Showing papers in "Pacific Journal of Mathematics in 2016"

Bergman theory of certain generalized Hartogs triangles

Abstract: The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_{\gamma}$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $\mathbb{B}_{\gamma}(z,z)$.

Cohomology and extensions of braces

Abstract: Braces and linear cycle sets are algebraic structures playing a major role in the classification of involutive set-theoretic solutions to the Yang-Baxter equation. This paper introduces two versions of their (co)homology theories. These theories mix the Harrison (co)homology for the abelian group structure and the (co)homology theory for general cycle sets, developed earlier by the authors. Different classes of brace extensions are completely classified in terms of second cohomology groups.

Topological Molino’s theory

Abstract: Molino's description of Riemannian foliations on compact manifolds is generalized to the setting of compact equicontinuous foliated spaces, in the case where the leaves are dense. In particular, a structural local group is associated to such a foliated space. As an application, we obtain a partial generalization of results by Carriere and Breuillard-Gelander, relating the structural local group to the growth of the leaves.

E-polynomial of the SL(3,C)-character variety of free groups.

Abstract: We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3,C). Expanding upon techniques of Logares, Munoz and Newstead (Rev. Mat. Complut. 26:2 (2013), 635-703), we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give a new proof of results of Cavazos and Lawton (Int. J. Math. 25:6 (2014), 1450058).

Solitons for the inverse mean curvature flow

Abstract: We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Minkowski integral formula to prove the rigidity of the hypersphere in the class of compact expanders of codimension one. We also establish that the moduli space of compact expanding surfaces of codimension two is big. Finally, we update the list of Huisken-Ilmanen's rotational expanders by constructing new examples of complete expanders with rotational symmetry, including topological hypercylinders, called infinite bottles, that interpolate between two concentric round hypercylinders.

Stable capillary hypersurfaces in a wedge

Abstract: Let $\Sigma$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $\Sigma$ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if $\partial \Sigma$ is embedded for $n=2$, or if $\partial\Sigma$ is convex for $n\geq3$, then $\Sigma$ is part of the sphere. And the same is true for $\Sigma$ in the half-space of $\mathbb R^{n+1}$ with connected boundary $\partial\Sigma$.

Complex hyperbolic (3,3,n)-triangle groups.

Abstract: Let p,q,rp,q,r be positive integers. Complex hyperbolic (p,q,r)(p,q,r) triangle groups are representations of the hyperbolic (p,q,r)(p,q,r) reflection triangle group to the holomorphic isometry group of complex hyperbolic space H2CHℂ2, where the generators fix complex lines. In this paper, we obtain all the discrete and faithful complex hyperbolic (3,3,n)(3,3,n) triangle groups for n≥4n≥4. Our result solves a conjecture of Schwartz in the case when p=q=3p=q=3.

Primitively generated Hall algebras

Abstract: Author(s): Berenstein, A; Greenstein, J | Abstract: In the present paper we show that Hall algebras of finitary exact categories behave like quantum groups in the sense that they are generated by indecomposable objects. Moreover, for a large class of such categories, Hall algebras are generated by their primitive elements, with respect to the natural comultiplication, even for nonhereditary categories. Finally, we introduce certain primitively generated subalgebras of Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.

Multiplicative reduction and the cyclotomic main conjecture for GL2

Abstract: We show that the cyclotomic Iwasawa--Greenberg Main Conjecture holds for a large class of modular forms with multiplicative reduction at $p$, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument.

Sigma theory and twisted conjugacy-II: Houghton groups and pure symmetric automorphism groups

Abstract: We say that $x,y\in \Gamma$ are in the same $\phi$-twisted conjugacy class and write $x\sim_\phi y$ if there exists an element $\gamma\in \Gamma$ such that $y=\gamma x\phi(\gamma^{-1})$. This is an equivalence relation on $\Gamma$ called the $\phi$-twisted conjugacy. Let $R(\phi)$ denote the number of $\phi$-twisted conjugacy classes in $\Gamma$. If $R(\phi)$ is infinite for all $\phi\in Aut(\Gamma)$, we say that $\Gamma$ has the $R_\infty$-property. The purpose of this note is to show that the symmetric group $S_\infty$, the Houghton groups and the pure symmetric automorphism groups have the $R_\infty$-property. We show, also, that the Richard Thompson group $T$ has the $R_\infty$-property. We obtain a general result establishing the $R_\infty$-property of finite direct product of finitely generated groups.

