Theory of Groups and Symmetries

This book is a sequel to the book by the same authors entitled Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras.The presentation begins with the Dirac notation, which is illustrated by boson and fermion oscillator algebras and also Grassmann algebra. Then detailed account of finite-dimensional representations of groups SL(2, C) and SU(2) and their Lie algebras is presented. The general theory of finite-dimensional irreducible representations of simple Lie algebras based on the construction of highest weight representations is given. The classification of all finite-dimensional irreducible representations of the Lie algebras of the classical series sℓ(n, C), so(n, C) and sp(2r, C) is exposed.Finite-dimensional irreducible representations of linear groups SL(N, C) and their compact forms SU(N) are constructed on the basis of the Schur–Weyl duality. A special role here is played by the theory of representations of the symmetric group algebra C[Sr] (Schur–Frobenius theory, Okounkov–Vershik approach), based on combinatorics of Young diagrams and Young tableaux. Similar construction is given for pseudo-orthogonal groups O(p, q) and SO(p, q), including Lorentz groups O(1, N-1) and SO(1, N-1), and their Lie algebras, as well as symplectic groups Sp(p, q). The representation theory of Brauer algebra (centralizer algebra of SO(p, q) and Sp(p, q) groups in tensor representations) is discussed.Finally, the covering groups Spin(p, q) for pseudo-orthogonal groups SO↑(p, q) are studied. For this purpose, Clifford algebras in spaces Rp, q are introduced and representations of these algebras are discussed.Contents: PrefaceDirac NotationsFinite-Dimensional Representations of Lie Algebras su(2) and sℓ(2, ℂ) and Lie Groups SU(2) and SL(2, ℂ)Representations of Simple Lie Algebras. Weight TheoryFinite-Dimensional Representations of Algebras sℓ(N, ℂ), su(N) and Groups SL(N, ℂ) and SU(N)Finite-Dimensional Representations of Groups SO, Sp and Lie Algebras so, spGroups Spin(p, q) and Their Finite-Dimensional RepresentationsSolutions to Selected ProblemsSelected BibliographyReferencesIndex
Readership: Graduate students and researchers in theoretical physics and mathematical physics.Groups;Lie Algebras;Representation Theory;Dirac Notation;Boson and Fermion Oscillator Algebra;Grassmann Algebra;Schur–Weyl Duality;Schur–Frobenius Theory;Okounkov-vershik Approach;Young Diagram;Young Tableux;Brauer Algebra;Clifford Algebra00