Pal P. A Physicist's Introduction to Algebraic Structures: Vector Spaces, Groups, Topological spaces and more

Cambridge University Press, 2019. — 717 p. — ISBN: 978-1-108-49220-1.Algebraic structures play an important role in understanding crucial concepts of modern physics. Starting with mathematical logic, sets and functions, this text presents short descriptions of different algebraic structures such as metric spaces, rings, fields, Boolean algebra, groups, vector spaces and measure spaces. A review of the basic ideas of vector space, linear independence of vectors, basis, dual space, operators on finite dimensional vector spaces, and so on, is followed up by a detailed discussion of the types of matrices, that are especially important for physics, for example, Hermitian and unitary matrices. Infinite dimensional vector spaces, also known as function spaces, and operators on such spaces are discussed.Group Theory, starting with general properties of groups and their representations, is elaborated. Finite groups, as well as representation of finite groups, are discussed in detail: permutation groups take the central role in the discussion. Some finite groups that have easy geometry, such as symmetry groups of polygons and polyhedrons as well as crystal symmetry groups, are interpreted. Likewise, continuous groups or Lie groups in particular are discussed. The representation of unitary groups and orthogonal groups is elaborated, followed by a chapter on the Lorentz group. Some other Lie groups, such as the symplectic groups and exceptional groups, are discussed briefly.Weights and root vectors, with enumeration of the number of compact Lie algebras using the Dynkin diagrams, are presented.The last part of the text deals with topology. Continuity of functions, and the idea of topological spaces, is taken up. The idea of homotopy of maps is introduced and the fundamental group of topological spaces is discussed in detail. Higher homotopy groups and homology groups have also been visited.This book will be useful for graduate students and researchers interested in picking up mathematical concepts, which are useful in many branches of physics.True PDF