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Applications of Symmetry in Mathematics, Physics & Chemistry - download pdf or read online

By Omer Cabrera

ISBN-10: 8132343484

ISBN-13: 9788132343486

Desk of Contents
Chapter 1 - Symmetry
Chapter 2 - workforce (Mathematics)
Chapter three - crew Action
Chapter four - usual Polytope
Chapter five - Lie element Symmetry

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Additional info for Applications of Symmetry in Mathematics, Physics & Chemistry

Example text

A group action is then nothing but a functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. A morphism between G-sets is then a natural transformation between the group action functors. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category. Without using the language of categories, one can extend the notion of a group action on a set X by studying as well its induced action on the power set of X.

A group action is a flexible generalization of the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets, such as the set of vertices, the set of edges and the set of faces of the polyhedron. If G is a group and X is a set then a group action may be defined as a group homomorphism from G to the symmetric group of X.

It studies the group by its group actions on other spaces. e. the group is acting on a vector space, such as the three-dimensional Euclidean space R3. A representation of G on an n-dimensional real vector space is simply a group homomorphism ρ: G → GL(n, R) from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations. Given a group action, this gives further means to study the object being acted on On the other hand, it also yields information about the group.

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Applications of Symmetry in Mathematics, Physics & Chemistry by Omer Cabrera


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