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Vertex Algebras And Integral Bases For The Enveloping Algebras Of Affine Lie Algebras
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Book Synopsis Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras by : Shari A. Prevost
Download or read book Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras written by Shari A. Prevost and published by American Mathematical Soc.. This book was released on 1992 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a new proof of the identities needed to exhibit an explicit [bold]Z-basis for the universal enveloping algebra associated to an affine Lie algebra. We then use the explicit [bold]Z-bases to extend Borcherds' description, via vertex operator representations, of a [bold]Z-form of the enveloping algebras for the simply-laced affine Lie algebras to the enveloping algebras associated to the unequal root length affine Lie algebras.
Book Synopsis Integral Bases for Affine Lie Algebras and Their Universal Enveloping Algebras by : David Mitzman
Download or read book Integral Bases for Affine Lie Algebras and Their Universal Enveloping Algebras written by David Mitzman and published by American Mathematical Soc.. This book was released on 1985 with total page 159 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work is a revised version of the author's Ph.D. thesis written under the supervision of J. Lepowsky at Rutgers University in 1983.
Book Synopsis Projective Modules over Lie Algebras of Cartan Type by : Daniel Ken Nakano
Download or read book Projective Modules over Lie Algebras of Cartan Type written by Daniel Ken Nakano and published by American Mathematical Soc.. This book was released on 1992 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph focuses on extending theorems for the classical Lie algebras in order to determine the structure and representation theory for Lie algebras of Cartan type. More specifically, Nakano investigates the block theory for the restricted universal enveloping algebras of the Lie algebras of Cartan type. The first section employs techniques developed by Holmes and Nakano to prove a Brauer-Humphreys reciprocity law for graded restricted Lie algebras and also to find the decompositions for the intermediate (Verma) modules used in the reciprocity law. The second section uses this information to investigate the structure of projective modules for the Lie algebras of types W and K. The restricted enveloping algebras for these Lie algebras are shown to have one block. Furthermore, Nakano provides a procedure for computing the Cartan invariants for Lie algebras of types W and K, given knowledge about the decomposition of the generalized Verma modules and about the Jantzen matrix of the classical/reductive zero component. Noteworthy for its readability and the continuity of its theme and purpose, this monograph appeals to graduate students and researchers interested in Lie algebras.
Book Synopsis Invariant Subsemigroups of Lie Groups by : Karl-Hermann Neeb
Download or read book Invariant Subsemigroups of Lie Groups written by Karl-Hermann Neeb and published by American Mathematical Soc.. This book was released on 1993 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work presents the first systematic treatment of invariant Lie semi groups. Because these semi groups provide interesting models for space times in general relativity, this work will be useful to both mathematicians and physicists. It will also appeal to engineers interested in bi-invariant control systems on Lie groups. Neeb investigates closed invariant subsemigroups of Lie groups which are generated by one-parameter semi groups and the sets of infinitesimal generators of such semi groups - invariant convex cones in Lie algebras.In addition, a characterization of those finite-dimensional real Lie algebras containing such cones is obtained. The global part of the theory deals with globality problems (Lie's third theorem for semi groups), controllability problems, and the facial structure of Lie semi groups. Neeb also determines the structure of the universal compactification of an invariant Lie semigroup and shows that the lattice of idempotents is isomorphic to a lattice of faces of the cone dual to the cone of infinitesimal generators.
Book Synopsis On Axiomatic Approaches to Vertex Operator Algebras and Modules by : Igor Frenkel
Download or read book On Axiomatic Approaches to Vertex Operator Algebras and Modules written by Igor Frenkel and published by American Mathematical Soc.. This book was released on 1993 with total page 64 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of vertex operator algebra arises naturally in the vertex operator construction of the Monster - the largest sporadic finite simple group. From another perspective, the theory of vertex operator algebras and their modules forms the algebraic foundation of conformal field theory. Vertex operator algebras and conformal field theory are now known to be deeply related to many important areas of mathematics. This essentially self-contained monograph develops the basic axiomatic theory of vertex operator algebras and their modules and intertwining operators, following a fundamental analogy with Lie algebra theory. The main axiom, the 'Jacobi(-Cauchy) identity', is a far-reaching analog of the Jacobi identity for Lie algebras.The authors show that the Jacobi identity is equivalent to suitably formulated rationality, commutativity, and associativity properties of products of quantum fields. A number of other foundational and useful results are also developed. This work was originally distributed as a preprint in 1989, and in view of the current widespread interest in the subject among mathematicians and theoretical physicists, its publication and availability should prove no less useful than when it was written.
