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In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a (skew) field with a complete valuation, as proposed by Jacques Tits. This second part builds upon the results of the first part and deals with the remaining cases. [ABSTRACT FROM AUTHOR]

In this paper, we first define a new family of Yoneda algebras, called F-Auslander-Yoneda algebras, in triangulated categories by introducing the notion of admissible sets F in N, which includes higher cohomologies indexed by F, and then present a general method to construct a family of new derived equivalences for these F-Auslander-Yoneda algebras (not necessarily Artin algebras), where the choices of the parameters F are rather abundant. Among applications of our method are the following results: (1) if A is a self-injective Artin algebra, then, for any A-module X and for any admissible set F in N, the F-Auslander-Yoneda algebras of A ⊕ X and A ⊕ OA(X) are derived equivalent, where O is the Heller loop operator. (2) Suppose that A and B are representation-finite self-injective algebras with additive generators AX and BY, respectively. If A and B are derived equivalent, then so are the F-Auslander-Yoneda algebras of X and Y for any admissible set F. In particular, the Auslander algebras of A and B are derived equivalent. The converse of this statement is open. Further, motivated by these derived equivalences between F-Auslander-Yoneda algebras, we show, among other results, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factoring out socles. [ABSTRACT FROM AUTHOR]

This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called \textit{arithmetic non-degeneracy}. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right} $ called the \textit{arithmetic Newton polygon} of $f(x,y)$, and introduce the notion of \textit{arithmetic non-degeneracy with respect to }$\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left[ x,y \right] $ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$. [ABSTRACT FROM AUTHOR]

Let $\mathcal{S}$ be a finite set of powers of $p$ containing 1. It is known that for some choices of $\mathcal{S}$, if $P$ is a finite $p$-group whose set of character degrees is $\mathcal{S}$, then the nilpotence class of $P$ is bounded by some integer that depends on $\mathcal{S}$, while for some other choices of $\mathcal{S}$ such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set $\mathcal{S}$ is class bounding if and only if $p\notin\mathcal{S}$. In this article we provide a new approach to this problem. Our main result shows the relevance of certain $p$-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets $\mathcal{S}$ such that $p\notin\mathcal{S}$. [ABSTRACT FROM AUTHOR]

We add extra data to the affine Dynkin diagrams to classify all the finite order automorphisms on real simple Lie algebras. As applications, we study the extensions of automorphisms on the maximal compact subalgebras and also study the fixed point sets of automorphisms. [ABSTRACT FROM AUTHOR]