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There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given sixterm exact sequence of K-groups by splicing together smaller graphs whose C*- algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigation have more than one ideal and where there are currently no relevant classification theories available. [ABSTRACT FROM AUTHOR]

In this paper we prove relationships between two generalizations of commutative valuation theory for noncommutative central simple algebras: (1) Dubrovin valuation rings; and (2) the value functions called gauges introduced by Tignol and Wadsworth. We show that if v is a valuation on a field F with associated valuation ring V and v is defectless in a central simple F-algebra A, and C is a subring of A, then the following are equivalent: (a) C is the gauge ring of some minimal v-gauge on A, i.e., a gauge with the minimal number of simple components of C/J(C); (b) C is integral over V with C = B1 ∩...∩ Bξ where each Bi is a Dubrovin valuation ring of A with center V, and the Bi satisfy Gräter's Intersection Property. Along the way we prove the existence of minimal gauges whenever possible and we show how gauges on simple algebras are built from gauges on central simple algebras. [ABSTRACT FROM AUTHOR]

In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz- Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C*-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems. [ABSTRACT FROM AUTHOR]

Let F be a quadratic form in N ≥ 2 variables defined on a vector space V ⊆ KN over a global field K, and Ƶ ⊆ KN be a finite union of varieties defined by families of homogeneous polynomials over K. We show that if V \ Ƶ contains a nontrivial zero of F, then there exists a linearly independent collection of small-height zeros of F in V \ Ƶ, where the height bound does not depend on the height of Ƶ, only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace W of the quadratic space (V, F) such that W is not contained in Ƶ. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit. [ABSTRACT FROM AUTHOR]

Let H be a hypergroup with left Haar measure. The amenability of H can be characterized by the existence of nets of positive, norm one functions in L¹(H) which tend to left invariance in any of several ways. In this paper we present a characterization of the amenability of H using configuration equations. Extending work of Rosenblatt and Willis, we construct, for a certain class of hypergroups, nets in L1(H) which tend to left invariance weakly, but not in norm. We define the semidirect product of H with a locally compact group. We show that the semidirect product of an amenable hypergroup and an amenable locally compact group is an amenable hypergroup and show how to construct Reiter nets for this semidirect product. These results are generalized to Lau algebras, providing a new characterization of left amenability of a Lau algebra and a notion of a semidirect product of a Lau algebra with a locally compact group. The semidirect product of a left amenable Lau algebra with an amenable locally compact group is shown to be a left amenable Lau algebra. [ABSTRACT FROM AUTHOR]

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 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if A is a separable algebra which is either simple or stable, then the corona of A has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit. [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]

In an earlier paper, we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras. Here, we consider full quivers (as well as pseudo-quivers) as a combinatoric tool in order to describe PI-varieties of algebras. Each full quiver is naturally associated to a polynomial that encapsulates trace-like properties of the underlying algebra. [ABSTRACT FROM AUTHOR]

Let G be a topological group. Then the based loopspace OG of G is an algebra over the cacti operad, while the double loopspace O2BG of the classifying space of G is an algebra over the framed little discs operad. This paper shows that these two algebras are equivalent, in the sense that they are weakly equivalent E-algebras, where E is an operad weakly equivalent to both framed little discs and cacti. We recover the equivalence between cacti and framed little discs, and Menichi's isomorphism between the BV-algebras H*(OG) and H*(O2BG). [ABSTRACT FROM AUTHOR]

In this paper we study the stack τg of smooth triple covers of a conic; when g ≥ 5 this stack is embedded Mg as the locus of trigonal curves. We show that τ is a quotient [Ug/Γg/], where Γg/ is a certain algebraic group and Ug/ is an open subscheme of a Γg/-equivariant vector bundle over an open subscheme of a representation of Γg/. Using this, we compute the integral Picard group of τg when g/ > 1. τhe main tools are a result of Miranda that describes a flat finite triple cover of a scheme S as given by a locally free sheaf E of rank two on S, with a section of Sym3E ⊗detEv, and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack. [ABSTRACT FROM AUTHOR]

The modular variety of nonsingular and complete hyperelliptic curves with level-two structure of genus $ 3$ It has 36 irreducible (isomorphic) components. One of the purposes of this paper will be to describe the equations of one of these components. Two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with $ 8$, uses the semistable degenerated point configurations in $ (P^1)^8$ $ Y=\overline{\mathcal{B}/\Gamma[1-{\textrm i}]}.$ We use the standard notation $ \bar M_{0,8}$}[rr]& &X\;.} \end{displaymath} --> $\displaystyle \xymatrix{ &\bar M_{0,8}\ar[dl]\ar[dr]&\\ Y\ar@{-->}[rr]& &X\;.}$ The horizontal arrow is only birational but not everywhere regular. In this paper we find another realization of this triangle which uses the fact that there are graded algebras (closely related to algebras of modular forms) $ A,B$ $ X=\mathop{\rm proj}\nolimits(A), Y=\mathop{\rm proj}\nolimits(B).$ [ABSTRACT FROM AUTHOR]

