Invariant Differential Operators For Quantum Symmetric Spaces

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Invariant Differential Operators for Quantum Symmetric Spaces

Gail Letzter
Invariant Differential Operators for Quantum Symmetric Spaces Book PDF

Author : Gail Letzter
Publisher : American Mathematical Soc.
Page : 104 pages
File Size : 55,5 Mb
Release : 2008
Category : Quantum groups
ISBN : 9780821841310

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Invariant Differential Operators for Quantum Symmetric Spaces by Gail Letzter Pdf

This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.

Unitary Invariants in Multivariable Operator Theory

Gelu Popescu
Unitary Invariants in Multivariable Operator Theory Book PDF

Author : Gelu Popescu
Publisher : American Mathematical Soc.
Page : 105 pages
File Size : 40,5 Mb
Release : 2009-06-05
Category : Mathematics
ISBN : 9780821843963

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Unitary Invariants in Multivariable Operator Theory by Gelu Popescu Pdf

This paper concerns unitary invariants for $n$-tuples $T:=(T_1,\ldots, T_n)$ of (not necessarily commuting) bounded linear operators on Hilbert spaces. The author introduces a notion of joint numerical radius and works out its basic properties. Multivariable versions of Berger's dilation theorem, Berger-Kato-Stampfli mapping theorem, and Schwarz's lemma from complex analysis are obtained. The author studies the joint (spatial) numerical range of $T$ in connection with several unitary invariants for $n$-tuples of operators such as: right joint spectrum, joint numerical radius, euclidean operator radius, and joint spectral radius. He also proves an analogue of Toeplitz-Hausdorff theorem on the convexity of the spatial numerical range of an operator on a Hilbert space, for the joint numerical range of operators in the noncommutative analytic Toeplitz algebra $F_n^\infty$.

Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

Volkmar Liebscher
Random Sets and Invariants for Type II Continuous Tensor Product Systems of Hilbert Spaces Book PDF

Author : Volkmar Liebscher
Publisher : American Mathematical Soc.
Page : 124 pages
File Size : 49,7 Mb
Release : 2009-04-10
Category : Mathematics
ISBN : 9780821843185

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Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces by Volkmar Liebscher Pdf

In a series of papers Tsirelson constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So the author connects each continuous tensor product system of Hilbert spaces with measure types of distributions of random (closed) sets in $[0,1]$ or $\mathbb R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the Tsirelson examples, that the classification scheme for product systems into types $\mathrm{I}_n$, $\mathrm{II}_n$ and $\mathrm{III}$ is not complete. Moreover, based on a detailed study of this kind of measure types, the author constructs for each stationary factorising measure type a continuous tensor product system of Hilbert spaces such that this measure type arises as the before mentioned invariant.

Cohomological Invariants: Exceptional Groups and Spin Groups

Skip Garibaldi
Cohomological Invariants Exceptional Groups and Spin Groups Book PDF

Author : Skip Garibaldi
Publisher : American Mathematical Soc.
Page : 102 pages
File Size : 42,8 Mb
Release : 2009-06-05
Category : Mathematics
ISBN : 9780821844045

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Cohomological Invariants: Exceptional Groups and Spin Groups by Skip Garibaldi Pdf

This volume concerns invariants of $G$-torsors with values in mod $p$ Galois cohomology--in the sense of Serre's lectures in the book Cohomological invariants in Galois cohomology--for various simple algebraic groups $G$ and primes $p$. The author determines the invariants for the exceptional groups $F_4$ mod 3, simply connected $E_6$ mod 3, $E_7$ mod 3, and $E_8$ mod 5. He also determines the invariants of $\mathrm{Spin}_n$ mod 2 for $n \leq 12$ and constructs some invariants of $\mathrm{Spin}_{14}$. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfister's results on 10- and 12-dimensional quadratic forms and Rost's theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of $\mathrm{Spin}_n$ is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.

Classical And Quantum Systems: Foundations And Symmetries - Proceedings Of The 2nd International Wigner Symposium

Heinz-dietrich Doebner,F Schroeck Jr,W Scherer
Classical And Quantum Systems Foundations And Symmetries Proceedings Of The 2nd International Wigner Symposium Book PDF

Author : Heinz-dietrich Doebner,F Schroeck Jr,W Scherer
Publisher : World Scientific
Page : 818 pages
File Size : 46,5 Mb
Release : 1993-01-19
Category : Electronic
ISBN : 9789814554398

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Classical And Quantum Systems: Foundations And Symmetries - Proceedings Of The 2nd International Wigner Symposium by Heinz-dietrich Doebner,F Schroeck Jr,W Scherer Pdf

The Wigner Symposium series is focussed on fundamental problems and new developments in physics and their experimental, theoretical and mathematical aspects. Particular emphasis is given to those topics which have developed from the work of Eugene P Wigner. The 2nd Wigner symposium is centered around notions of symmetry and geometry, the foundations of quantum mechanics, quantum optics and particle physics. Other fields like dynamical systems, neural networks and physics of information are also represented.This volume brings together 19 plenary lectures which survey latest developments and more than 130 contributed research reports.

Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules

AndrŽ Martinez,Vania Sordoni
Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules Book PDF

Author : AndrŽ Martinez,Vania Sordoni
Publisher : American Mathematical Soc.
Page : 96 pages
File Size : 46,6 Mb
Release : 2009-06-05
Category : Mathematics
ISBN : 9780821842966

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Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules by AndrŽ Martinez,Vania Sordoni Pdf

The authors construct an abstract pseudodifferential calculus with operator-valued symbol, suitable for the treatment of Coulomb-type interactions, and they apply it to the study of the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case of the electronic Hamiltonian admitting a local gap in its spectrum. In particular, they show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, they study the propagation of certain wave packets up to long time values of Ehrenfest order.

Heisenberg Calculus and Spectral Theory of Hypoelliptic Operators on Heisenberg Manifolds

Raphael Ponge
Heisenberg Calculus and Spectral Theory of Hypoelliptic Operators on Heisenberg Manifolds Book PDF

Author : Raphael Ponge
Publisher : American Mathematical Soc.
Page : 150 pages
File Size : 42,7 Mb
Release : 2008
Category : Calculus
ISBN : 9780821841488

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Heisenberg Calculus and Spectral Theory of Hypoelliptic Operators on Heisenberg Manifolds by Raphael Ponge Pdf

This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, including CR and contact manifolds. In this context the main differential operators at stake include the Hormander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian, the CR conformal operators of Gover-Graham and the contact Laplacian. These operators cannot be elliptic and the relevant pseudodifferential calculus to study them is provided by the Heisenberg calculus of Beals-Greiner andTaylor.

The Topological Dynamics of Ellis Actions

Ethan Akin,Joseph Auslander,Eli Glasner
The Topological Dynamics of Ellis Actions Book PDF

Author : Ethan Akin,Joseph Auslander,Eli Glasner
Publisher : American Mathematical Soc.
Page : 166 pages
File Size : 43,9 Mb
Release : 2008
Category : Topological semigroups
ISBN : 9780821841884

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The Topological Dynamics of Ellis Actions by Ethan Akin,Joseph Auslander,Eli Glasner Pdf

An Ellis semigroup is a compact space with a semigroup multiplication which is continuous in only one variable. An Ellis action is an action of an Ellis semigroup on a compact space such that for each point in the space the evaluation map from the semigroup to the space is continuous. At first the weak linkage between the topology and the algebra discourages expectations that such structures will have much utility. However, Ellis has demonstrated that these actions arise naturallyfrom classical topological actions of locally compact groups on compact spaces and provide a useful tool for the study of such actions. In fact, via the apparatus of the enveloping semigroup the classical theory of topological dynamics is subsumed by the theory of Ellis actions. The authors'exposition describes and extends Ellis' theory and demonstrates its usefulness by unifying many recently introduced concepts related to proximality and distality. Moreover, this approach leads to several results which are new even in the classical setup.

A Proof of Alon's Second Eigenvalue Conjecture and Related Problems

Joel Friedman
A Proof of Alon s Second Eigenvalue Conjecture and Related Problems Book PDF

Author : Joel Friedman
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 53,9 Mb
Release : 2008
Category : Eigenvalues
ISBN : 9780821842805

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A Proof of Alon's Second Eigenvalue Conjecture and Related Problems by Joel Friedman Pdf

A $d$-regular graph has largest or first (adjacency matrix) eigenvalue $\lambda_1=d$. Consider for an even $d\ge 4$, a random $d$-regular graph model formed from $d/2$ uniform, independent permutations on $\{1,\ldots,n\}$. The author shows that for any $\epsilon>0$ all eigenvalues aside from $\lambda_1=d$ are bounded by $2\sqrt{d-1}\;+\epsilon$ with probability $1-O(n^{-\tau})$, where $\tau=\lceil \bigl(\sqrt{d-1}\;+1\bigr)/2 \rceil-1$. He also shows that this probability is at most $1-c/n^{\tau'}$, for a constant $c$ and a $\tau'$ that is either $\tau$ or $\tau+1$ (``more often'' $\tau$ than $\tau+1$). He proves related theorems for other models of random graphs, including models with $d$ odd.

