Download Affine Lie Algebras and Quantum Groups: An Introduction, by Jürgen A. Fuchs PDF

By Jürgen A. Fuchs

This is often an creation to the idea of affine Lie algebras, to the idea of quantum teams, and to the interrelationships among those fields which are encountered in conformal box thought. the outline of affine algebras covers the type challenge, the relationship with loop algebras, and illustration thought together with modular houses. the required historical past from the speculation of semisimple Lie algebras can also be supplied. The dialogue of quantum teams concentrates on deformed enveloping algebras and their illustration thought, yet different points similar to R-matrices and matrix quantum teams also are handled. This ebook may be of curiosity to researchers and graduate scholars in theoretical physics and utilized arithmetic.

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Additional resources for Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory

Example text

1: Comparing the geometry of U (1) as S 1 to the geometry of SU (2) as S 3 . 3 Dynamics of a two-state system Recall that the time dependence of states in quantum mechanics is given by the Schr¨ odinger equation d |ψ(t) = −iH|ψ(t) dt where H is a particular self-adjoint linear operator on H, the Hamiltonian operator. Considering the case of H time-independent, the most general such operator H on C2 will be given by H = h0 1 + h1 σ1 + h2 σ2 + h3 σ3 for four real parameters h0 , h1 , h2 , h3 . The solution to the Schr¨odinger equation is then given by exponentiation: |ψ(t) = U (t)|ψ(0) where U (t) = e−itH The h0 1 term in H contributes an overall phase factor e−ih0 t , with the remaining factor of U (t) an element of the group SU (2) rather than the larger group U (2) of all 2 by 2 unitaries.

Assume that M commutes with all the π(g). We want to show that (π, V ) irreducible implies M = λ1. Since we are working over the field C (this doesn’t work for R), we can always solve the eigenvalue equation det(M − λ1) = 0 to find the eigenvalues λ of M . The eigenspaces Vλ = {v ∈ V : M v = λv} are non-zero vector subspaces of V and can also be described as ker(M − λ1), the kernel of the operator M −λ1. Since this operator and all the π(g) commute, we have v ∈ ker(M − λ1) =⇒ π(g)v ∈ ker(M − λ1) so ker(M − λ1) ⊂ V is a representation of G.

2 Exponentials of Pauli matrices: unitary transformations of the two-state system We saw in chapter 2 that in the U (1) case, knowing the observable operator Q on H determined the representation of U (1), with the representation matrices found by exponentiating iθQ. Here we will find the representation corresponding to the two-state system observables by exponentiating the observables in a similar way. Taking the identity matrix first, multiplication by iθ and exponentiation gives the diagonal unitary matrix eiθ1 = eiθ 0 0 eiθ This is exactly the case studied in chapter 2, for a U (1) group acting on H = C2 , with 1 0 Q= 0 1 This matrix commutes with any other 2 by 2 matrix, so we can treat its action on H independently of the action of the σj .

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