Book Synopsis The Subregular Germ of Orbital Integrals by : Thomas Callister Hales
Download or read book The Subregular Germ of Orbital Integrals written by Thomas Callister Hales and published by American Mathematical Soc.. This book was released on 1992 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: Langlands theory predicts deep relationships between representations of different reductive groups over a local or global field. The trace formula attempts to reduce many such relationships to problems concerning conjugacy classes and integrals over conjugacy classes (orbital integrals) on $p$-adic groups. It is possible to reformulate these problems as ones in algebraic geometry by associating a variety $Y$ to each reductive group. Using methods of Igusa, the geometrical properties of the variety give detailed information about the asymptotic behavior of integrals over conjugacy classes.This monograph constructs the variety $Y$ and describes its geometry. As an application, the author uses the variety to give formulas for the leading terms (regular and subregular germs) in the asymptotic expansion of orbital integrals over $p$-adic fields. The final chapter shows how the properties of the variety may be used to confirm some predictions of Langlands theory on orbital integrals, Shalika germs, and endoscopy.
Book Synopsis Lie Algebras, Vertex Operator Algebras, and Related Topics by : Katrina Barron
Download or read book Lie Algebras, Vertex Operator Algebras, and Related Topics written by Katrina Barron and published by American Mathematical Soc.. This book was released on 2017-08-15 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the conference on Lie Algebras, Vertex Operator Algebras, and Related Topics, celebrating the 70th birthday of James Lepowsky and Robert Wilson, held from August 14–18, 2015, at the University of Notre Dame, Notre Dame, Indiana. Since their seminal work in the 1970s, Lepowsky and Wilson, their collaborators, their students, and those inspired by their work, have developed an amazing body of work intertwining the fields of Lie algebras, vertex algebras, number theory, theoretical physics, quantum groups, the representation theory of finite simple groups, and more. The papers presented here include recent results and descriptions of ongoing research initiatives representing the broad influence and deep connections brought about by the work of Lepowsky and Wilson and include a contribution by Yi-Zhi Huang summarizing some major open problems in these areas, in particular as they pertain to two-dimensional conformal field theory.
Book Synopsis Extensions of the Jacobi Identity for Vertex Operators, and Standard $A^{(1)}_1$-Modules by : Cristiano Husu
Download or read book Extensions of the Jacobi Identity for Vertex Operators, and Standard $A^{(1)}_1$-Modules written by Cristiano Husu and published by American Mathematical Soc.. This book was released on 1993 with total page 85 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book extends the Jacobi identity, the main axiom for a vertex operator algebra, to multi-operator identities. Based on constructions of Dong and Lepowsky, relative ${\mathbf Z}_2$-twisted vertex operators are then introduced, and a Jacobi identity for these operators is established. Husu uses these ideas to interpret and recover the twisted Z -operators and corresponding generating function identities developed by Lepowsky and Wilson for the construction of the standard $A^{(1)}_1$-modules. The point of view of the Jacobi identity also shows the equivalence between these twisted Z-operator algebras and the (twisted) parafermion algebras constructed by Zamolodchikov and Fadeev. The Lepowsky-Wilson generating function identities correspond to the identities involved in the construction of a basis for the space of C-disorder fields of such parafermion algebras.
