The Finite Irreducible Linear 2-Groups of Degree 4
The Finite Irreducible Linear 2-Groups of Degree 4

Author: Dane Laurence Flannery

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 93

ISBN-13: 0821806254

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This memoir contains a complete classification of the finite irreducible 2-subgroups of GL(4, C). Specifically, the author provides a parametrized list of representatives for the conjugacy classes of such groups, where each representative is defined by generating a set of monomial matrices. The problem is treated by a variety of techniques, including: elementary character theory; a method for describing Hasse diagrams of submodule lattices; and calculation of 2-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence. Related questions concerning isomorphism between the listed groups and Schur indices of their defining characters are also considered

Matching of Orbital Integrals on $GL(4)$ and $GSp(2)$
Matching of Orbital Integrals on $GL(4)$ and $GSp(2)$

Author: Yuval Zvi Flicker

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 127

ISBN-13: 0821809598

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The trace formula is the most powerful tool currently available to establish liftings of automorphic forms, as predicted by Langlands principle of functionality. The geometric part of the trace formula consists of orbital integrals, and the lifting is based on the fundamental lemma. The latter is an identity of the relevant orbital integrals for the unit elements of the Hecke algebras. This volume concerns a proof of the fundamental lemma in the classically most interesting case of Siegel modular forms, namely the symplectic group Sp(2). These orbital integrals are compared with those on GL(4), twisted by the transpose inverse involution. The technique of proof is elementary. Compact elements are decomposed into their absolutely semi-simple and topologically unipotent parts also in the twisted case; a double coset decomposition of the form H\ G/K--where H is a subgroup containing the centralizer--plays a key role.

Diagram Groups
Diagram Groups

Author: Victor Guba

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 130

ISBN-13: 0821806394

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Diagram groups are groups consisting of spherical diagrams (pictures) over monoid presentations. They can be also defined as fundamental groups of the Squier complexes associated with monoid presentations. The authors show that the class of diagram groups contains some well-known groups, such as the R. Thompson group F. This class is closed under free products, finite direct products, and some other group-theoretical operations. The authors develop combinatorics on diagrams similar to the combinatorics on words. This helps in finding some structure and algorithmic properties of diagram groups. Some of these properties are new even for R. Thompson's group F. In particular, the authors describe the centralizers of elements in F, prove that it has solvable conjugacy problems, etc.

Lie Groups and Subsemigroups with Surjective Exponential Function
Lie Groups and Subsemigroups with Surjective Exponential Function

Author: Karl Heinrich Hofmann

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 189

ISBN-13: 0821806416

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In the structure theory of real Lie groups, there is still information lacking about the exponential function. Most notably, there are no general necessary and sufficient conditions for the exponential function to be surjective. It is surprising that for subsemigroups of Lie groups, the question of the surjectivity of the exponential function can be answered. Under nature reductions setting aside the "group part" of the problem, subsemigroups of Lie groups with surjective exponential function are completely classified and explicitly constructed in this memoir. There are fewer than one would think and the proofs are harder than one would expect, requiring some innovative twists. The main protagonists on the scene are SL(2, R) and its universal covering group, almost abelian solvable Lie groups (ie. vector groups extended by homotheties), and compact Lie groups. This text will also be of interest to those working in algebra and algebraic geometry.

Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras
Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras

Author: Michael David Weiner

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 121

ISBN-13: 0821808664

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Begins with the bosonic construction of four level -1/2 irreducible representations of the symplectic affine Kac-Moody Lie algebra Cl. The direct sum of two of these is given the structure of a vertex operator algebra (VOA), and the direct sum of the other two is given the structure of a twisted VOA-module. The dissertation includes the bosonic analog of the fermionic construction of a vertex operator superalgebra from the four level 1 irreducible modules of type Dl. No index. Annotation copyrighted by Book News, Inc., Portland, OR

Hopf Algebras, Polynomial Formal Groups, and Raynaud Orders
Hopf Algebras, Polynomial Formal Groups, and Raynaud Orders

Author: Lindsay Childs

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 133

ISBN-13: 0821810774

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This volume gives two new methods for constructing $p$-elementary Hopf algebra orders over the valuation ring $R$ of a local field $K$ containing the $p$-adic rational numbers. One method constructs Hopf orders using isogenies of commutative degree 2 polynomial formal groups of dimension $n$, and is built on a systematic study of such formal group laws. The other method uses an exponential generalization of a 1992 construction of Greither. Both constructions yield Raynaud orders as iterated extensions of rank $p$ Hopf algebras; the exponential method obtains all Raynaud orders whose invariants satisfy a certain $p$-adic condition.

Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable
Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable

Author: Kazuyoshi Kiyohara

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 159

ISBN-13: 0821806408

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Two classes of manifolds whose geodesic flows are integrable are defined, and their global structures are investigated. They are called Liouville manifolds and Kahler-Liouville manifolds respectively. In each case, the author finds several invariants with which they are partly classified. The classification indicates, in particular, that these classes contain many new examples of manifolds with integrable geodesic flow.

Wandering Vectors for Unitary Systems and Orthogonal Wavelets
Wandering Vectors for Unitary Systems and Orthogonal Wavelets

Author: Xingde Dai

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 82

ISBN-13: 0821808001

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Investigates topological and structural properties of the set W(U) of all complete wandering vectors for a system U of unitary operators acting on a Hilbert space. The authors parameterize W(U) in terms of a fixed vector y and the set of all unitary operators which locally commute with U at y. No index. Annotation copyrighted by Book News, Inc., Portland, OR

Annihilating Fields of Standard Modules of $\mathfrak {sl}(2, \mathbb {C})^\sim $ and Combinatorial Identities
Annihilating Fields of Standard Modules of $\mathfrak {sl}(2, \mathbb {C})^\sim $ and Combinatorial Identities

Author: Arne Meurman

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 105

ISBN-13: 0821809237

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In this volume, the authors show that a set of local admissible fields generates a vertex algebra. For an affine Lie algebra $\tilde{\frak g}$, they construct the corresponding level $k$ vertex operator algebra and show that level $k$ highest weight $\tilde{\frak g}$-modules are modules for this vertex operator algebra. They determine the set of annihilating fields of level $k$ standard modules and study the corresponding loop $\tilde{\frak g}$-module--the set of relations that defines standard modules. In the case when $\tilde{\frak g}$ is of type $A{(1)} 1$, they construct bases of standard modules parameterized by colored partitions, and as a consequence, obtain a series of Rogers-Ramanujan type combinatorial identities.