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‣ Vertex algebras generated by primary fields of low conformal weight

De Sole, Alberto, 1975-
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 144 p.; 5438427 bytes; 5438236 bytes; application/pdf; application/pdf
Português
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476.38938%
We classify a certain class of vertex algebras, finitely generated by a Virasoro field and even (resp. odd) primary fields of conformal weight 1 (resp 3/2). This is the first interesting case to consider when looking at finitely generated vertex algebras containing a Virasoro field (the most interesting from the point of view of physics). By the axioms of vertex algebras it follows that the space g of fields with conformal weight 1 is a Lie algebra, and the space U of fields with conformal weight 3/2 is a g-module with a symmetric invariant bilinear form. One of the main observations is that, under the assumption of existence of a quasi- classical limit (which basically translates to the existence of a one parameter family of vertex algebras, the free parameter being the Kac-Moody level k), the complex connected algebraic group G corresponding to the Lie algebra 0 acts transitively on the quadric ... This generalizes a similar result of Kac in the case of conformal algebras. Using this observation, we will classify vertex algebras satisfying the above assumptions, by using the classification of connected compact subgroups of SON acting transitively on the unit sphere. The solution is given by the following list ... However, if one removes the assumption of existence of quasi-classical limit...

‣ New examples of four dimensional AS-regular algebras

Caines, Ian
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 49 p.; 2019440 bytes; 2023142 bytes; application/pdf; application/pdf
Português
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This thesis deals with AS-regular algebras, first defined by Michael Artin and William Schelter in Graded Algebras of Global Dimension 3. All such algebras of dimension three have been classified, but the corresponding problem in higher dimensions remains open. We construct new examples of four dimensional AS-regular algebras, and provide some information about their module structure. Results are provided for proving the regularity of such algebras. In addition we classify the AS-regular algebras of dimension four satisfying certain conditions.; by Ian Caines.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.; Includes bibliographical references (p. 48-49).

‣ Double affine Hecke algebras and noncommutative geometry

Oblomkov, Alexei
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 96 p.; 4016575 bytes; 4027564 bytes; application/pdf; application/pdf
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In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, 1) of type An-1 is an integral Cohen-Macaulay algebra isomorphic to the center Z of H(t, 1), and H(t, 1)e is a Cohen-Macaulay eH(t, 1)e-module with the property H(t, 1) = EndeH(t,tl)(H(t, 1)e). This implies the classification of the finite dimensional representations of the algebras. In the second part we study the algebraic properties of the five-parameter family H(tl, t2, t3, t4; q) of double affine Hecke algebras of type CVC1, which control Askey- Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. We prove that the only fiat de- formations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove several results on the universality of the five-parameter family H(tl...

‣ Finite dimensional representations of sympletic reflection algebras for wreath products

Montarani, Silvia
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 137 p.
Português
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472.376%
Symplectic reflection algebras are attached to any finite group G of automorphisms of a symplectic vector space V , and are a multi-parameter deformation of the smash product TV ?G, where TV is the tensor algebra. Their representations have been studied in connection with different subjects, such as symplectic quotient singularities, Hilbert scheme of points in the plane and combinatorics. Let ... SL(2,C) be a finite subgroup, and let Sn be the symmetric group on n letters. We study finite dimensional representations of the wreath product symplectic reflection algebra ... of rank n, attached to the wreath product group ... and to the parameters (k, c), where k is a complex number (occurring only for n > 1), and c a class function on the set of nontrivial elements of ... In particular, we construct, for the first time, families of irreducible finite dimensional modules when ... is not cyclic, n > 1, and (k, c) vary in some linear subspace of the space of parameters. The method is deformation theoretic and uses properties of the Hochschild cohomology of H1,k,c(...), and a Morita equivalence, established by Crawley-Boevey and Holland, between the rank one algebra H1, ... and the deformed preprojective algebra ?Q), where Q is the extended Dynkin quiver attached to ?? via the McKay correspondence. We carry out a similar construction for continuous wreath product symplectic reflection algebras...

‣ Nonstandard hulls of Banach-Lie groups and algebras

Pestov, Vladimir G.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/05/1992 Português
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We propose a new construction of Banach-Lie groups and algebras relying on nonstandard analysis. A major standard application is the Local Theorem which to certain extent reduces the problem of associating a Lie group to a given banach-Lie algebra to a similar problem for finitely generated Lie subalgebras. We discuss possible applications, e.g., to gauge theories.; Comment: 12 pages

‣ Sparsity and Spatial Localization Measures for Spatially Distributed Systems

Motee, Nader; Sun, Qiyu
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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476.30496%
We consider the class of spatially decaying systems, where the underlying dynamics are spatially decaying and the sensing and controls are spatially distributed. This class of systems arise in various applications where there is a notion of spatial distance with respect to which couplings between the subsystems can be quantified using a class of coupling weight functions. We exploit spatial decay property of the underlying dynamics of the system to introduce a class of sparsity and spatial localization measures for spatially distributed systems. We develop a new methodology based on concepts of $q$-Banach algebras of spatially decaying operators that enable us to establish a relationship between spatial decay properties of spatially decaying systems and their sparsity and spatial localization features. Moreover, it is shown that the inverse-closedness property of operator algebras plays a central role in exploiting various structural properties of spatially decaying systems. We characterize conditions for exponentially stability of spatially decaying system over $q$-Banach algebras and prove that the unique solutions of the Lyapunov and Riccati equations over a proper $q$-Banach algebra also belong to the same $q$-Banach algebra. It is shown that the quadratically optimal state feedback controllers for spatially decaying systems are sparse and spatially localized in the sense that they have near-optimal sparse information structures.; Comment: 20 pages...

