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Lie algebras are efficient tools for analyzing the properties of physical systems. Concrete applications comprise the formulation of symmetries of Hamiltonian systems, the description of atomic, molecular and nuclear spectra, the physics of elementary particles and many others. This work gives an introduction to the properties and the structure of the Lie algebras su(n). First, characteristic quantities such as structure constants, the Killing form and functions of Lie algebras are introduced. The properties of the algebras su(2), su(3) and su(4) are investigated in detail. Geometric models of the representations are developed. A lot of care is taken over the use of the term "multiplet of an algebra".The book features an elementary (matrix) access to su(N)-algebras, and gives a first insight into Lie algebras. Student readers should be enabled to begin studies on physical su(N)-applications, instructors will profit from the detailed calculations and examples.

Lie algebras. --- Nonassociative rings.

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Nonassociative rings --- Nonassociative algebras --- Data processing. --- Algèbres non associatives --- Algèbres non associatives. --- Algèbres non associatives

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Ordered algebraic structures --- 512.55 --- Associative algebras --- -Associative rings --- -Galois theory --- -Nonassociative algebras --- -Nonassociative rings --- -Rings (Algebra) --- Algebras, Non-associative --- Algebras, Nonassociative --- Non-associative algebras --- Algebra, Abstract --- Algebras, Linear --- Equations, Theory of --- Group theory --- Number theory --- Rings (Algebra) --- Algebras, Associative --- Algebra --- Rings and modules --- Congresses --- -Rings and modules --- 512.55 Rings and modules --- -512.55 Rings and modules --- Associative rings --- Galois theory --- Nonassociative algebras --- Nonassociative rings --- Associative rings - Congresses --- Nonassociative rings - Congresses --- Associative algebras - Congresses --- Nonassociative algebras - Congresses --- Galois theory - Congresses

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This book focuses on recent developments in the theory of vertex algebras, with particular emphasis on affine vertex algebras, affine W-algebras, and W-algebras appearing in physical theories such as logarithmic conformal field theory. It is widely accepted in the mathematical community that the best way to study the representation theory of affine Kac–Moody algebras is by investigating the representation theory of the associated affine vertex and W-algebras. In this volume, this general idea can be seen at work from several points of view. Most relevant state of the art topics are covered, including fusion, relationships with finite dimensional Lie theory, permutation orbifolds, higher Zhu algebras, connections with combinatorics, and mathematical physics. The volume is based on the INdAM Workshop Affine, Vertex and W-algebras, held in Rome from 11 to 15 December 2017. It will be of interest to all researchers in the field.

Lie algebras --- Nonassociative rings. --- Rings (Algebra). --- Mathematical physics. --- Non-associative Rings and Algebras. --- Mathematical Physics. --- Physical mathematics --- Physics --- Rings (Algebra) --- Algebraic rings --- Ring theory --- Algebraic fields --- Mathematics

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For the past ten years, alternative loop rings have intrigued mathematicians from a wide cross-section of modern algebra. As a consequence, the theory of alternative loop rings has grown tremendously. One of the main developments is the complete characterization of loops which have an alternative but not associative, loop ring. Furthermore, there is a very close relationship between the algebraic structures of loop rings and of group rings over 2-groups. Another major topic of research is the study of the unit loop of the integral loop ring. Here the interaction between loop rings and grou

Alternative rings --- Group rings --- Loops (Group theory) --- 512.55 --- 512.55 Rings and modules --- Rings and modules --- Group theory --- Rings (Algebra) --- Loop groups --- Nonassociative rings --- Alternative rings. --- Group rings.