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Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras
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.
The Cohen-Macaulay and Gorenstein Rees Algebras Associated to Filtrations
At first, this volume was intended to be an investigation of symbolic blow-up rings for prime ideals defining curve singularities. The motivation for that has come from the recent 3-dimensional counterexamples to Cowsik's question, given by the authors and Watanabe: it has to be helpful, for further researches on Cowsik's question and a related problem of Kronecker, to generalize their methods to those of a higher dimension. However, while the study was progressing, it proved apparent that the framework of Part I still works, not only for the rather special symbolic blow-up rings but also in the study of Rees algebras R(F) associated to general filtrations F = {F[subscript]n} [subscript]n [subscript][set membership symbol][subscript bold]Z of ideals. This observation is closely explained in Part II of this volume, as a general ring-theory of Rees algebras R(F). We are glad if this volume will be a new starting point for the further researchers on Rees algebras R(F) and their associated graded rings G(F).
Semistability of Amalgamated Products and HNN-Extensions
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.
Phantom Homology
This book uses a powerful new technique, tight closure, to provide insight into many different problems that were previously not recognized as related. The authors develop the notion of weakly Cohen-Macaulay rings or modules and prove some very general acyclicity theorems. These theorems are applied to the new theory of phantom homology, which uses tight closure techniques to show that certain elements in the homology of complexes must vanish when mapped to well-behaved rings. These ideas are used to strengthen various local homological conjectures. Initially, the authors develop the theory in positive characteristic, but it can be extended to characteristic 0 by the method of reduction to characteristic $p$. The book would be suitable for use in an advanced graduate course in commutative algebra.
The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations
This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centres on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coicides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. a number of new examples illustrate the effectiveness of this approach.
Completely Prime Maximal Ideals and Quantization
Let [Fraktur lowercase]g be a complex simple Lie algebra of classical type, [italic capital]U([Fraktur lowercase]g) its enveloping algebra. We classify the completely prime maximal spectrum of [italic capital]U([Fraktur lowercase]g). We also construct some interesting algebra extensions of primitive quotients of [italic capital]U([Fraktur lowercase]g), and compute their Goldie ranks, lengths as bimodules, and characteristic cycles. Finally, we study the relevance of these algebras to D. Vogan's program of "quantizing" covers of nilpotent orbits [script]O in [Fraktur lowercase]g[superscript]*.
Neumann Systems for the Algebraic AKNS Problem
This work is concerned with an algebraically completely integrable Hamiltonian system whose solutions may be used to describe the finite gap solutions of the AKNS spectral problem, a first order two-by-two matrix linear system. Trace formulas, constraints, Lax paris, and constants of motion are obtained using Krichever's algebraic inverse spectral transform. Computations are carried out explicityly over the class of spectral problems with square matrix coefficients.
The Kinematic Formula in Riemannian Homogeneous Spaces
This memoir investigates a method that generalizes the Chern-Federer kinematic formula to arbitrary homogeneous spaces with an invariant Riemannian metric, and leads to new formulas even in the case of submanifolds of Euclidean space.
