The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppas discovery of algebraic geometry codes around 1980, many applications of function fields were found in different areas of mathematics and information theory. This book presents survey articles on some of these new developments. The topics focus on material which has not yet been presented in other books or survey articles.
The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppas discovery of algebraic geometry ...
Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields. The first stage of this development of the theory is associated with its founder, J.F. Ritt (1893-1951), and R. Cohn, whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrown the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations...
Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields. The first stage of this develo...
A principal ingredient in the proof of the Moonshine Theorem, connecting the Monster group to modular forms, is the infinite dimensional Lie algebra of physical states of a chiral string on an orbifold of a 26 dimensional torus, called the Monster Lie algebra. It is a Borcherds-Kac-Moody Lie algebra with Lorentzian root lattice; and has an associated automorphic form having a product expansion describing its structure. Lie superalgebras are generalizations of Lie algebras, useful for depicting supersymmetry - the symmetry relating fermions and bosons. Most known examples of Lie...
A principal ingredient in the proof of the Moonshine Theorem, connecting the Monster group to modular forms, is the infinite dimensional Lie algebr...
The ?rst aim of this work is to present the main results and methods of the theory of Noetherian semigroup algebras. These general results are then applied and illustrated in the context of certain interesting and important concrete classes of algebras that arise in a variety of areas and have been recently intensively studied. One of the main motivations for this project has been the growing int- est in the class of semigroup algebras (and their deformations) and in the application of semigroup theoretical methods. Two factors seem to be the cause for this. First, this ?eld covers several...
The ?rst aim of this work is to present the main results and methods of the theory of Noetherian semigroup algebras. These general results are then ap...
The theory of table algebras was introduced in 1991 by Z. Arad and H. Blau in order to treat, in a uniform way, products of conjugacy classes and irreducible characters of finite groups. Today, table algebra theory is a well-established branch of modern algebra with various applications, including the representation theory of finite groups, algebraic combinatorics and fusion rules algebras. This book presents the latest developments in this area. Its main goal is to give a classification of the Normalized Integral Table Algebras (Fusion Rings) generated by a faithful non-real element of...
The theory of table algebras was introduced in 1991 by Z. Arad and H. Blau in order to treat, in a uniform way, products of conjugacy classes and irre...
Polycyclic groups are built from cyclic groups in a specific way. They arise in many contexts within group theory itself but also more generally in algebra, for example in the theory of Noetherian rings. They also touch on some aspects of topology, geometry and number theory. The first half of this book develops the standard group theoretic techniques for studying polycyclic groups and the basic properties of these groups. The second half then focuses specifically on the ring theoretic properties of polycyclic groups and their applications, often to purely group theoretic situations.
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Polycyclic groups are built from cyclic groups in a specific way. They arise in many contexts within group theory itself but also more generally in...
Let FG be the group ring of a group G over a field F. Write U(FG) for the group of units of FG. It is an important problem to determine the conditions under which U(FG) satisfies a group identity. In the mid 1990s, a conjecture of Hartley was verified, namely, if U(FG) satisfies a group identity, and G is torsion, then FG satisfies a polynomial identity. Necessary and sufficient conditions for U(FG) to satisfy a group identity soon followed.
Let FG be the group ring of a group G over a field F. Write U(FG) for the group of units of FG. It is an important problem to determine the conditions...
A Mathematician Said Who Can Quote Me a Theorem that's True? For the ones that I Know Are Simply not So, When the Characteristic is Two This pretty limerick ?rst came to my ears in May 1998 during a talk by T.Y. Lam 1 on ?eld invariants from the theory of quadratic forms. It is--poetic exaggeration allowed--a suitable motto for this monograph. What is it about? At the beginning of the seventies I drew up a specialization theoryofquadraticandsymmetricbilinear formsover ?elds 32].Let?: K? L be a place. Then one can assign a form? (?)toaform? over K in a meaningful way ? if? has "good...
A Mathematician Said Who Can Quote Me a Theorem that's True? For the ones that I Know Are Simply not So, When the Characteristic is Two This pretty l...
Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. Many excellent texts present, with different goals and perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely the group SL2(Fq), which not only provides the simplicity required for a complete description of the theory, but also the richness needed for illustrating the most delicate aspects.
The development of Deligne-Lusztig theory was inspired by Drinfeld's example in 1974, and...
Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. ...
The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This book describes some fundamental facts about the representation theory of the Virasoro algebra in a self-contained manner. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and the rational vertex operator algebras associated to the so-called minimal series representations.
Covering a wide range of material, this book has three appendices which...
The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This ...
