Wide Subalgebras of Semisimple Lie Algebras

Let G be a connected semisimple algebraic group over \({\mathbb C}\) , with Lie algebra \({\mathfrak g}\) . Let \({\mathfrak h}\) be a subalgebra of \({\mathfrak g}\) . A simple finite-dimensional \({\mathfrak g}\) -module \({\mathbb V}\) is said to be \({\mathfrak h}\) -indecomposable if it cannot be written as a direct sum of two proper \({\mathfrak h}\) -submodules. We say that \({\mathfrak h}\) is wide, if all simple finite-dimensional \({\mathfrak g}\) -modules are \({\mathfrak h}\) -indecomposable. Some very special examples of indecomposable modules and wide subalgebras appear recently in the literature. In this paper, we describe several large classes of wide subalgebras of \({\mathfrak g}\) and initiate their systematic study. Our approach is based on the study of idempotents in the associative algebra of \({\mathfrak h}\) -invariant endomorphisms of \({\mathbb V}\) . We also discuss a relationship between wide subalgebras and epimorphic subgroups.     

Spectra of tensor triangulated categories over category algebras

Let \({\mathcal{C}}\) be a finite EI category and k be a field. We consider the category algebra \({k\mathcal{C}}\) . Suppose \({\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}\) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When \({\mathcal{C}=G \propto \mathcal{P}}\) is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category \({\underline{\sf{CM}} k(G \propto \mathcal{P})}\) , of maximal Cohen–Macaulay modules, as a quotient category of \({{\sf{K}}(G \propto \mathcal{P})}\) . Since \({\underline{\sf{CM}} k(G\propto\mathcal{P})}\) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

{\mathfrak {sl}}_3-Web bases,intermediate crystal bases and categorification

We give an explicit graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra \(K_S\) . In order to do this, we identify Kuperberg’s basis for the \({\mathfrak {sl}}_3\) -web space \(W_S\) with a version of Leclerc–Toffin’s intermediate crystal basis and we identify Brundan, Kleshchev and Wang’s degree of tableaux with the weight of flows on webs and the \(q\) -degree of foams. We use these observations to give a “foamy” version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra. We restrict ourselves to the \({\mathfrak {sl}}_3\) case over \(\mathbb {C}\) here, but our approach should, up to the combinatorics of \({\mathfrak {sl}}_N\) -webs, work for all \(N>1\) or over \(\mathbb {Z}\) .     

Solid Extensions of the Cesàro Operator on ℓ p and c 0

We introduce and study the largest Banach lattice (for the coordinate-wise order) which is a solid subspace of \({\mathbb{C}^\mathbb{N}}\) and to which the classical Cesàro operator \({\mathcal{C}\colon\ell^p \to \ell^p}\) (a positive operator) can be continuously extended while still maintaining its values in ? ^p . Properties of this optimal Banach lattice \({[\mathcal{C}, \ell^p]_s}\) are presented. In addition, all continuous convolution operators of \({[\mathcal{C}, \ell^p]_s}\) into itself are identified and the spectrum of \({\mathcal{C}\colon[\mathcal{C}, \ell^p]_s \to[\mathcal{C}, \ell^p]_s}\) is determined. A similar investigation is undertaken for the Cesàro operator \({\mathcal{C}\colon c_0\to c_0}\) .     

Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X,A) be the algebra of all continuous A-valued functions on X equipped with the topology of \({\mathcal{K}(X)}\) -convergence. Moreover, let \({\mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(A)}\) denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X,A) and show that there exist bijections from \({\mathfrak{M}_{\ell}(C(X, A))}\) onto \({X \times \mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(C(X, A))}\) onto \({X \times \mathfrak{M}(A)}\) . We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand–Mazur algebra with jointly continuous multiplication.     

The Algebraicity of Ill-Distributed Sets

We show that every set \({S \subseteq [N]^d}\) occupying \({\ll p^{\kappa}}\) residue classes for some real number \({0 \leq \kappa < d}\) and every prime p, must essentially lie in the solution set of a polynomial equation of degree \({\ll ({\rm log} N)^C}\) , for some constant C depending only on \({\kappa}\) and d. This provides the first structural result for arbitrary \({\kappa < d}\) and S.     

Orthoproducts and f-representations of archimedean {\ell}-groups

Let G be an archimedean \({\ell}\) -group. By an f-representation of G we mean an orthomorphism-valued group homomorphism S on G for which (Sf)g =  (Sg)f for all \({f, g \in G}\) . We prove that the set \({\mathfrak{Rep}(G)}\) of all f-representations in G is an archimedean \({\ell}\) -group with respect to pointwise addition and ordering. Furthermore, we define an orthoproduct on G to be a bilinear map on G which is an orthomorphism in each variable separately. It turns out that the set \({\mathfrak{Opro}(G)}\) is an archimedean \({\ell}\) -group G with the set \({\mathfrak{Mult}(G)}\) of f-multiplications in G as a positive cone. Moreover, we show that \({\mathfrak{Opro}(G)}\) and \({\mathfrak{Rep}(G)}\) are isomorphic as \({\ell}\) -groups. In spite of that, we get a representation theorem for f-multiplications in an \({\ell}\) -subgroup of an archimedean f-ring R with unit element. This allows us to find an example of an archimedean \({\ell}\) -group with no nontrivial structure of an f-ring and another which cannot be a reduced f-ring.     

