Chevalley Supergroups

Author : Rita Fioresi,Fabio Gavarini
Publisher : American Mathematical Soc.
Page : 77 pages
File Size : 41,8 Mb
Release : 2012
Category : Chevalley groups
ISBN : 9780821853009

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Chevalley Supergroups by Rita Fioresi,Fabio Gavarini Pdf

In the framework of algebraic supergeometry, the authors give a construction of the scheme-theoretic supergeometric analogue of split reductive algebraic group-schemes, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular, all Lie superalgebras of both basic and strange types are considered. This provides a unified approach to most of the algebraic supergroups considered so far in the literature, and an effective method to construct new ones. The authors' method follows the pattern of a suitable scheme-theoretic revisitation of Chevalley's construction of semisimple algebraic groups, adapted to the reductive case. As an intermediate step, they prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for their associated Kostant superalgebras.

Author : Sergio Ferrara,Rita Fioresi,V.S. Varadarajan
Publisher : Springer
Page : 279 pages
File Size : 44,6 Mb
Release : 2011-08-27
Category : Mathematics
ISBN : 9783642217449

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Supersymmetry in Mathematics and Physics by Sergio Ferrara,Rita Fioresi,V.S. Varadarajan Pdf

Supersymmetry was created by the physicists in the 1970's to give a unified treatment of fermions and bosons, the basic constituents of matter. Since then its mathematical structure has been recognized as that of a new development in geometry, and mathematicians have busied themselves with exploring this aspect. This volume collects recent advances in this field, both from a physical and a mathematical point of view, with an accent on a rigorous treatment of the various questions raised.

Author : Leonid A. Bokut',Aleksej Ivanovič Kostrikin
Publisher : American Mathematical Soc.
Page : 734 pages
File Size : 41,9 Mb
Release : 1992
Category : Algebra
ISBN : 9780821851371

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Proceedings of the International Conference on Algebra Dedicated to the Memory of A. I. Mal$'$cev by Leonid A. Bokut',Aleksej Ivanovič Kostrikin Pdf

Author : Nicolás Andruskiewitsch,Gongxiang Liu,Susan Montgomery,Yinhuo Zhang
Publisher : American Mathematical Soc.
Page : 359 pages
File Size : 44,9 Mb
Release : 2021-07-06
Category : Education
ISBN : 9781470456245

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Hopf Algebras, Tensor Categories and Related Topics by Nicolás Andruskiewitsch,Gongxiang Liu,Susan Montgomery,Yinhuo Zhang Pdf

The articles highlight the latest advances and further research directions in a variety of subjects related to tensor categories and Hopf algebras. Primary topics discussed in the text include the classification of Hopf algebras, structures and actions of Hopf algebras, algebraic supergroups, representations of quantum groups, quasi-quantum groups, algebras in tensor categories, and the construction method of fusion categories.

Author : Frank Nielsen,Frédéric Barbaresco
Publisher : Springer
Page : 764 pages
File Size : 40,8 Mb
Release : 2019-08-19
Category : Computers
ISBN : 9783030269807

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Geometric Science of Information by Frank Nielsen,Frédéric Barbaresco Pdf

This book constitutes the proceedings of the 4th International Conference on Geometric Science of Information, GSI 2019, held in Toulouse, France, in August 2019. The 79 full papers presented in this volume were carefully reviewed and selected from 105 submissions. They cover all the main topics and highlights in the domain of geometric science of information, including information geometry manifolds of structured data/information and their advanced applications.

Author : Rita Fioresi,María Antonia Lledó
Publisher : World Scientific
Page : 364 pages
File Size : 41,5 Mb
Release : 2015-03-09
Category : Science
ISBN : 9789814566353

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The Minkowski and Conformal Superspaces by Rita Fioresi,María Antonia Lledó Pdf