Book Synopsis Unraveling the Integral Knot Concordance Group by : Neal W. Stoltzfus
Download or read book Unraveling the Integral Knot Concordance Group written by Neal W. Stoltzfus and published by American Mathematical Soc.. This book was released on 1977 with total page 91 pages. Available in PDF, EPUB and Kindle. Book excerpt: The group of concordance classes of high dimensional homotopy spheres knotted in codimension two in the standard sphere has an intricate algebraic structure which this paper unravels. The first level of invariants is given by the classical Alexander polynomial. By means of a transfer construction, the integral Seifert matrices of knots whose Alexander polynomial is a power of a fixed irreducible polynomial are related to forms with the appropriate Hermitian symmetry on torsion free modules over an order in the algebraic number field determined by the Alexander polynomial. This group is then explicitly computed in terms of standard arithmetic invariants. In the symmetric case, this computation shows there are no elements of order four with an irreducible Alexander polynomial. Furthermore, the order is not necessarily Dedekind and non-projective modules can occur. The second level of invariants is given by constructing an exact sequence relating the global concordance group to the individual pieces described above. The integral concordance group is then computed by a localization exact sequence relating it to the rational group computed by J. Levine and a group of torsion linking forms.
Book Synopsis Abelian Coverings of the Complex Projective Plane Branched along Configurations of Real Lines by : Eriko Hironaka
Download or read book Abelian Coverings of the Complex Projective Plane Branched along Configurations of Real Lines written by Eriko Hironaka and published by American Mathematical Soc.. This book was released on 1993 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work studies abelian branched coverings of smooth complex projective surfaces from the topological viewpoint. Geometric information about the coverings (such as the first Betti numbers of a smooth model or intersections of embedded curves) is related to topological and combinatorial information about the base space and branch locus. Special attention is given to examples in which the base space is the complex projective plane and the branch locus is a configuration of lines.
Book Synopsis Duality and Definability in First Order Logic by : Michael Makkai
Download or read book Duality and Definability in First Order Logic written by Michael Makkai and published by American Mathematical Soc.. This book was released on 1993 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt: Using the theory of categories as a framework, this book develops a duality theory for theories in first order logic in which the dual of a theory is the category of its models with suitable additional structure. This duality theory resembles and generalizes M. H. Stone's famous duality theory for Boolean algebras. As an application, the author derives a result akin to the well-known definability theorem of E. W. Beth. This new definability theorem is related to theorems of descent in category theory and algebra and can also be stated as a result in pure logic without reference to category theory. Containing novel techniques as well as applications of classical methods, this carefuly written book shows an attention to both organization and detail and will appeal to mathematicians and philosophers interested in category theory.
Book Synopsis Semistability of Amalgamated Products and HNN-Extensions by : Michael L. Mihalik
Download or read book Semistability of Amalgamated Products and HNN-Extensions written by Michael L. Mihalik and published by American Mathematical Soc.. This book was released on 1992 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, the authors show that amalgamated products and HNN-extensions of finitely presented semistable at infinity groups are also semistable at infinity. A major step toward determining whether all finitely presented groups are semistable at infinity, this result easily generalizes to finite graphs of groups. The theory of group actions on trees and techniques derived from the proof of Dunwoody's accessibility theorem are key ingredients in this work.
Book Synopsis The Cohen-Macaulay and Gorenstein Rees Algebras Associated to Filtrations by : Shirō Gotō
Download or read book The Cohen-Macaulay and Gorenstein Rees Algebras Associated to Filtrations written by Shirō Gotō and published by American Mathematical Soc.. This book was released on 1994 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph consists of two parts. Part I investigates the Cohen-Macaulay and Gorenstein properties of symbolic Rees algebras for one-dimensional prime ideals in Cohen-Macaulay local rings. Practical criteria for these algebras to be Cohen-Macaulay and Gorenstein rings are described in terms of certain elements in the prime ideals. This framework is generalized in Part II to Rees algebras $R(F)$ and graded rings $G(F)$ associated to general filtrations of ideals in arbitrary Noetherian local rings. Goto and Nishida give certain cohomological characterizations for algebras $R(F)$ to be Cohen-Macaulay or Gorenstein rings in connection with the corresponding ring-theoretic properties of $G(F)$. In this way, readers follow a history of the development of the ring theory of Rees algebras. The book raises many important open questions.