Torus Fibrations, Gerbes, and Duality

Ron Donagi,Tony Pantev
Torus Fibrations Gerbes and Duality Book PDF

Author : Ron Donagi,Tony Pantev
Publisher : American Mathematical Soc.
Page : 104 pages
File Size : 52,8 Mb
Release : 2008
Category : Calabi-Yau manifolds
ISBN : 9780821840924

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Torus Fibrations, Gerbes, and Duality by Ron Donagi,Tony Pantev Pdf

Let $X$ be a smooth elliptic fibration over a smooth base $B$. Under mild assumptions, the authors establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an $\mathcal{O} DEGREES{\times}$ gerbe over a genus one fibration which is a twisted form

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

William Mark Goldman,Eugene Zhu Xia
Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces Book PDF

Author : William Mark Goldman,Eugene Zhu Xia
Publisher : American Mathematical Soc.
Page : 86 pages
File Size : 52,9 Mb
Release : 2008
Category : Geometry, Algebraic
ISBN : 9780821841365

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Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces by William Mark Goldman,Eugene Zhu Xia Pdf

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkähler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of $X$. The authors describe the moduli spaces and their geometry in terms of the Riemann period matrix of $X$.

Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories

Dominic Verity
Complicial Sets Characterising the Simplicial Nerves of Strict omega Categories Book PDF

Author : Dominic Verity
Publisher : American Mathematical Soc.
Page : 208 pages
File Size : 51,5 Mb
Release : 2008
Category : Algebraic topology
ISBN : 9780821841426

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Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories by Dominic Verity Pdf

The primary purpose of this work is to characterise strict $\omega$-categories as simplicial sets with structure. The author proves the Street-Roberts conjecture in the form formulated by Ross Street in his work on Orientals, which states that they are exactly the ``complicial sets'' defined and named by John Roberts in his handwritten notes of that title (circa 1978). On the way the author substantially develops Roberts' theory of complicial sets itself and makes contributions to Street's theory of parity complexes. In particular, he studies a new monoidal closed structure on the category of complicial sets which he shows to be the appropriate generalisation of the (lax) Gray tensor product of 2-categories to this context. Under Street's $\omega$-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others.

Toroidal Dehn Fillings on Hyperbolic 3-Manifolds

Cameron Gordon,Ying-Qing Wu
Toroidal Dehn Fillings on Hyperbolic 3 Manifolds Book PDF

Author : Cameron Gordon,Ying-Qing Wu
Publisher : American Mathematical Soc.
Page : 154 pages
File Size : 54,9 Mb
Release : 2008
Category : Dehn surgery
ISBN : 9780821841679

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Toroidal Dehn Fillings on Hyperbolic 3-Manifolds by Cameron Gordon,Ying-Qing Wu Pdf

The authors determine all hyperbolic $3$-manifolds $M$ admitting two toroidal Dehn fillings at distance $4$ or $5$. They show that if $M$ is a hyperbolic $3$-manifold with a torus boundary component $T 0$, and $r,s$ are two slopes on $T 0$ with $\Delta(r,s) = 4$ or $5$ such that $M(r)$ and $M(s)$ both contain an essential torus, then $M$ is either one of $14$ specific manifolds $M i$, or obtained from $M 1, M 2, M 3$ or $M {14}$ by attaching a solid torus to $\partial M i - T 0$.All the manifolds $M i$ are hyperbolic, and the authors show that only the first three can be embedded into $S3$. As a consequence, this leads to a complete classification of all hyperbolic knots in $S3$ admitting two toroidal surgeries with distance at least $4$.

The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations

Salah-Eldin A. Mohammed,Salah-Eldin Mohammed,Tusheng Zhang,Huaizhong Zhao
The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations Book PDF

Author : Salah-Eldin A. Mohammed,Salah-Eldin Mohammed,Tusheng Zhang,Huaizhong Zhao
Publisher : American Mathematical Soc.
Page : 120 pages
File Size : 45,9 Mb
Release : 2008
Category : Evolution equations
ISBN : 9780821842508

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The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations by Salah-Eldin A. Mohammed,Salah-Eldin Mohammed,Tusheng Zhang,Huaizhong Zhao Pdf

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations and stochastic partial differential equations near stationary solutions.

Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints

Sergiu Aizicovici,Nikolaos Socrates Papageorgiou,Vasile Staicu
Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints Book PDF

Author : Sergiu Aizicovici,Nikolaos Socrates Papageorgiou,Vasile Staicu
Publisher : American Mathematical Soc.
Page : 84 pages
File Size : 49,8 Mb
Release : 2008
Category : Differential equations, Elliptic
ISBN : 9780821841921

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Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints by Sergiu Aizicovici,Nikolaos Socrates Papageorgiou,Vasile Staicu Pdf

In this paper the authors examine the degree map of multivalued perturbations of nonlinear operators of monotone type and prove that at a local minimizer of the corresponding Euler functional, this degree equals one.