The Idempotents of the TL n -Module {\otimes^{n}\mathbb{C}2} in Terms of Elements of {{\rm U}_{q} \mathfrak{sl}_2}

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\) . The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\) -module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL _n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\) ) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\) . When q = q _c is a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\) .     

Some Integral Representations and Singular Integral over Plane in Clifford Analysis

In this paper, Cauchy type integral and singular integral over hyper-complex plane \({\prod}\) are considered. By using a special Möbius transform, an equivalent relation between \({\widehat{H}^\mu}\) class functions over \({\prod}\) and \({H^\mu}\) class functions over the unit sphere is shown. For \({\widehat{H}^\mu}\) class functions over \({\prod}\) , we prove the existence of Cauchy type integral and singular integral over \({\prod}\) . Cauchy integral formulas as well as Poisson integral formulas for monogenic functions in upper-half and lower-half space are given respectively. By using Möbius transform again, the relation between the Cauchy type integrals and the singular integrals over \({\prod}\) and unit sphere is built.     

A Clifford Bundle Approach to the Differential Geometry of Branes

We first recall using the Clifford bundle formalism (CBF) of differential forms and the theory of extensors acting on \({\mathcal{C}\ell}\) (M, g) (the Clifford bundle of differential forms) the formulation of the intrinsic geometry of a differential manifold M equipped with a metric field g of signature (p, q) and an arbitrary metric compatible connection \({\nabla}\) introducing the torsion (2?1)-extensor field \({\tau}\) , the curvature (2 ? 2) extensor field \({\Re}\) and (once fixing a gauge) the connection (1?2)-extensor \({\omega}\) and the Ricci operator \({\partial \bigwedge \partial}\) (where \({\partial}\) is the Dirac operator acting on sections of \({\mathcal{C}\ell(M, g)}\) ) which plays an important role in this paper. Next, using the CBF we give a thoughtful presentation the Riemann or the Lorentzian geometry of an orientable submanifold M (dim M =  m) living in a manifold M? (such that M? \({\simeq \mathbb{R}^n}\) is equipped with a semi- Riemannian metric g? with signature (p?, q?) and p?+q? = n and its Levi- Civita connection D?) and where there is defined a metric g =  i*g?, where \({i : M \rightarrow}\) M? is the inclusion map. We prove several equivalent forms for the curvature operator \({\Re}\) of M. Moreover we show a very important result, namely that the Ricci operator of M is the (negative) square of the shape operator S of M (object obtained by applying the restriction on M of the Dirac operator ?? of \({\mathcal{C}\ell}\) (M?, g?) to the projection operator P). Also we disclose the relationship between the (1?2)-extensor \({\omega}\) and the shape biform \({\mathcal{S}}\) (an object related to S). The results obtained are used to give a mathematical formulation to Clifford’s theory of matter. It is hoped that our presentation will be useful for differential geometers and theoretical physicists interested, e.g., in string and brane theories and relativity theory by divulging, improving and expanding very important and so far unfortunately largely ignored results appearing in reference [13].     

On the Relaxed Colouring Game and the Unilateral Colouring Game

In the paper we introduce the new game—the unilateral \({\mathcal{P}}\) -colouring game which can be used as a tool to study the r-colouring game and the (r, d)-relaxed colouring game. Let be given a graph G, an additive hereditary property \({\mathcal {P}}\) and a set C of r colours. In the unilateral \({\mathcal {P}}\) -colouring game similarly as in the r-colouring game, two players, Alice and Bob, colour the uncoloured vertices of the graph G, but in the unilateral \({\mathcal {P}}\) -colouring game Bob is more powerful than Alice. Alice starts the game, the players play alternately, but Bob can miss his move. Bob can colour the vertex with an arbitrary colour from C, while Alice must colour the vertex with a colour from C in such a way that she cannot create a monochromatic minimal forbidden subgraph for the property \({\mathcal {P}}\) . If after |V(G)| moves the graph G is coloured, then Alice wins the game, otherwise Bob wins. The \({\mathcal {P}}\) -unilateral game chromatic number, denoted by \({\chi_{ug}^\mathcal {P}(G)}\) , is the least number r for which Alice has a winning strategy for the unilateral \({\mathcal {P}}\) -colouring game with r colours on G. We prove that the \({\mathcal {P}}\) -unilateral game chromatic number is monotone and is the upper bound for the game chromatic number and the relaxed game chromatic number. We give the winning strategy for Alice to play the unilateral \({\mathcal {P}}\) -colouring game. Moreover, for k ≥  2 we define a class of graphs \({\mathcal {H}_k =\{G|{\rm every \;block \;of\;}G \; {\rm has \;at \;most}\; k \;{\rm vertices}\}}\) . The class \({\mathcal {H}_k }\) contains, e.g., forests, Husimi trees, line graphs of forests, cactus graphs. Let \({\mathcal {S}_d}\) be the class of graphs with maximum degree at most d. We find the upper bound for the \({\mathcal {S}_2}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_3}\) and we study the \({\mathcal {S}_d}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_4}\) for \({d \in \{2,3\}}\) . As the conclusion from these results we obtain the result for the d-relaxed game chromatic number: if \({G \in \mathcal {H}_k}\) , then \({\chi_g^{(d)}(G) \leq k + 2-d}\) , for \({k \in \{3, 4\}}\) and \({d \in \{0, \ldots, k-1\}}\) . This generalizes a known result for trees.     

