Extended Graphical Calculus For Categorified Quantum Sl 2
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Extended Graphical Calculus For Categorified Quantum Sl 2
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Author :
language : en
Publisher:
Release Date : 2012
Extended Graphical Calculus For Categorified Quantum Sl 2 written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Categories categories.
A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements. These formulas have integral coefficients and imply that one of the main results of Lauda's paper--identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)--also holds when the 2-category is defined over the ring of integers rather than over a field. A new diagrammatic description of Schur functions is also given and it is shown that the the Jacobi-Trudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).
Extended Graphical Calculus For Categorified Quantum Sl 2
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Author : Mikhail Khovanov
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Extended Graphical Calculus For Categorified Quantum Sl 2 written by Mikhail Khovanov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
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
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Author : Anna Beliakova
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-02-21
Categorification And Higher Representation Theory written by Anna Beliakova and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-02-21 with Mathematics categories.
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.
Non Cooperative Equilibria Of Fermi Systems With Long Range Interactions
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Author : Jean-Bernard Bru
language : en
Publisher: American Mathematical Soc.
Release Date : 2013-06-28
Non Cooperative Equilibria Of Fermi Systems With Long Range Interactions written by Jean-Bernard Bru and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-28 with Mathematics categories.
The authors define a Banach space $\mathcal{M}_{1}$ of models for fermions or quantum spins in the lattice with long range interactions and make explicit the structure of (generalized) equilibrium states for any $\mathfrak{m}\in \mathcal{M}_{1}$. In particular, the authors give a first answer to an old open problem in mathematical physics--first addressed by Ginibre in 1968 within a different context--about the validity of the so-called Bogoliubov approximation on the level of states. Depending on the model $\mathfrak{m}\in \mathcal{M}_{1}$, the authors' method provides a systematic way to study all its correlation functions at equilibrium and can thus be used to analyze the physics of long range interactions. Furthermore, the authors show that the thermodynamics of long range models $\mathfrak{m}\in \mathcal{M}_{1}$ is governed by the non-cooperative equilibria of a zero-sum game, called here thermodynamic game.
Infinite Dimensional Representations Of 2 Groups
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Author : John C. Baez
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Infinite Dimensional Representations Of 2 Groups written by John C. Baez and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
Just as groups can have representations on vector spaces, 2-groups have representations on 2-vector spaces, but Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. Therefore, Crane, Sheppeard, and Yetter introduced certain infinite-dimensional 2-vector spaces, called measurable categories, to study infinite-dimensional representations of certain Lie 2-groups, and German and North American mathematicians continue that work here. After introductory matters, they cover representations of 2-groups, and measurable categories, representations on measurable categories. There is no index. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).
Elliptic Integrable Systems
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Author : Idrisse Khemar
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Elliptic Integrable Systems written by Idrisse Khemar and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
In this paper, the author studies all the elliptic integrable systems, in the sense of C, that is to say, the family of all the $m$-th elliptic integrable systems associated to a $k^\prime$-symmetric space $N=G/G_0$. The author describes the geometry behind this family of integrable systems for which we know how to construct (at least locally) all the solutions. The introduction gives an overview of all the main results, as well as some related subjects and works, and some additional motivations.
Hopf Algebras And Congruence Subgroups
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Author : Yorck Sommerhäuser
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Hopf Algebras And Congruence Subgroups written by Yorck Sommerhäuser and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
We prove that the kernel of the action of the modular group on the center of a semisimple factorizable Hopf algebra is a congruence subgroup whenever this action is linear. If the action is only projective, we show that the projective kernel is a congruence subgroup. To do this, we introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.
Finite Order Automorphisms And Real Forms Of Affine Kac Moody Algebras In The Smooth And Algebraic Category
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Author : Ernst Heintze
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Finite Order Automorphisms And Real Forms Of Affine Kac Moody Algebras In The Smooth And Algebraic Category written by Ernst Heintze and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
Heintze and Gross discuss isomorphisms between smooth loop algebras and of smooth affine Kac-Moody algebras in particular, and automorphisms of the first and second kinds of finite order. Then they consider involutions of the first and second kind, and make the algebraic case. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).
Character Identities In The Twisted Endoscopy Of Real Reductive Groups
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Author : Paul Mezo
language : en
Publisher: American Mathematical Soc.
Release Date : 2013-02-26
Character Identities In The Twisted Endoscopy Of Real Reductive Groups written by Paul Mezo and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-02-26 with Mathematics categories.
Suppose $G$ is a real reductive algebraic group, $\theta$ is an automorphism of $G$, and $\omega$ is a quasicharacter of the group of real points $G(\mathbf{R})$. Under some additional assumptions, the theory of twisted endoscopy associates to this triple real reductive groups $H$. The Local Langlands Correspondence partitions the admissible representations of $H(\mathbf{R})$ and $G(\mathbf{R})$ into $L$-packets. The author proves twisted character identities between $L$-packets of $H(\mathbf{R})$ and $G(\mathbf{R})$ comprised of essential discrete series or limits of discrete series.
Pseudo Differential Operators With Discontinuous Symbols Widom S Conjecture
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Author : Aleksandr Vladimirovich Sobolev
language : en
Publisher: American Mathematical Soc.
Release Date : 2013-02-26
Pseudo Differential Operators With Discontinuous Symbols Widom S Conjecture written by Aleksandr Vladimirovich Sobolev and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-02-26 with Mathematics categories.
Relying on the known two-term quasiclassical asymptotic formula for the trace of the function $f(A)$ of a Wiener-Hopf type operator $A$ in dimension one, in 1982 H. Widom conjectured a multi-dimensional generalization of that formula for a pseudo-differential operator $A$ with a symbol $a(\mathbf{x}, \boldsymbol{\xi})$ having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs on a hyperplane. The present paper provides a proof of Widom's Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.