An introduction to groups, groupoids and their representations. (2019)

• Record Type: Book
• Further Information: Note: Alberto Ibort, Miguel A. Rodriguez.
• Authors: Ibort, Alberto; Rodriguez, Miguel A
• Contents: I WORKING WITH CATEGORIES AND GROUPOIDS 1. Categories: basic notions and examples ; Introducing the main characters; Categories: formal definitions; A categorical definition of groupoids and groups; Historical notes and additional comments 2. Groups ; Groups, subgroups and normal subgroups: basic notions; The symmetric group; Group homomorphisms and Cayley's theorem The alternating group; Products of groups; Historical notes and additional comments 3. Groupoids ; Groupoids: basic concepts; Puzzles and groupoids 4. Actions of groups and groupoids; Symmetries, groups and groupoids; The action groupoid; Symmetries and groupoids Weinstein's tilings; Cayley's theorem for groupoids 5. Functors and transformations; Functors; An interlude: categories and databases; Homomorphisms of groupoids; Equivalence: Natural transformations 6. The structure of groupoids; Normal subgroupoids; Simple groupoids The structure of groupoids: second structure theorem; Classification of groupoids up to order 20; Groupoids with Abelian isotropy group II REPRESENTATIONS OF FINITE GROUPS AND GROUPOIDS 7. Linear representations of groups; Linear and unitary representations of groups; Irreducible representations; Unitary representations of groups; Schur's lemmas for groups 8. Characters; Orthogonality relations; Characters; Orthogonality relations of characters; Inequivalent representations and irreducibility criteria; Decomposition of the regular representation; Tensor products of representations of groups; Tables of characters; Canonical decomposition; An application in quantum mechanics: spectrum degeneracy 9. Linear representations of categories ; Linear representations of categories; Properties of representations of categories; Linear representations of groupoids 10. Algebras and groupoids; Algebras; The algebra of a category; The algebra of a groupoid; Representations of Algebras; Representations of groupoids and modules 11. Semi-simplicity ; Irreducible representations of algebras; Semi-simple modules; The Jordan-Holder theorem; Semi-simple algebras: the Jacobson radical; Characterizations of semi-simplicity; The algebra of a finite groupoid is semi-simple 12. Representations of groupoids ; Characters again; Operations with groupoids and representations; The left and right regular representations of a finite groupoid; Some simple examples; Discussion III APPENDICES A Glossary of Linear Algebra B Generators and relations C Schwinger Algebra Bibliography Index ;
• Edition: 1st
• Publisher Details: Boca Raton : CRC Press
• Publication Date: 2019
• Extent: 1 online resource
• Subjects: 512.2; Groupoids; Group theory
• Languages: English
• ISBNs: 9781351869560; 9781351869577; 9781351869553; 9781315232942
• Related ISBNs: 9781138035867
• Notes: Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
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