This book is aimed at graduate students and researchers in physics and mathematics who seek to understand the basics of supersymmetry from a mathematical point of view. It provides a bridge between the physical and mathematical approaches to the superworld. The physicist who is devoted to learning the basics of supergeometry can find a friendly approach here, since only the concepts that are strictly necessary are introduced. On the other hand, the mathematician who wants to learn from physics will find that all the mathematical assumptions are firmly rooted in physical concepts. This may open up a channel of communication between the two communities working on different aspects of supersymmetry. Starting from special relativity and Minkowski space, the idea of conformal space and superspace is built step by step in a mathematically rigorous way, and always connecting with the ideas and notation used in physics. While the book is mainly devoted to these important physical examples of superspaces, it can also be used as an introduction to the field of supergeometry, where a reader can ease into the subject without being overwhelmed with the technical difficulties. Contents:Introduction to SupergeometryThe Ordinary Minkowski and Conformal SpacesSupersymmetry in PhysicsThe Minkowski and Conformal SuperspacesDeformations of the Minkowski and Conformal SuperspacesAppendices:CategoriesRepresentability CriterionLie Superalgebras and Lie Supergroups of Classical TypeSuper Harish-Chandra PairsQuantum Supergroups Readership: Graduate students and researchers in mathematics and physics interested in supersymmetry. Keywords:Superspaces;Supersymmetry;Supergeometry;Quantum Groups;Conformal Group;Minkowski;SuperminkowskiKey Features:Comprehensible introduction to supergeometry: why the algebraic geometric framework is essentialThe superspaces chosen in this book allow us to introduce gradually concepts such as supergroups and homogeneous superspaces while, at the same time working out particular cases that are, both, non-trivial from a mathematical point of view and very important from the physical point of viewA further step is taken towards non-commutative geometry, although not introducing all its machinery: already at the level of quantum deformation the problem is involved and interesting. The construction may well show a path towards the formulation of physical theories on quantum spaces

Author : Aleksandr Sergeevich Kleshchëv,Vladimir Shchigolev
Publisher : American Mathematical Soc.
Page : 123 pages
File Size : 47,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821874318

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Modular Branching Rules for Projective Representations of Symmetric Groups and Lowering Operators for the Supergroup $Q(n)$ by Aleksandr Sergeevich Kleshchëv,Vladimir Shchigolev Pdf

There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup $Q(n)$ via appropriate Schur (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic $p$ to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type $A_{p-1}^{(2)}$. The goal of this work is to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups.

Author : Claudio Carmeli,Lauren Caston,Rita Fioresi
Publisher : European Mathematical Society
Page : 308 pages
File Size : 46,9 Mb
Release : 2011
Category : Lie algebras
ISBN : 3037190973

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Mathematical Foundations of Supersymmetry by Claudio Carmeli,Lauren Caston,Rita Fioresi Pdf

Supersymmetry is a highly active area of considerable interest among physicists and mathematicians. It is not only fascinating in its own right, but there is also indication that it plays a fundamental role in the physics of elementary particles and gravitation. The purpose of the book is to lay down the foundations of the subject, providing the reader with a comprehensive introduction to the language and techniques, as well as detailed proofs and many clarifying examples. This book is aimed ideally at second-year graduate students. After the first three introductory chapters, the text is divided into two parts: the theory of smooth supermanifolds and Lie supergroups, including the Frobenius theorem, and the theory of algebraic superschemes and supergroups. There are three appendices. The first introduces Lie superalgebras and representations of classical Lie superalgebras, the second collects some relevant facts on categories, sheafification of functors and commutative algebra, and the third explains the notion of Frechet space in the super context.

Author : Weinan E,Jianfeng Lu
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 50,5 Mb
Release : 2013-01-25
Category : Mathematics
ISBN : 9780821875605

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The Kohn-Sham Equation for Deformed Crystals by Weinan E,Jianfeng Lu Pdf

The solution to the Kohn-Sham equation in the density functional theory of the quantum many-body problem is studied in the context of the electronic structure of smoothly deformed macroscopic crystals. An analog of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure of the deformed crystal under the following physical conditions: (1) the band structure of the undeformed crystal has a gap, i.e. the crystal is an insulator, (2) the charge density waves are stable, and (3) the macroscopic dielectric tensor is positive definite. The effective equation governing the piezoelectric effect of a material is rigorously derived. Along the way, the authors also establish a number of fundamental properties of the Kohn-Sham map.

Author : Daniel Allcock
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 45,9 Mb
Release : 2012-10-31
Category : Mathematics
ISBN : 9780821869116

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The Reflective Lorentzian Lattices of Rank 3 by Daniel Allcock Pdf

"November 2012, volume 220, Number 1033 (first of 4 numbers)."