This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on $\mathbb C^n$. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1). [ABSTRACT FROM AUTHOR]

Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of $F$-rationality under flat base change. [ABSTRACT FROM AUTHOR]

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring. [ABSTRACT FROM AUTHOR]

An $SL_n$-character of a group $G$ is the trace of an $SL_n$-represen\-ta\-tion of $G.$ We show that all algebraic relations between $SL_n$-characters of $G$ can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space $X,$ with $\pi_1(X)=G.$ We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of $SL_n$-representations of groups. \par The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of $M$ which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the $SL_2$-character variety of $\pi_1(M).$ This paper provides a generalization of this result to all $SL_n$-character varieties. [ABSTRACT FROM AUTHOR]

To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories. This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type $A$ and $B$, giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra. [ABSTRACT FROM AUTHOR]

In this paper we give some topological characterizations of bounded Baire-1 functions using some ranks. Kechris and Louveau classified the Baire-1 functions to the subclasses $\mathbb{B}^\xi_1(K)$ for every $\xi<\omega_1$ (where $K$ is a compact metric space). The first basic result of this paper is that for $\xi<\omega$, $f\in \mathbb{B}^{\xi+1}_1(K)$ iff there exists a sequence $(f_n)$ of differences of bounded semicontinuous functions on $K$ with $f_n\to f$ pointwise and $\gamma((f_n))\le \omega^\xi$ (where ``$\gamma$'' denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for $\xi=1$. We also show that the result fails for $\xi\ge \omega$. The second basic result of the paper involves the introduction of a new ordinal-rank on sequences $(f_n)$, called the $\delta$-rank, which is smaller than the convergence rank $\gamma$. This result yields the following characterization of $\mathbb{B}^\xi_1(K): f\in \mathbb{B}^\xi_1(K)$ iff there exists a sequence $(f_n)$ of continuous functions with $f_n\to f$ pointwise and $\delta((f_n))\le \omega^{\xi-1}$ if $1\le \xi<\omega$, resp. $\delta((f_n))\le \omega^\xi$ if $\xi\ge \omega$. [ABSTRACT FROM AUTHOR]

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras $A(+)$, $A(-)$, and $A({\mathbf{a}})$, where ${\mathbf{a}}\in \BbP^{2}$, which were first defined in an earlier paper. We omit a set $S\subset \BbP^2$ consisting of 11 specified points where the algebras $A({\mathbf{a}})$ become too degenerate to be regular. \begin{thm} Let $A$ represent $A(+)$, $A(-)$ or $A({\mathbf{a}})$, where ${\mathbf{a}} \in \BbP^2\setminus S$. Then $A$ is an Artin-Schelter regular algebra of global dimension three. Moreover, $A$ is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. \end{thm} This, combined with our earlier results, completes the classification. [ABSTRACT FROM AUTHOR]

{ An (ordinary) $P$-partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandard tableaux associated with Schur's $S$-functions. In this paper, we introduce and develop a theory of enriched $P$-partitions; like ordinary $P$-partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched $P$-partitions given here are the tableaux associated with Schur's $Q$-functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented. } [ABSTRACT FROM AUTHOR]

Let $G$ be a reductive $ \mathfrak{p}$-adic group. In his paper, ``The Plancherel Formula for Reductive $\mathfrak{p}$-adic Groups", Harish-Chandra summarized the theory underlying the Plancherel formula for $G$ and sketched a proof of the Plancherel theorem for $G$. One step in the proof, stated as Theorem 11 in Harish-Chandra's paper, has seemed an elusively difficult step for the reader to supply. In this paper we prove the Plancherel theorem, essentially, by proving a special case of Theorem 11. We close by deriving a version of Theorem 11 from the Plancherel theorem. [ABSTRACT FROM AUTHOR]

We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method actually provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial. [ABSTRACT FROM AUTHOR]

The article presents a study to examine the solvability of second-order equations with partially BMO coefficients. It analyses the Lp-solvability of second-order elliptic and parabolic equations in nondivergence form using divergence-form equations. It mentions the leading coefficients has been assumed to be in locally BMO spaces.