‣ On Hopf Algebras and Their Generalizations

Karaali, Gizem
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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471.201%
We survey Hopf algebras and their generalizations. In particular, we compare and contrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (Hopf monads and hopfish algebras). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.

‣ Classification of $N$-(super)-extended Poincar\'e algebras and bilinear invariants of the spinor representation of $Spin(p,q)$

Alekseevsky, Dmitry V.; Cortés, Vicente
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/11/1995 Português
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469.6194%
We classify extended Poincar\'e Lie super algebras and Lie algebras of any signature (p,q), that is Lie super algebras and Z_2-graded Lie algebras g = g_0 + g_1, where g_0 = so(V) + V is the (generalized) Poincar\'e Lie algebra of the pseudo Euclidean vector space V = R^{p,q} of signature (p,q) and g_1 = S is the spinor so(V)-module extended to a g_0-module with kernel V. The remaining super commutators {g_1,g_1} (respectively, commutators [g_1, g_1]) are defined by an so(V)-equivariant linear mapping vee^2 g_1 -> V (respectively, wedge^2 g_1 -> V). Denote by P^+(n,s) (respectively, P^-(n,s)) the vector space of all such Lie super algebras (respectively, Lie algebras), where n = p + q = dim V and s = p - q is the signature. The description of P^+-(n,s) reduces to the construction of all so(V)-invariant bilinear forms on S and to the calculation of three Z_2-valued invariants for some of them. This calculation is based on a simple explicit model of an irreducible Clifford module S for the Clifford algebra Cl_{p,q} of arbitrary signature (p,q). As a result of the classification, we obtain the numbers L^+-(n,s) = \dim P^+-(n,s) of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity...

‣ Affine transformation crossed product like algebras and noncommutative surfaces

Arnlind, Joakim; Silvestrov, Sergei
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/03/2009 Português
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Several classes of *-algebras associated to the action of an affine transformation are considered, and an investigation of the interplay between the different classes of algebras is initiated. Connections are established that relate representations of *-algebras, geometry of algebraic surfaces, dynamics of affine transformations, graphs and algebras coming from a quantization procedure of Poisson structures. In particular, algebras related to surfaces being inverse images of fourth order polynomials (in R^3) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.

‣ Semi-direct products of Lie algebras and their invariants

Panyushev, Dmitri I.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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473.7523%
The goal of this paper is to extend the standard invariant-theoretic design, well-developed in the reductive case, to the setting of representation of certain non-reductive groups. This concerns the following notions and results: the existence of generic stabilisers and generic isotropy groups for finite-dimensional representations; structure of the fields and algebras of invariants; quotient morphisms and structure of their fibres. One of the main tools for obtaining non-reductive Lie algebras is the semi-direct product construction. We observe that there are surprisingly many non-reductive Lie algebras whose adjoint representation has a polynomial algebra of invariants. We extend results of Takiff, Geoffriau, Rais-Tauvel, and Levasseur-Stafford concerning Takiff Lie algebras to a wider class of semi-direct products. This includes $Z_2$-contractions of simple Lie algebras and generalised Takiff algebras.; Comment: 49 pages, title changed, section 11 is shortened, numerous minor corrections; accepted version, to appear in Publ. RIMS 43(2007)

‣ Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of Lagrangian frames

Quiroga-Barranco, Raul; Sanchez-Nungaray, Armando
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We prove that the quasi-homogenous symbols on the projective space $\mathbb{P}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball $\mathbb{B}^n$. These algebras are Banach but not $C^*$. We prove the existence of a strong link between such symbols and algebras with the geometry of $\mathbb{P}^n(\mathbb{C})$. The latter is also proved for the corresponding symbols and algebras on $\mathbb{B}^n$.; Comment: Corrected version

‣ Non-associative algebras, Yang-Baxter equations and quantum computers

Iordanescu, Radu; Nichita, Florin F.; Nichita, Ion M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/08/2014 Português
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472.5102%
Non-associtive algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we present a unification for associative algebras, Jordan algebras and Lie algebras. The (quantum) Yang-Baxter equation and related structures are interesting topics, because they have applications in many areas of mathematics, physics and computer science. Several new interpretations and results are presented below.; Comment: 7 pages

‣ Generalization of C*-algebra methods via real positivity for operator and Banach algebras

