Affine ADE bundles over surfaces with $$ p _{g}=0$$

Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle \(\mathcal {E}\) over X. When \( p _{g}(X)=0\), we construct deformations of holomorphic structures on \(\mathcal {E}\) such that the new bundle is trivial over any ADE curve \( C^{\prime }\) inside C and therefore descends to the singular surface obtained by contracting \(C^{\prime }\).     

Classification of Klein four symmetric pairs of holomorphic type for $$\mathrm {E}_{6(-14)}$$

The author classifies Klein four symmetric pairs of holomorphic type for non-compact Lie group \(\mathrm {E}_{6(-14)}\), which gives a class of pairs \((G,G')\) of real reductive Lie group G and its reductive subgroup \(G'\) such that there exist irreducible unitary representations \(\pi \) of G, which are admissible upon restriction to \(G'\).     

Bochner representable operators on Banach function spaces

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, \(E'\) the Köthe dual of E and \((X,\Vert \cdot \Vert _X)\) be a Banach space. It is shown that every Bochner representable operator \(T:E\rightarrow X\) maps relatively \(\sigma (E,E')\)-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is order continuous, then every Bochner representable operator \(T:E\rightarrow X\) is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, where \(\gamma _E\) stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.     

Analytic sets and extension of holomorphic maps of positive codimension

Let D, \(D'\) be arbitrary domains in \({\mathbb C}^n\) and \({\mathbb C}^N\) respectively, \(1<n\le N\), both possibly unbounded and \(M \subseteq \partial D\), \(M'\subseteq \partial D'\) be open pieces of the boundaries. Suppose that \(\partial D\) is smooth real-analytic and minimal in an open neighborhood of \({\bar{M}}\) and \(\partial D'\) is smooth real-algebraic and minimal in an open neighborhood of \({\bar{M}'}\). Let \(f: D\rightarrow D'\) be a holomorphic mapping such that the cluster set \(\mathrm{cl}_{f}(M)\) does not intersect \(D'\). It is proved that if the cluster set \(\mathrm{cl}_{f}(p)\) of some point \(p\in M\) contains some point \(q\in M'\) and the graph of f extends as an analytic set to a neighborhood of \((p, q)\in {\mathbb {C}}^n \times {\mathbb C}^N\), then f extends as a holomorphic map to a dense subset of some neighborhood of p. If in addition, \(M =\partial D\), \(M'=\partial D'\) and \(M'\) is compact, then f extends holomorphically across an open dense subset of \(\partial D\).     

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

Linear sets in the projective line over the endomorphism ring of a finite field

Let \({{\mathrm{{PG}}}}(1,E)\) be the projective line over the endomorphism ring \( E={{\mathrm{End}}}_q({\mathbb F}_{q^t})\) of the \({\mathbb F}_q\)-vector space \({\mathbb F}_{q^t}\). As is well known, there is a bijection \(\varPsi :{{\mathrm{{PG}}}}(1,E)\rightarrow {\mathcal G}_{2t,t,q}\) with the Grassmannian of the \((t-1)\)-subspaces in \({{\mathrm{{PG}}}}(2t-1,q)\). In this paper along with any \({\mathbb F}_q\)-linear set L of rank t in \({{\mathrm{{PG}}}}(1,q^t)\), determined by a \((t-1)\)-dimensional subspace \(T^\varPsi \) of \({{\mathrm{{PG}}}}(2t-1,q)\), a subset \(L_T\) of \({{\mathrm{{PG}}}}(1,E)\) is investigated. Some properties of linear sets are expressed in terms of the projective line over the ring E. In particular, the attention is focused on the relationship between \(L_T\) and the set \(L'_T\), corresponding via \(\varPsi \) to a collection of pairwise skew \((t-1)\)-dimensional subspaces, with \(T\in L'_T\), each of which determine L. This leads among other things to a characterization of the linear sets of pseudoregulus type. It is proved that a scattered linear set L related to \(T\in {{\mathrm{{PG}}}}(1,E)\) is of pseudoregulus type if and only if there exists a projectivity \(\varphi \) of \({{\mathrm{{PG}}}}(1,E)\) such that \(L_T^\varphi =L'_T\).     

Algebraic properties of Zappa–Szép products of semigroups and monoids

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that S is a retract of \(E\bowtie S\). We also find a subset T of \(E \bowtie S\) which is left restriction.     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit I

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.     

Persistence of Banach lattices under nonlinear order isomorphisms

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection \(T: E\rightarrow F\) such that \(x\ge y\) if and only if \(Tx \ge Ty\) for all \(x,y \in E\). We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to \(c_{0}\), and for Banach lattices that contain a closed sublattice order isomorphic to \(c_{0}\).     

A finite spectral representation of the partition function

Let \(E_{/\mathbb {Q}}\!\) be an elliptic curve, \(p>3\) a good ordinary prime for E, and \(K_\infty \) a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for E over a tower of extensions \(K_n/k\) inside \(K_\infty \); we obtain lower bounds on the former, and upper bounds on the latter’s size.     

