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Homological Invariants in the Representation Theory of Finite Dimensional Algebras

2011
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Release : 2011
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Book Rating : 414/5 ( reviews)

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Book Synopsis Homological Invariants in the Representation Theory of Finite Dimensional Algebras by : Thomas Troy Howard

Download or read book Homological Invariants in the Representation Theory of Finite Dimensional Algebras written by Thomas Troy Howard. This book was released on 2011. Available in PDF, EPUB and Kindle. Book excerpt: A lattice-valued version of complexity is introduced for finitely generated representations of finite dimensional algebras which measures the dimension growth of their syzygies. The spectrum of complexities over a finite dimensional algebra is shown to be invariant under derived equivalences in many instances. Directed graphs called "syzygy quivers" are used to explicitly compute complexity for representations of monomial algebras and to confirm that the spectrum of complexities of a monomial algebra is a derived invariant. "Relative complexity spectra" are used to show that this complexity spectrum is a derived invariant for any finite dimensional algebra which satisfies certain "translation invariance" properties. Examples of such algebras include Gorenstein algebras, local algebras, and commutative algebras; no example is known of an algebra which lacks this property.

Representation Theory

2014-08-15 Mathematics
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Release : 2014-08-15
Genre : Mathematics
Kind : eBook
Book Rating : 689/5 ( reviews)

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Book Synopsis Representation Theory by : Alexander Zimmermann

Download or read book Representation Theory written by Alexander Zimmermann. This book was released on 2014-08-15. Available in PDF, EPUB and Kindle. Book excerpt: Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field. Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced. Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.

Trends in Representation Theory of Algebras and Related Topics

2008 Representations of algebras
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Release : 2008
Genre : Representations of algebras
Kind : eBook
Book Rating : 623/5 ( reviews)

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Book Synopsis Trends in Representation Theory of Algebras and Related Topics by : Andrzej Skowroński

Download or read book Trends in Representation Theory of Algebras and Related Topics written by Andrzej Skowroński. This book was released on 2008. Available in PDF, EPUB and Kindle. Book excerpt: This book is concerned with recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, quantum groups, homological algebra, invariant theory, combinatorics, model theory and theoretical physics. The collection of articles, written by leading researchers in the field, is conceived as a sort of handbook providing easy access to the present state of knowledge and stimulating further development. The topics under discussion include diagram algebras, Brauer algebras, cellular algebras, quasi-hereditary algebras, Hall algebras, Hecke algebras, symplectic reflection algebras, Cherednik algebras, Kashiwara crystals, Fock spaces, preprojective algebras, cluster algebras, rank varieties, varieties of algebras and modules, moduli of representations of quivers, semi-invariants of quivers, Cohen-Macaulay modules, singularities, coherent sheaves, derived categories, spectral representation theory, Coxeter polynomials, Auslander-Reiten theory, Calabi-Yau triangulated categories, Poincare duality spaces, selfinjective algebras, periodic algebras, stable module categories, Hochschild cohomologies, deformations of algebras, Galois coverings of algebras, tilting theory, algebras of small homological dimensions, representation types of algebras, and model theory. This book consists of fifteen self-contained expository survey articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. They contain a large number of open problems and give new perspectives for research in the field.

Representation Theory

2020-05
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Release : 2020-05
Genre :
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Representation Theory by : Alice Fialowski

Download or read book Representation Theory written by Alice Fialowski. This book was released on 2020-05. Available in PDF, EPUB and Kindle. Book excerpt: Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field. Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given - such as the structure of blocks of cyclic defect groups - whenever appropriate. Overall, many methods from the representation theory of algebras are introduced. Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.

Homological Theory of Representations

2021-11-18 Mathematics
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Release : 2021-11-18
Genre : Mathematics
Kind : eBook
Book Rating : 898/5 ( reviews)

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Book Synopsis Homological Theory of Representations by : Henning Krause

Download or read book Homological Theory of Representations written by Henning Krause. This book was released on 2021-11-18. Available in PDF, EPUB and Kindle. Book excerpt: This book for advanced graduate students and researchers discusses representations of associative algebras and their homological theory.

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