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The book presents the main approaches in study of algebraic
structures of symmetries in models of theoretical and mathematical
physics, namely groups and Lie algebras and their deformations. It
covers the commonly encountered quantum groups (including
Yangians). The second main goal of the book is to present a
differential geometry of coset spaces that is actively used in
investigations of models of quantum field theory, gravity and
statistical physics. The third goal is to explain the main ideas
about the theory of conformal symmetries, which is the basis of the
AdS/CFT correspondence.The theory of groups and symmetries is an
important part of theoretical physics. In elementary particle
physics, cosmology and related fields, the key role is played by
Lie groups and algebras corresponding to continuous symmetries. For
example, relativistic physics is based on the Lorentz and Poincare
groups, and the modern theory of elementary particles — the
Standard Model — is based on gauge (local) symmetry with the
gauge group SU(3) x SU(2) x U(1). This book presents constructions
and results of a general nature, along with numerous concrete
examples that have direct applications in modern theoretical and
mathematical physics.
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 s (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.
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