Extended Graphical Calculus for Categorified Quantum sl(2) - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Extended Graphical Calculus for Categorified Quantum sl(2) write by Mikhail Khovanov. This book was released on 2012. Extended Graphical Calculus for Categorified Quantum sl(2) available in PDF, EPUB and Kindle. In an earlier paper, Aaron D. Lauda constructed a categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2); here he, Khovanov, Marco Mackaay, and Marko Stosic enhance the graphical calculus he introduced to include two-morphisms between divided powers one-morphisms and their compositions. They obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms, which are in a bijection with the Lusztig canonical basis elements. Their results show that one of Lauda's main results holds when the 2-category is defined over the ring of integers rather than over a field. The study is not indexed. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).
Categorification and Higher Representation Theory
Categorification and Higher Representation Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Categorification and Higher Representation Theory write by Anna Beliakova. This book was released on 2017-02-21. Categorification and Higher Representation Theory available in PDF, EPUB and Kindle. The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory. This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory. The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.
Dualizable Tensor Categories
Dualizable Tensor Categories - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Dualizable Tensor Categories write by Christopher L. Douglas. This book was released on 2021-06-18. Dualizable Tensor Categories available in PDF, EPUB and Kindle. We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with cer-tain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-dimensional 3-framed local field theories. We also show that all finite tensor cat-egories are 2-dualizable, and yield categorified 2-dimensional 3-framed local field theories. On the other hand, topological properties of 3-framed manifolds deter-mine algebraic equations among functors of tensor categories. We show that the 1-dimensional loop bordism, which exhibits a single full rotation, acts as the double dual autofunctor of a tensor category. We prove that the 2-dimensional belt-trick bordism, which unravels a double rotation, operates on any finite tensor category, and therefore supplies a trivialization of the quadruple dual. This approach pro-duces a quadruple-dual theorem for suitably dualizable objects in any symmetric monoidal 3-category. There is furthermore a correspondence between algebraic structures on tensor categories and homotopy fixed point structures, which in turn provide structured field theories; we describe the expected connection between piv-otal tensor categories and combed fixed point structures, and between spherical tensor categories and oriented fixed point structures.
Knot Invariants and Higher Representation Theory
Knot Invariants and Higher Representation Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Knot Invariants and Higher Representation Theory write by Ben Webster. This book was released on 2018-01-16. Knot Invariants and Higher Representation Theory available in PDF, EPUB and Kindle. The author constructs knot invariants categorifying the quantum knot variants for all representations of quantum groups. He shows that these invariants coincide with previous invariants defined by Khovanov for sl and sl and by Mazorchuk-Stroppel and Sussan for sl . The author's technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is sl , the author shows that these categories agree with certain subcategories of parabolic category for gl .
Frobenius Algebras and 2-D Topological Quantum Field Theories
Frobenius Algebras and 2-D Topological Quantum Field Theories - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Frobenius Algebras and 2-D Topological Quantum Field Theories write by Joachim Kock. This book was released on 2004. Frobenius Algebras and 2-D Topological Quantum Field Theories available in PDF, EPUB and Kindle. This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.