Converting homotopies to isotopies and dividing homotopies in half in an effective way

We prove two theorems about homotopies of curves on two-dimensional Riemannian manifolds. We show that, for any \({\epsilon > 0}\) , if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by \({L + \epsilon}\) . If the manifold is orientable, then for any \({\epsilon > 0}\) we show that, if we can contract a curve \({\gamma}\) traversed twice through curves of length bounded by L, then we can also contract \({\gamma}\) through curves bounded in length by \({L + \epsilon}\) . Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.     

Multiplicity of solutions to a nonlinear elliptic problem with nonlinear boundary conditions

We study the problem $$\left\{\begin{array}{ll}\Delta_p u = |u|^{q-2}u, & \quad x \in \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}= \lambda |u|^{p-2}u, &\quad x \in \partial \Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N}\) is a bounded smooth domain, \({\nu}\) is the outward unit normal at \({\partial \Omega}\) and \({\lambda > 0}\) is regarded as a bifurcation parameter. When p = 2 and in the superlinear regime q > 2, we show existence of n nontrivial solutions for all \({\lambda > \lambda_n}\) , \({\lambda_n}\) being the n-th Steklov eigenvalue. It is proved in addition that bifurcation from the trivial solution takes place at all \({\lambda_n}\) ’s. Similar results are obtained in the sublinear case 1 < q < 2. In this case, bifurcation from infinity takes place in those \({\lambda_n}\) with odd multiplicity. Partial extensions of these features are shown in the nonlinear diffusion case \({p \neq 2}\) and related problems under spatially heterogeneous reactions are also addressed.     

On the number of reducible polynomials of bounded naive height

We prove an asymptotical formula for the number of reducible integer polynomials of degree d and of naive height at most T when \({T \to \infty}\) . The main term turns out to be of the form \({\kappa_d T^d}\) for each \({d \geq 3}\) , where the constant \({\kappa_d}\) is given in terms of some infinite Dirichlet series involving the volumes of symmetric convex bodies in \({\mathbb{R}^d}\) . For d = 2, we prove that there are asymptotically \({\kappa_2 T^2 \,\text{log} T}\) of such polynomials, where \({\kappa_2:=6(3\sqrt{5}+2\,\text{log} (1+\sqrt{5}) -2 \,\text{log}\, 2)/\pi^2}\) . Earlier results in this direction were given by van der Waerden, Pólya and Szegö, Dörge, Chela, and Kuba.     

Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

Let (M, g) and \({(K, \kappa)}\) be two Riemannian manifolds of dimensions m and k, respectively. Let \({\omega \in C^{2} (N), \omega > 0}\) . The warped product \({M \times_\omega K}\) is the (m +  k)-dimensional product manifold \({M \times K}\) furnished with metric \({g + \omega^{2} \kappa}\) . We prove that the supercritical problem $$- \Delta_{g + \omega^{2} \kappa} u + hu = u^{\frac{m+2}{m-2} \pm \varepsilon} ,\quad u > 0,\quad {\rm in}\,\, (M \times_{\omega} K, g + \omega^{2} \kappa)$$ has a solution concentrated along a k-dimensional minimal submanifold \({\Gamma}\) of \({M \times_{\omega } N}\) as the real parameter \({\varepsilon}\) goes to zero, provided the function h and the sectional curvatures along \({\Gamma}\) satisfy a suitable condition.     

The Cubic Complex Moment Problem

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\) . Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\) , then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\) . The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.     

Automorphisms of {{\rm M_{n}}(\mathbb{D})}, Partially Ordered by Rank Subtractivity Ordering

Let \({\mathbb{D}}\) be an arbitrary division ring and \({{\rm M_{n}}(\mathbb{D})}\) be the set of all n × n matrices over \({\mathbb{D}}\) . We define the rank subtractivity or minus partial order on \({{\rm M_{n}}(\mathbb{D})}\) as defined on \({{\rm M_{n}}(\mathbb{C})}\) , i.e., \({A \leqslant B}\) iff rank(B) = rank(A) + rank(B?A). We describe the structure of maps Φ on \({{\rm M_{n}}(\mathbb{D})}\) such that \({A\leqslant B}\) iff \({\Phi(A)\leqslant \Phi(B) (A, B\in {\rm M_{n}}(\mathbb{D}) )}\) .