### Aspects of affine toda field theory

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[For the entire collection see Zbl 0742.00067.]We are interested in partial differential equations on domains in ${\mathcal{C}}^{n}$. One of the most natural questions is that of analytic continuation of solutions and domains of holomorphy. Our aim is to describe the domains of holomorphy for solutions of the complex Laplace and Dirac equations. We call them cells of harmonicity. We deduce their properties mostly by examining geometrical properties of the characteristic surface (which is the same for both equations),...

[For the entire collection see Zbl 0742.00067.]For the purpose of providing a comprehensive model for the physical world, the authors set up the notion of a Clifford manifold which, as mentioned below, admits the usual tensor structure and at the same time a spin structure. One considers the spin space generated by a Clifford algebra, namely, the vector space spanned by an orthonormal basis $\{{e}_{j}:j=1,\cdots ,n\}$ satisfying the condition $\{{e}_{i},{e}_{j}\}\equiv {e}_{i}{e}_{j}={e}_{j}{e}_{i}=2I{\eta}_{ij}$, where $I$ denotes the unit scalar of the algebra and (${\eta}_{ij}$) the nonsingular Minkowski...

Summary: [For the entire collection see Zbl 0742.00067.]A general theory of fibre bundles structured by an arbitrary differential-geometric category is presented. It is proved that the structured bundles of finite type coincide with the classical associated bundles.

[For the entire collection see Zbl 0742.00067.]Let $G$ be a connected semisimple Lie group with finite center. In this review article the author describes first the geometric realization of the discrete series representations of $G$ on Dolbeault cohomology spaces and the tempered series of representations of $G$ on partial Dolbeault cohomology spaces. Then he discusses his joint work with Wilfried Schmid on the construction of maximal globalizations of standard Zuckerman modules via geometric quantization....

[For the entire collection see Zbl 0742.00067.]The author formulates several theorems about invariant orders in Lie groups (without proofs). The main theorem: a simply connected Lie group $G$ admits a continuous invariant order if and only if its Lie algebra $L\left(G\right)$ contains a pointed invariant cone. V. M. Gichev has proved this theorem for solvable simply connected Lie groups (1989). If $G$ is solvable and simply connected then all pointed invariant cones $W$ in $L\left(G\right)$ are global in $G$ (a Lie wedge $W\subset L\left(G\right)$ is said to...

[For the entire collection see Zbl 0742.00067.]This paper is devoted to a method permitting to determine explicitly all multilinear natural operators between vector-valued differential forms and between sections of several other natural vector bundles.

[For the entire collection see Zbl 0742.00067.]Let ${\U0001d524}_{k}$ be the Lie algebra $\U0001d524l(k,\mathcal{C})$, and let ${U}_{k}$ be the universal enveloping algebra for ${\U0001d524}_{k}$. Let ${Z}_{k}$ be the center of ${U}_{k}$. The authors consider the chain of Lie algebras ${\U0001d524}_{n}\supset {\U0001d524}_{n-1}\supset \cdots \supset {\U0001d524}_{1}$. Then $Z=\langle {Z}_{k}\mid k=1,2,\cdots n\rangle $ is an associative algebra which is called the Gel’fand-Zetlin subalgebra of ${U}_{n}$. A ${\U0001d524}_{n}$ module $V$ is called a $GZ$-module if $V={\sum}_{x}\oplus V\left(x\right)$, where the summation is over the space of characters of $Z$ and $V\left(x\right)=\{v\in V\mid {(a-x\left(a\right))}^{m}v=0$, $m\in {\mathcal{Z}}_{+}$, $a\in \mathcal{Z}\}$. The authors describe several properties of $GZ$- modules. For example, they prove that if $V\left(x\right)=0$ for some $x$...

[For the entire collection see Zbl 0742.00067.]In the first part some general results on Hecke algebras are recalled; the structure constants corresponding to the standard basis are defined; in the following the example of the commuting algebra of the Gelfand- Graev representation of the general linear group $GL(2,F)$ is examined; here $F$ is a finite field of $q$ elements; the structure constants are explicitly determined first for the standard basis and then for a new basis obtained via a Mellin-transformation....

[For the entire collection see Zbl 0742.00067.]The Tanaka-Krein type equivalence between Hopf algebras and functored monoidal categories provides the heuristic strategy of this paper. The author introduces the notion of a double cross product of monoidal categories as a generalization of double cross product of Hopf algebras, and explains some of the motivation from physics (the representation theory for double quantum groups).The Hopf algebra constructions are formulated in terms of monoidal categories...