n) = e(3) and e(3) is not solvable. On the other hand the subalgebra q = {R3' PI' P2 , P3} introduced above is solvable because q(2) is zero.
The procedures used here will be established rigorously in Chapter 3. We begin with su(2) and generalize. Typical elements of the Lie group SU(2) are U1 (a) = e- i(a/2)(1I, U2 (/3) = e- i(P/2)(12, U3 (Y) = e- i(y/2)(13; the corresponding elements of the Lie algebra -~ -~ . -~ are E1 = -2-' E2 = -2-' E3 = -2-' Clearly the Ej can be obtallled by differentiating the group elements at the identity: for example This method works for any linear Lie group (fj. Start with a curve get) in the group that passes through the identity at t = O.
Bt)AIO) is annihilated by J+, J_ andJo so it is an su(2) singlet. That is, it gives a one-dimensional (trivial) representation of su(2). But it is an eigenvector of P with eigenvalue 2A. By contrast, the only singlet of u(2) which we obtained without the b's was the vacuum state 10). Now we have an infinite set of distinct one dimensional, non-trivial representations of u(2). More generally, the states IA,n1,n z ) = NAnln2(atb! )n210) where the normalization constant is given by NZ _ (nl + nz)!
Differential Function Fields and Moduli of Algebraic Varieties by Alexandru Buium
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