Blecher, David P.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/08/2015 Português
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474.6403%
With Charles Read we have introduced and studied a new notion of (real) positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. As motivation note that the `completely' real positive maps on C*-algebras or operator systems are precisely the completely positive maps in the usual sense; however with real positivity one may develop a useful order theory for more general spaces and algebras. This is intimately connected to new relationships between an operator algebra and the C*-algebra it generates. We have continued this work together with Read, and also with Matthew Neal. Recently with Narutaka Ozawa we have investigated the parts of the theory that generalize further to Banach algebras. In the present paper we describe some of this work which is connected with generalizing various C*-algebraic techniques initiated by Richard V. Kadison. In particular Section 2 is in part a tribute to him in keeping with the occasion of this volume, and also discusses some of the origins of the theory of positivity (in our sense) in the setting of algebras, which the later parts of our paper develops further. The most recent work will be emphasized.; Comment: 32 pages

‣ Proportions of Cyclic Matrices in Maximal Reducible Matrix Groups and Algebras

Brown, Scott; Praeger, Cheryl E.; Giudici, Michael
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/05/2011 Português
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554.6545%
A matrix is said to be {\it cyclic} if its characteristic polynomial is equal to its minimal polynomial. Cyclic matrices play an important role in some algorithms for matrix group computation, such as the Cyclic Meataxe developed by P. M. Neumann and C. E. Praeger in 1999. In that year also, G. E. Wall and J. E. Fulman independently found the limiting proportion of cyclic matrices in general linear groups over a finite field of fixed order q as the dimension n approaches infinity, namely $(1-q^{-5}) \prod_{i=3}^\infty (1-q^{-i}) = 1 - q^{-3} + O(q^{-4}).$ We study cyclic matrices in a maximal reducible matrix group or algebra, that is, in the largest subgroup or subalgebra that leaves invariant some proper nontrivial subspace. We modify Wall's generating function approach to determine the limiting proportions of cyclic matrices in maximal reducible matrix groups and algebras over a field of order q, as the dimension of the underlying vector space increases while that of the invariant subspace remains fixed. The limiting proportion in a maximal reducible group is proved to be $1 - q^{-2} + O(q^{-3})$; note the change of the exponent of q in the second term of the expansion. Moreover, we exhibit in each maximal reducible matrix group a family of noncyclic matrices whose proportion is $q^{-2} + O(q^{-3})$.; Comment: 62 pages PhD thesis of first author available at http://theses.library.uwa.edu.au/adt-WU2006.0079/

‣ The mathematics of Donald Gordon Higman

Bannai, Eiichi; Griess, Jr., Robert L.; Praeger, Cheryl; Scott, Leonard
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/01/2009 Português
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554.44734%
This is about the mathematics and life of Donald Gordon Higman, 1928-2006. He did important work in representation theory of groups and algebras and in algebraic combinatorics. Charles C. Sims and Donald Higman discovered and constructed one of the sporadic simple groups.; Comment: 43 pages; submitted to Michigan Math Journal for their special memorial issue on Donald Higman (expected in 2009)

‣ Operads, Algebras and Modules in General Model Categories

Spitzweck, Markus
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/01/2001 Português
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468.9316%
In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. In a second part we develop the thoery of S-modules of [EKMM]., which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.; Comment: 48 pages, part of PHD thesis

‣ Operator Algebras of Functions

Mittal, Meghna; Paulsen, Vern
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 29/07/2009 Português
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474.9079%
We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.; Comment: 33 pages

‣ Noncommutative physics on Lie algebras, Z_2^n lattices and Clifford algebras

Majid, S.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, $\theta$-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice $(\Z_2)^n$ and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell's equations. We exactly quantize noncommutative U(1)-Yang-Mills theory on $\Z_2\times\Z_2$ in a path integral approach.; Comment: Final version to appear in Clifford Algebras: Application to Mathematics, Physics, and Engineering, ed. R. Ablamowicz, Birkhauser (2003); added a couple of references and fixed typos (no significant revision). 24 pages, 1 .eps figure

‣ K-Theory and peiodic cyclic homology of some noncompact quantum algebras

Diep, Do Ngoc; Kuku, Aderemi O.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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469.12312%
We prove in this paper that the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of connected and simply connected Lie group, are isomorphic to the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of compact maximal subgroups, without localization. Some noncompact quantum groups and algebras were constructed and their irreducible representations were classified in recent works of Do Ngoc Diep and Nguyen Viet Hai, and Do Duc Hanh by using deformation quantization. In this paper we compute their K-groups, periodic cyclic homology groups and their Chern characters.; Comment: 34 pages, no pictures, LaTeX2e

‣ Some Exceptional Cases in Mathematics: Euler Characteristic, Division Algebras, Cross Vector Product and Fano Matroid

Nieto, J. A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 08/11/2011 Português
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468.9316%
We review remarkable results in several mathematical scenarios, including graph theory, division algebras, cross product formalism and matroid theory. Specifically, we mention the following subjects: (1) the Euler relation in graph theory, and its higher-dimensional generalization, (2) the dimensional theorem for division algebras and in particular the Hurwitz theorem for normed division algebras, (3) the vector cross product dimensional possibilities, (4) some theorems for graphs and matroids. Our main goal is to motivate a possible research work in these four topics, putting special interest in their possible links.; Comment: 14 pages, Latex