Iwahori-Hecke algebras for Kac-Moody groups over local fields

Abstract: We define the Iwahori-Hecke algebra for an almost split Kac-Moody group over a local non-archimedean field. We use the hovel associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The fixer K of some chamber in the standard apartment plays the role of the Iwahori subgroup. We can define the Iwahori-Hecke algebra as the algebra of some K-bi-invariant functions on the group with support consisting of a finite union of double classes. As two chambers in the hovel are not always in a same apartment, this support has to be in some large subsemigroup of the Kac-Moody group. In the split case, we prove that the structure constants of the multiplication in this algebra are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We give a presentation of this algebra, similar to the Bernstein-Lusztig presentation in the reductive case, and embed it in a greater algebra, algebraically defined by the Bernstein-Lusztig presentation. In the affine case, this algebra contains the Cherednik's double affine Hecke algebra. Actually, our results apply to abstract ``locally finite'' hovels, so that we can define the Iwahori-Hecke algebra with unequal parameters.

On seaweed subalgebras and meander graphs in type C

Abstract: In 2000, Dergachev and Kirillov introduced subalgebras of ”seaweed type” in gl n (or sln) and computed their index using certain graphs. In this article, those graphs are called type-A meander graphs. Then the subalgebras of seaweed type, or just ”seaweeds”, have been defined by Panyushev (2001) for arbitrary simple Lie algebras. Namely, if p1, p2 ⊂ g are parabolic subalgebras such that p1 + p2 = g, then q = p1 ∩ p2 is a seaweed in g. If p1 and p2 are “adapted” to a fixed triangular decomposition of g, then q is said to be standard. The number of standard seaweeds is finite. A general algebraic formula for the index of seaweeds has been proposed by Tauvel and Yu (2004) and then proved by Joseph (2006). In this paper, elaborating on the “graphical” approach of Dergachev and Kirillov, we introduce the type-C meander graphs, i.e., the graphs associated with the standard seaweed subalgebras of sp 2n , and give a formula for the index in terms of these graphs. We also note that the very same graphs can be used in case of the odd orthogonal Lie algebras. Recall that q is called Frobenius, if the index of q equals 0. We provide several applications of our formula to Frobenius seaweeds in sp 2n . In particular, using a natural partition of the set Fn of standard Frobenius seaweeds, we prove that #Fn strictly increases for the passage from n to n + 1. The similar monotonicity question is open for the standard Frobenius seaweeds in sln, even for the passage from n to n+ 2.

A note on nonunital absorbing extensions

Abstract: Elliott and Kucerovsky stated that a nonunital extension of separable C -algebras with a stable ideal is nuclearly absorbing if and only if the extension is purely large. However, their proof was flawed. We give a counterexample to their theorem as stated, but establish an equivalent formulation of nuclear absorption under a very mild additional assumption to being purely large. In particular, if the quotient algebra is nonunital, then we show that the original theorem applies. We also examine how this affects results in classification theory.

Noncommutative differentials on Poisson–Lie groups and pre-Lie algebras

Abstract: We show that the quantisation of a connected simply connected Poisson‐Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a preLie algebra structure. As an example, we find a pre-Lie algebra structure underlying the standard 3-dimensional differential structure on CqTSU2U. At the noncommutative geometry level we show that the enveloping algebra U.m/ of a Lie algebra m, viewed as quantisation of m , admits a connected differential exterior algebra of classical dimension if and only if m admits a pre-Lie algebra structure. We give an example where m is solvable and we extend the construction to tangent and cotangent spaces of Poisson‐Lie groups by using bicross-sum and bosonisation of Lie bialgebras. As an example, we obtain a 6-dimensional left-covariant differential structure on the bicrossproduct quantum group CTSU2UIGU .su 2 /.

