Abstract
Representation theory is a deep and beautiful subject. Our goal in this chapter is to develop those concepts and results that we need for applications to Fourier analysis on compact groups and hence to probability theory. In the first part, we give a self-contained account of key aspects of the representation theory of compact groups, including proofs of Schur’s lemma, the Schur orthogonality relations and the Peter-Weyl theorem. We also introduce the Fourier transform for suitable functions on the group and establish some of its elementary properties. In the second part of the chapter, we introduce weights and roots and sketch proofs of the Weyl character and Weyl dimension formulae. This part is far less rigorous and many proofs are omitted. The key point that readers need to absorb is that irreducible representations are in one-to-one correspondence with highest weights and, as we will see in later chapters, this enables us to carry out a finer analysis of functions and measures in “Fourier space”. Finally we illustrate the abstract theory by finding all the irreducible representations of \(SU(2)\). (We use a lot of elementary Hilbert space ideas in this chapter. Readers requiring a reminder of key concepts should consult a standard text such as Debnath and Mikusinski [55] or Reed and Simon [166]).
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Notes
- 1.
These numbers have many important applications to quantum physics, see e.g. Jones [110] pp. 109–118.
- 2.
To see this, note that by Proposition 3.4.5 in Cohn [50], p. 110, we need only show that the \(\sigma \)-algebra \(\mathcal{B}(G)\) is countably generated. But if \(G\) is second countable, then it has a countable basis for its topology which generates \(\mathcal{B}(G)\).
- 3.
Of course, in a generic quantum group, there is no underlying group \(G\), only the Hopf algebra structure.
- 4.
A basis \(\mathcal{U}_{e}\) for a neighbourhood of the neutral element in \(\widehat{G}\) is given by the sets \(\{\chi \in \widehat{G}; |\chi (g) - 1| < \epsilon ~\text{ for } \text{ all }~g \in K\}\) where \(K\) is a compact subset of \(G\) and \(\epsilon > 0\). A basis for the topology of \(\widehat{G}\) is then given by the collection of all left translates of sets in \(\mathcal{U}_{e}\).
- 5.
In the representation theory literature, it is quite common to meet the alternative notation \(\pi (f):=\widehat{\tilde{f}}(\pi ) = \int _{G}f(g)\pi (g)dg\).
- 6.
This is more restrictive than necessary, but is all that we need here. In general we might want to consider infinite-dimensional representations.
- 7.
Note that both Helgason and Knapp work with complex semisimple Lie algebras and utilise the (non-degenerate) Killing form instead of an Ad-invariant inner product, but the proofs of those facts that we need transfer easily to our context.
- 8.
Of course, there is some abuse of notation here.
- 9.
This is the negation of the Casimir operator for \(\mathbf{su(2)}\), i.e. \(-C = \xi (L_{0})^{2} + \xi (L_{1})^{2} + \xi (L_{2})^{2}.\)
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Applebaum, D. (2014). Representations, Peter-Weyl Theory and Weights. In: Probability on Compact Lie Groups. Probability Theory and Stochastic Modelling, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-319-07842-7_2
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DOI: https://doi.org/10.1007/978-3-319-07842-7_2
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