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by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat­ egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap­ proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
A quantum group \(\mathfrak{G}\) is at present a purely mythical being to which one nevertheless associates a Hopf algebra R q [G]resembling the algebra of regular functions R[G]on a (genuine) affine algebraic group G. It is supposed that if \(\mathfrak{G}\) were to exist it would have the same representations as those of G.This is expressed by requiring R q [G] and R[G] to be isomorphic as coalgebras.
Anthony Joseph

Chapter 1. Hopf Algebras

Abstract
Let k be a field. For any k-vector space V the symbol 1dv (or simply, 1d) denotes the identity map on V. An associative k-algebra (or simply, algebra) is a vector space A together with a linear map \(\mu :A \otimes A \to A\) such that the diagram is commutative. Writing \(\mu (a \otimes b) = ab\) this may be expressed as (ab)c = a(bc) for all a, b, c, \( \in \) A. An identity of an algebra A is a k-linear map \(\eta :k \to A\) such that the diagramsare commutative. By (ii) and (ii)r it follows that \(\eta (1) = :{1_A}\) (or simply, 1) satisfies la = al = a for all \(a \in A\) Moreover \(\eta (\alpha )a = \alpha a\) and \(\eta (\alpha \beta ) = \alpha \eta (\beta ) = \eta (\alpha )\eta (\beta )\) so \(\eta \) is an algebra homomorphism. It is often convenient to omit \(\eta \). Unless otherwise specified the term algebra will mean associative algebra.
Anthony Joseph

Chapter 2. Excerpts from the Classical Theory

Abstract
Let A be a bialgebra. Here we examine part of the Hopf dual A* of A and show that it admits a Lie subalgebra.
Anthony Joseph

Chapter 3. Encoding the Cartan Matrix

Abstract
Unless otherwise indicated it is assumed in the remainder of this text that char k = 0 and that q is an indeterminate.
Anthony Joseph

Chapter 4. Highest Weight Modules

Abstract
The base field k is assumed of characteristic zero, with \(\overline {k(q)} \) the algebraic closure of k(q) The Cartan matrix is assumed symmetric, so then U q (MATH) is defined.
Anthony Joseph

Chapter 5. The Crystal Basis

Abstract
Here and in chapter 6, C is assumed integrable and one writes \({\mathfrak{g}_C}\) simply as \(\mathfrak{g}\).
Anthony Joseph

Chapter 6. The Global Bases

Abstract
The compatibility of B(λ) with the tensor product (5.2.4) already has some combinatorial consequences. A crucial additional fact is that the antiautomorphism ⋆ of U q (ndefined by MATH induces an involution of B(∞). Again the crystal bases B(λ), B(∞) can be lifted to global bases of V(λ) and U q (n - )respectively. This leads to a common basis theorem (6.2.19) which is central to the structure theory of U q (\(\mathfrak{g}\)). It also gives the Demazure character formula (6.3.15). In the last section Littelmann’s theory is described and compared to Kashiwara’s. The global bases, obtained here are now known to coincide with Lusztig’s canonical bases (6.5.1) and there are ideas here common to those of Lusztig particularly the use of ⋆, see [L7].
Anthony Joseph

Chapter 7. Structure Theorems for U q (g)

Abstract
The base field will once again be denoted by k and assumed to be of characteristic zero. Apart from this the conventions of Chapter 5 will be retained and in particular U q (g) is as defined as in 5.1.1. Notice that this means that the Cartan matrix is assumed integral.
Anthony Joseph

Chapter 8. The Primitive Spectrum of U q (g)

Abstract
In this chapter it is assumed throughout that the Cartan matrix C is of finite type. The base field k is assumed of characteristic zero and \(\overline {k(q)} \) denotes the algebraic closure of k(q) The goal is to determine Prim\(({U_q}(\mathfrak{g}){ \otimes _{k(q)}}\overline {k(q)} )\).
Anthony Joseph

Chapter 9. Structure Theorems for R q [G]

Abstract
A finite dimensional module over \(U = {U_q}({\mathfrak{g}_C})\) has a filtration whose quotients are simple integrable highest weight modules and hence by 4.3.10 is semisimple. Thus the Hopf dual U⋆ satisfies the direct sum decomposition of 1.4.13. By 7.1.15 (ii) this is only of interest when C is of finite type as will be henceforth assumed.
Anthony Joseph

Chapter 10. The Prime Spectrum of Rq[G]

Abstract
A highest weight module for R is a module generated by a one dimensional R + module. Their theory is superficially similar to highest weight modules for U. Thus there are universal highest weight modules (analogous to Verma modules) and these admit unique simple quotients (10.1.5). However not all characters on R + can give rise to such a module (10.1.3) and Prim R is not exhausted (10.1.7) by their annihilators. On the other hand they satisfy a remarkable tensor product theorem (10.1.18) which gives rise to a braid group action (10.2) on U.
Anthony Joseph

Backmatter

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