Degenerate flag varieties and Schubert varieties: a characteristic free approach

Abstract: We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.

Completely contractive projections on operator algebras

Abstract: The main goal of this paper is to find operator algebra variants of certain deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and Stormer, Robertson and Youngson, Youngson, and others, concerning projections on C*-algebras and their ranges. (See papers of these authors referenced in the bibliography.) In particular we investigate the `bicontractive projection problem' and related questions in the category of operator algebras. To do this, we will add the ingredient of `real positivity' from recent papers of the first author with Read.

Affine weakly regular tensor triangulated categories

Abstract: We prove that the Balmer spectrum of a tensor triangulated category is homeomorphic to the Zariski spectrum of its graded central ring, provided the triangulated category is generated by its tensor unit and the graded central ring is noetherian and regular in a weak sense. There follows a classification of all thick subcategories, and the result extends to the compactly generated setting to yield a classification of all localizing subcategories as well as the analog of the telescope conjecture. This generalizes results of Shamir for commutative ring spectra.

Exhausting curve complexes by finite rigid sets

Abstract: Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.

Natural commuting of vanishing cycles and the Verdier dual

Abstract: We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a natural isomorphism, even when the coefficients are not in a field.

Vector bundle valued differential forms on ℕQ-manifolds

Abstract: Geometric structures on NQ-manifolds, i.e. non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher ana- logues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. I describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, I use this description to present, in a unified way, novel proofs of known results, and new results about degree one NQ-manifolds equipped with certain geometric structures, namely symplectic structures, contact structures, involutive distributions (already present in liter- ature) and locally conformal symplectic structures, and generic vector bundle valued higher order forms, in particular multisymplectic structures (not yet present in literature).

The Fundamental Theorem of Tropical Differential Algebraic Geometry

Abstract: Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality $\text{trop}(V(I))=V(\text{trop}(I))$ between the tropicalization $\text{trop}(V(I))$ of the closed subscheme $V(I)\subset (K^*)^n$ and the tropical variety $V(\text{trop}(I))$ associated to the tropicalization of the ideal $\text{trop}(I)$. In this work we prove an analogous result for a differential ideal $G$ of the ring of differential polynomials $K[[t]]\{x_1,\ldots,x_n\}$, where $K$ is an uncountable algebraically closed field of characteristic zero. We define the tropicalization $\text{trop}(\text{Sol}(G))$ of the set of solutions $\text{Sol}(G)\subset K[[t]]^n$ of $G$, and the set of solutions associated to the tropicalization of the ideal $\text{trop}(G)$. These two sets are linked by a tropicalization morphism $\text{trop}:\text{Sol}(G)\longrightarrow \text{Sol}(\text{trop}(G))$. We show the equality $\text{trop}(\text{Sol}(G))=\text{Sol}(\text{trop}(G))$, answering a question raised by D. Grigoriev earlier this year.

On Yamabe type problems on Riemannian manifolds with boundary

Abstract: Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \\ \partial_ u u+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $ u$ is the outward pointing unit normal to $\partial M $ and $\varepsilon$ is a small positive parameter. We build solutions which blow-up at a point of the boundary as $\varepsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_g ,$ where $H_g$ is the boundary mean curvature.

Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image.

Abstract: This article is the second part of a series of three articles a bout compatible systems of symplectic Galois representations and applications to the inv erse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. We prove a classification result on those sub groups of a general symplectic group over a finite field that contain a nontrivial transvection. Tr anslating this group theoretic result into the language of symplectic representations whose image contains a nontrivial transvection, these fall into three very simply describable classes: the reduci ble ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result w ith the main result of the first part to obtain a strenghtened application to the inverse Galois pro blem. MSC (2010): 11F80 (Galois representations); 20G14 (Linear algebraic groups over finite fields), 12F12 (Inverse Galois theory).

A combinatorial approach to Voiculescu’s bi-free partial transforms

Abstract: In this paper, we present a combinatorial approach to the 2-variable bi-free partial $S$- and $T$-transforms recently discovered by Voiculescu. This approach produces an alternate definition of said transforms using $(\ell, r)$-cumulants.