Theory of Groups and Symmetries

The following sections are included:

• Dedication
• Contents
• Preface

In this chapter, we introduce convenient notations which we use throughout the book for describing basis vectors and matrix elements of operators in the spaces of representations of Lie groups and Lie algebras. These notations were introduced by P. Dirac and they are standard in quantum mechanics. We have used these notations in the first book. Here, we give the general definitions and rules. We illustrate the convenience of these notations by considering examples of coherent states for quantum oscillator algebra and free fermion algebra.

In this chapter, we study complex finite-dimensional representations of Lie algebras sl(2,ℂ), su(2) and Lie group SU(2), and also a restricted set of representations of Lie group SL(2,ℂ). Unless the opposite is stated, the representations that we analyze in this and following chapters are finite dimensional. The representations that we consider are in one-to-one correspondence with each other: any representation of algebra sl(2,ℂ) is at the same time a representation of algebra su(2) and is lifted to representation of group SU(2). These representations of SU(2) are conveniently studied by relating them to certain representations of Lie group SL(2,ℂ), namely, to those representations which are contained in tensor products of its defining representation T. Clearly, this class of representations does not exhaust all finite-dimensional representations of SL(2,ℂ). As an example, it does not include sub-representations of the tensor products T^⊗r₁ ⊗ (T^*)^⊗r₂, where T^* is complex conjugate to defining representation.

This chapter builds on material presented in Refs. [5]–[8]. Also, we heavily use the results given in the first book in Chapter I-6 “Root systems and classification of simple Lie algebras” (Sections I-6.1 and I-6.2). To avoid confusion, we emphasize again that we study finite-dimensional representations, unless we state the opposite.

We have already shown (see Proposition 3.4.8, Section 3.4.2) that all complex, finite-dimensional irreducible representations of Lie algebras sl(N,ℂ) and su(N), and hence representations of groups SU(N), can be thought of as subrepresentations in tensor products T^⊗r of defining representations T. These representations are conveniently studied as corresponding representations of group SL(N,ℂ). One should bear in mind, however, that the special class of representations of SL(N,ℂ) that we consider (and that is related to the representations of SU(N)) includes only those representations that are embedded in tensor products T^⊗r, where T is defining representation of SL(N,ℂ), and it does not include representations constructed by invoking complex conjugate representations T^*…

We consider in Sections 5.1–5.3 a class of complex finite-dimensional irreducible representations of groups of SO and Sp families, which are subrepresentations in tensor products of defining representations of these groups. The representations from this class exhaust all irreducible representations of groups of SO and Sp which are associated with representations of complex Lie algebras so and sp. In the case of Sp family, we do not get anything new when turning from groups to algebras sp. Indeed, as we have shown in Section 3.4.2 (item 5), all irreducible representations of Lie algebras sp are contained in tensor products of their defining representations and are in one-to-one correspondence with representations of groups Sp of the class we consider. So, our analysis covers all complex finite-dimensional irreducible representations of groups of Sp-series associated with representations of algebras of sp-series. This is not the case for algebras so: as we discussed in Section 3.4.2 (items 3 and 4), these algebras have spinor representations which cannot be embedded in tensor products of defining representations and do not have analogs among the representations of groups SO. Instead, the spinor representations of so algebras are generated by representations of groups Spin, which are locally isomorphic to groups SO. We begin the discussion of spinor representations of algebras so in Section 5.4 and consider groups Spin in some detail in Chapter 6.

Complex and real representations (including spinor) of real Lie algebras so(p, q) are important in physics applications. These representations are at the same time representations of groups Spin(p, q) which are locally isomorphic to groups SO(p, q) and cover connected component SO^↑(p, q) of group SO(p, q)…

The following sections are included:

• Problem 1.3.6 of Section 1.3.3
• Problem 1.3.7 of Section 1.3.3
• Problem 2.2.13 of Section 2.2.3
• Problem 2.3.4 of Section 2.3.2
• Problem 2.3.6 of Section 2.3.3
• Problem 2.3.7 of Section 2.3.3
• Problem 2.3.11 of Section 2.3.3
• Problem 2.5.4 of Section 2.5.2
• Problem 2.5.15 of Section 2.5.4
• Problem 3.3.1 of Section 3.3
• Problem 3.4.5 of Section 3.4.2
• Problems 4.3.2 and 4.3.3 of Section 4.3.1
• Problem 4.3.10 of Section 4.3.2
• Problem 4.3.11 of Section 4.3.2
• Problem 4.5.7 of Section 4.5.1
• Problem 4.6.9 of Section 4.6.5
• Problem 4.6.11 of Section 4.6.5
• Problem 4.7.1 of Section 4.7
• Problem 5.1.3 of Section 5.1.3
• Problem 5.1.5 of Section 5.1.3
• Problem 5.3.4 of Section 5.3
• Problem 6.2.2 of Section 6.2.1
• Problem 6.2.4 of Section 6.2.1
• Problem 6.3.5 of Section 6.3.2
• Problem 6.3.6 of Section 6.3.2
• Problem 6.3.10 of Section 6.3.2
• Problem 6.4.3 of Section 6.4.1
• Problem 6.4.13 of Section 6.4.3

The following sections are included:

• Selected Bibliography
• References
• Index