Quasi-classical generalized CRF structures

In an earlier paper, we studied manifolds M endowed with a generalized F structure \(\Phi \in \mathrm{End}(TM\oplus T^*M)\), skew-symmetric with respect to the pairing metric, such that \(\Phi ^3+\Phi =0\). Furthermore, if \(\Phi \) is integrable (in some well-defined sense), \(\Phi \) is a generalized CRF structure. In the present paper, we study quasi-classical generalized F and CRF structures, which may be seen as a generalization of the holomorphic Poisson structures (it is well known that the latter may also be defined via generalized geometry). The structures that we study are equivalent to a pair of tensor fields \((A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)\), where \(A^3+A=0\) and some relations between A and \(\pi \) hold. We establish the integrability conditions in terms of \((A,\pi )\). They include the facts that A is a classical CRF structure, \(\pi \) is a Poisson bivector field and \(\mathrm{im}\,A\) is a (non)holonomic Poisson submanifold of \((M,\pi )\). We discuss the case where either \(\mathrm{ker}\,A\) or \(\mathrm{im}\,A\) is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of \(\mathrm{im}\,A\) inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of \(\pi \), including an associated spectral sequence and a Dolbeault type grading.     

A group of continuous self-maps on a topological groupoid

The group of bisections of groupoids plays an important role in the study of Lie groupoids. In this paper another construction is introduced. Indeed, for a topological groupoid G, the set of all continuous self-maps f on G such that (x, f(x)) is a composable pair for every \(x\in G\), is denoted by \(S_G\). We show that \(S_G\) by a natural binary operation is a monoid. \(S_G(\alpha )\), the group of units in \(S_G\) precisely consists of those \(f\in S_G\) such that the map \(x\mapsto xf(x)\) is a bijection on G. Similar to the group of bisections, \(S_G(\alpha )\) acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that \(S_G(\alpha )\) with the compact- open topology inherited from C(G, G) is a left topological group. For a compact Hausdorff groupoid G it is proved that the group of bisections of \(G^2\) is isomorphic to the group \(S_G(\alpha )\) and the group of transitive bisections of G, \(Bis_T(G)\), is embedded in \(S_G(\alpha )\), where \(G^2\) is the groupoid of all composable pairs.     

On the Projection of Stein Domains in Holomorphic Fiber Bundles

If \(\pi :E\rightarrow X\) is a locally trivial holomorphic fiber bundle, with positive dimensional connected fiber and with the total space E being a Stein manifold, and \(D\subset X\) is an arbitrary connected open subset, then there exists an open, connected Stein subset \(\widetilde{D}\subset E\) such that \(\pi (\widetilde{D})=D\).     

On the density of images of the power maps in Lie groups

Let G be a connected Lie group. In this paper, we study the density of the images of individual power maps \(P_k:G\rightarrow G:g\mapsto g^k\). We give criteria for the density of \(P_k(G)\) in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if \(\mathrm{Reg}(G)\) is the set of regular elements of G, then \(P_k(G)\cap \mathrm{Reg}(G)\) is closed in \(\mathrm{Reg}(G)\). On the other hand, the weak exponentiality of G turns out to be equivalent to the density of all the power maps \(P_k\). In linear Lie groups, weak exponentiality reduces to the density of \(P_2(G)\). We also prove that the density of the image of \(P_k\) for G implies the same for any connected full rank subgroup.     

Cliques in $$C_4$$-free graphs of large minimum degree

A graph G is called \(C_4\)-free if it does not contain the cycle \(C_4\) as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd?s) a peculiar property of \(C_4\)-free graphs: \(C_4\)-free graphs with n vertices and average degree at least cn contain a complete subgraph (clique) of size at least \(c'n\) (with \(c'= 0.1c^2\)). We prove here better bounds \(\big ({c^2n\over 2+c}\) in general and \((c-1/3)n\) when \( c \le 0.733\big )\) from the stronger assumption that the \(C_4\)-free graphs have minimum degree at least cn. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2k-regular \(C_4\)-free graphs on \(4k+1\) vertices contain a clique of size \(k+1\). This is the best possible as shown by the kth power of the cycle \(C_{4k+1}\).     

Fredholmness of fringe operators over the bidisk

Let M be an invariant subspace of \(H^2\) over the bidisk. Associated with M, we have the fringe operator \(F^M_z\) on \(M\ominus w M\). For \(A\subset H^2\), let [A] denote the smallest invariant subspace containing A. Assume that \(F^M_z\) is Fredholm. If h is a bounded analytic function on \(\mathbb {D}^2\) satisfying \(h(0,0)\not =0\), then \(F^{[h M]}_z\) is Fredholm and \(\mathrm{ind}\,F^{[h M]}_z=\mathrm{ind}\,F^M_z\).     

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset \(\Gamma \subset G\) that contains a semi-simple subgroup \(G_{1}\) of G. If one can show that \( \Gamma \) does not leave invariant a contractible subset on any flag manifold of G, then \(\Gamma \) generates G if \(\mathrm {Ad}\left( \Gamma \right) \) generates a Zariski dense subgroup of the algebraic group \(\mathrm {Ad}\left( G\right) \). The proof is reduced to check that some specific closed orbits of \(G_{1}\) in the flag manifolds of G are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups G and subgroups \(G_{1}\subset G\).