Book Synopsis Gorenstein Quotient Singularities in Dimension Three by : Stephen Shing-Toung Yau
Download or read book Gorenstein Quotient Singularities in Dimension Three written by Stephen Shing-Toung Yau and published by American Mathematical Soc.. This book was released on 1993 with total page 88 pages. Available in PDF, EPUB and Kindle. Book excerpt: If $G$ is a finite subgroup of $G\!L(3,{\mathbb C})$, then $G$ acts on ${\mathbb C}^3$, and it is known that ${\mathbb C}^3/G$ is Gorenstein if and only if $G$ is a subgroup of $S\!L(3,{\mathbb C})$. In this work, the authors begin with a classification of finite subgroups of $S\!L(3,{\mathbb C})$, including two types, (J) and (K), which have often been overlooked. They go on to present a general method for finding invariant polynomials and their relations to finite subgroups of $G\!L(3,{\mathbb C})$. The method is, in practice, substantially better than the classical method due to Noether. Some properties of quotient varieties are presented, along with a proof that ${\mathbb C}^3/G$ has isolated singularities if and only if $G$ is abelian and 1 is not an eigenvalue of $g$ for every nontrivial $g \in G$. The authors also find minimal quotient generators of the ring of invariant polynomials and relations among them.
Book Synopsis On Sets Not Belonging to Algebras of Subsets by : Leonid Š. Grinblat
Download or read book On Sets Not Belonging to Algebras of Subsets written by Leonid Š. Grinblat and published by American Mathematical Soc.. This book was released on 1992 with total page 111 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main results of this work can be formulated in such an elementary way that it is likely to attract mathematicians from a broad spectrum of specialties, though its main audience will likely be combintorialists, set-theorists, and topologists. The central question is this: Suppose one is given an at most countable family of algebras of subsets of some fixed set such that, for each algebra, there exists at least one set that is not a member of that algebra. Can one then assert that there is a set that is not a member of any of the algebras? Although such a set clearly exists in the case of one or two algebras, it is very easy to construct an example of three algebras for which no such set can be found. Grinblat's principal concern is to determine conditions that, if imposed on the algebras, will insure the existence of a set not belonging to any of them. If the given family of algebras is finite, one arrives at a purely combinatorial problem for a finite set of ultrafilters. If the family is countably infinite, however, one needs not only combinatorics of ultrafilters but also set theory and general topology.
Book Synopsis An Index of a Graph with Applications to Knot Theory by : Kunio Murasugi
Download or read book An Index of a Graph with Applications to Knot Theory written by Kunio Murasugi and published by American Mathematical Soc.. This book was released on 1993 with total page 101 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a remarkable application of graph theory to knot theory. In knot theory, there are a number of easily defined geometric invariants that are extremely difficult to compute; the braid index of a knot or link is one example. The authors evaluate the braid index for many knots and links using the generalized Jones polynomial and the index of a graph, a new invariant introduced here. This invariant, which is determined algorithmically, is likely to be of particular interest to computer scientists.
Book Synopsis On the Existence of Feller Semigroups with Boundary Conditions by : Kazuaki Taira
Download or read book On the Existence of Feller Semigroups with Boundary Conditions written by Kazuaki Taira and published by American Mathematical Soc.. This book was released on 1992 with total page 65 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a careful and accessible exposition of functional analytic methods in stochastic analysis. The author focuses on the relationship among three subjects in analysis: Markov processes, Feller semigroups, and elliptic boundary value problems. The approach here is distinguished by the author's extensive use of the theory of partial differential equations. Filling a mathematical gap between textbooks on Markov processes and recent developments in analysis, this work describes a powerful method capable of extensive further development. The book would be suitable as a textbook in a one-year, advanced graduate course on functional analysis and partial differential equations, with emphasis on their strong interrelations with probability theory.