Author : Aleksandr Vladimirovich Sobolev
Publisher : American Mathematical Soc.
Page : 104 pages
File Size : 47,6 Mb
Release : 2013-02-26
Category : Mathematics
ISBN : 9780821884874

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Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture by Aleksandr Vladimirovich Sobolev Pdf

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.

Author : Dominique Lecomte
Publisher : American Mathematical Soc.
Page : 83 pages
File Size : 48,8 Mb
Release : 2013-01-25
Category : Mathematics
ISBN : 9780821875575

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Potential Wadge Classes by Dominique Lecomte Pdf

Let $\bf\Gamma$ be a Borel class, or a Wadge class of Borel sets, and $2\!\leq\! d\!\leq\!\omega$ be a cardinal. A Borel subset $B$ of ${\mathbb R}^d$ is potentially in $\bf\Gamma$ if there is a finer Polish topology on $\mathbb R$ such that $B$ is in $\bf\Gamma$ when ${\mathbb R}^d$ is equipped with the new product topology. The author provides a way to recognize the sets potentially in $\bf\Gamma$ and applies this to the classes of graphs (oriented or not), quasi-orders and partial orders.

Author : Paul Mezo
Publisher : American Mathematical Soc.
Page : 94 pages
File Size : 53,8 Mb
Release : 2013-02-26
Category : Mathematics
ISBN : 9780821875650

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Character Identities in the Twisted Endoscopy of Real Reductive Groups by Paul Mezo Pdf

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.

Author : David A. Cox,Andrew R. Kustin,Claudia Polini,Bernd Ulrich
Publisher : American Mathematical Soc.
Page : 116 pages
File Size : 49,9 Mb
Release : 2013-02-26
Category : Mathematics
ISBN : 9780821887431

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A Study of Singularities on Rational Curves Via Syzygies by David A. Cox,Andrew R. Kustin,Claudia Polini,Bernd Ulrich Pdf

Consider a rational projective curve $\mathcal{C}$ of degree $d$ over an algebraically closed field $\pmb k$. There are $n$ homogeneous forms $g_{1},\dots, g_{n}$ of degree $d$ in $B=\pmb k[x, y]$ which parameterize $\mathcal{C}$ in a birational, base point free, manner. The authors study the singularities of $\mathcal{C}$ by studying a Hilbert-Burch matrix $\varphi$ for the row vector $[g_{1},\dots, g_{n}]$. In the ``General Lemma'' the authors use the generalized row ideals of $\varphi$ to identify the singular points on $\mathcal{C}$, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let $p$ be a singular point on the parameterized planar curve $\mathcal{C}$ which corresponds to a generalized zero of $\varphi$. In the `'triple Lemma'' the authors give a matrix $\varphi'$ whose maximal minors parameterize the closure, in $\mathbb{P}^{2}$, of the blow-up at $p$ of $\mathcal{C}$ in a neighborhood of $p$. The authors apply the General Lemma to $\varphi'$ in order to learn about the singularities of $\mathcal{C}$ in the first neighborhood of $p$. If $\mathcal{C}$ has even degree $d=2c$ and the multiplicity of $\mathcal{C}$ at $p$ is equal to $c$, then he applies the Triple Lemma again to learn about the singularities of $\mathcal{C}$ in the second neighborhood of $p$. Consider rational plane curves $\mathcal{C}$ of even degree $d=2c$. The authors classify curves according to the configuration of multiplicity $c$ singularities on or infinitely near $\mathcal{C}$. There are $7$ possible configurations of such singularities. They classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity $c$ singularities on, or infinitely near, a fixed rational plane curve $\mathcal{C}$ of degree $2c$ is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix $\varphi$ for a parameterization of $\mathcal{C}$.

Author : Matthias Lesch,Henri Moscovici,Markus Pflaum
Publisher : American Mathematical Soc.
Page : 92 pages
File Size : 42,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821872963

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Connes-Chern Character for Manifolds with Boundary and Eta Cochains by Matthias Lesch,Henri Moscovici,Markus Pflaum Pdf

"November 2012, volume 220, number (end of volume)."