On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

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.     

Relationships Between the 2-Metric Dimension and the 2-Adjacency Dimension in the Lexicographic Product of Graphs

Given a connected simple graph \(G=(V(G),E(G))\), a set \(S\subseteq V(G)\) is said to be a 2-metric generator for G if and only if for any pair of different vertices \(u,v\in V(G)\), there exist at least two vertices \(w_1,w_2\in S\) such that \(d_G(u,w_i)\ne d_G(v,w_i)\), for every \(i\in \{1,2\}\), where \(d_G(x,y)\) is the length of a shortest path between x and y. The minimum cardinality of a 2-metric generator is the 2-metric dimension of G, denoted by \(\dim _2(G)\). The metric \(d_{G,2}: V(G)\times V(G)\longmapsto {\mathbb {N}}\cup \{0\}\) is defined as \(d_{G,2}(x,y)=\min \{d_G(x,y),2\}\). Now, a set \(S\subseteq V(G)\) is a 2-adjacency generator for G, if for every two vertices \(x,y\in V(G)\) there exist at least two vertices \(w_1,w_2\in S\), such that \(d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i)\) for every \(i\in \{1,2\}\). The minimum cardinality of a 2-adjacency generator is the 2-adjacency dimension of G, denoted by \({\mathrm {adim}}_2(G)\). In this article, we obtain closed formulae for the 2-metric dimension of the lexicographic product \(G\circ H\) of two graphs G and H. Specifically, we show that \(\dim _2(G\circ H)=n\cdot {\mathrm {adim}}_2(H)+f(G,H),\) where \(f(G,H)\ge 0\), and determine all the possible values of f(G, H).     

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 \(C^{\prime }\).     

Complete classification of pseudo <Emphasis Type="Italic">H</Emphasis>-type Lie algebras: I

Let \({\mathscr {N}}\) be a 2-step nilpotent Lie algebra endowed with a non-degenerate scalar product \(\langle .\,,.\rangle \), and let \({\mathscr {N}}=V\oplus _{\perp }Z\), where Z is the centre of the Lie algebra and V its orthogonal complement. We study classification of the Lie algebras for which the space V arises as a representation space of the Clifford algebra \({{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\), and the representation map \(J:{{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(V)\) is related to the Lie algebra structure by \(\langle J_zv,w\rangle =\langle z,[v,w]\rangle \) for all \(z\in {\mathbb {R}}^{r,s}\) and \(v,w\in V\). The classification depends on parameters r and s and is completed for the Clifford modules V having minimal possible dimension, that are not necessary irreducible. We find necessary conditions for the existence of a Lie algebra isomorphism according to the range of the integer parameters \(0\le r,s<\infty \). We present a constructive proof for the isomorphism maps for isomorphic Lie algebras and determine the class of non-isomorphic Lie algebras.     

Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle \({\mathcal {E}}\) on X. Let \(\mathfrak {h}\) be the Lie algebra of H. Let \(\mathcal {S}(X,{\mathcal {E}})\) be the space of Schwartz sections of \({\mathcal {E}}\). We prove that \(\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})\) is a closed subspace of \(\mathcal {S}(X,{\mathcal {E}})\) of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let \(\pi \) be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants \(V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}\) is an isomorphism if and only if any linear \(\mathfrak {h}\)-invariant functional on V is continuous in the topology induced from \(\pi \). The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.     

INTEGRABILITY OF GEODESIC FLOWS FOR METRICS ON SUBORBITS OF THE ADJOINT ORBITS OF COMPACT GROUPS

Let G/K be an orbit of the adjoint representation of a compact connected Lie group G, σ be an involutive automorphism of G and \( \tilde{G} \) be the Lie group of fixed points of σ. We find a sufficient condition for the complete integrability of the geodesic ow of the Riemannian metric on \( \tilde{G}/\left(\tilde{G}\cap K\right) \) which is induced by the bi-invariant Riemannian metric on \( \tilde{G} \). The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when G is the unitary group U(n) and σ is its automorphism determined by the complex conjugation.     

Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups

If G is a compact Lie group endowed with a left invariant metric g, then G acts via pullback by isometries on each eigenspace of the associated Laplace operator \(\Delta _g\). We establish algebraic criteria for the existence of left invariant metrics g on G such that each eigenspace of \(\Delta _g\), regarded as the real vector space of the corresponding real eigenfunctions, is irreducible under the action of G. We prove that generic left invariant metrics on the Lie groups \(G={ SU}(2)\times \cdots \times { SU}(2)\times T\), where T is a (possibly trivial) torus, have the property just described. The same holds for quotients of such groups G by discrete central subgroups. In particular, it also holds for \({ SO}(3)\), \({ U}(2)\), \({ SO}(4)\).     

On Borcherds type Lie algebras

For each even lattice \({\mathcal L}\), there is a canonical way to construct an infinite-dimensional Lie algebra via lattice vertex operator algebra theory, we call this Lie algebra and its subalgebras the Borcherds type Lie algebras associated to \({\mathcal L}\). In this paper, we apply this construction to even lattices arising from representation theory of finite-dimensional associative algebras. This is motivated by the different realizations of Kac-Moody algebras by Borcherds using lattice vertex operators and by Peng-Xiao using Ringel-Hall algebras respectively. For any finite-dimensional algebra \(A\) of finite global dimension, we associate a Borcherds type Lie algebra \(\mathfrak {BL}(A)\) to \(A\). In contrast to the Ringel-Hall Lie algebra approach, \(\mathfrak {BL}(A)\) only depends on the symmetric Euler form or Tits form but not the full representation theory of \(A\). However, our results show that for certain classes of finite-dimensional algebras whose representation theory is ’controlled’ by the Euler bilinear forms or Tits forms, their Borcherds type Lie algebras do have close relations with the representation theory of these algebras. Beyond the class of hereditary algebras, these algebras include canonical algebras, representation-directed algebras and incidence algebras of finite prinjective types.     

Some infinite families of Ramsey ($${\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}$$)-minimal trees

For any given two graphs G and H, the notation \(F\rightarrow \) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if \(F\rightarrow \) (G, H) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The class of all Ramsey (G, H)-minimal graphs is denoted by \(\mathcal {R}(G,H)\). In this paper, we construct some infinite families of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n=8\) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n\ge 10\).     

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 \(k-1\).     

Zigzag Strip Bundle Realization of B(Λ0) over $U_{q}(C_{n}^{(1)})$

Zigzag strip bundles are new combinatorial models realizing the crystals B(∞) for the quantum affine algebras \(U_{q}(\mathfrak {g})\), where \(\mathfrak {g}=B_{n}^{(1)},D_{n}^{(1)}\), \(D_{n+1}^{(2)}\), \(C_{n}^{(1)}\), \(A_{2n-1}^{(2)}\), \(A_{2n}^{(2)}\). Recently, these models were used to the realization of highest weight crystals except for the highest weight crystal B(Λ₀) over the quantum affine algebra \(U_{q}(C_{n}^{(1)})\). In this paper, we construct the highest weight crystal B(Λ₀) over the quantum affine algebra \(U_{q}(C_{n}^{(1)})\) using zigzag strip bundles, which completes the realizations of all highest weight crystals over \(U_{q}(\mathfrak {g})\).     

Packing Chromatic Number of Base-3 Sierpiński Graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least \(i+1\). Let \(S^n\) be the base-3 Sierpiński graph of dimension n. It is proved that \(\chi _{\rho }(S^1) = 3\), \(\chi _{\rho }(S^2) = 5\), \(\chi _{\rho }(S^3) = \chi _{\rho }(S^4) = 7\), and that \(8\le \chi _\rho (S^n) \le 9\) holds for any \(n\ge 5\).     

Induced Cycles in Graphs

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by \(c_\mathrm{ind}(G)\). We prove that if G is an r-regular graph of order n, then \(c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}\) and we prove that if G is a cubic, claw-free graph on order n, then \(c_\mathrm{ind}(G) > \frac{13}{20}n\) and this bound is asymptotically best possible.     

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 \(G_{1}\subset G\).     

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$-modules

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in (0,1)\cup (1,+\,\infty )\). As an application, we prove that for an adjontable operator A if \(\mathscr {R}(A)\) is nonclosed, then \(\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty \). Finally, we show that for an adjointable operator A if \( \overline{\mathscr {R}(A^*) } \) is orthogonally complemented in \( \mathscr {E} \), then under certain coditions there exists an idempotent C and a unique operator X such that \( XAX=X, AXA=CA, AX=C \) and \( XA=P_{A^*} \), where \( P_{A^*} \) is the orthogonal projection of \( \mathscr {E} \) onto \( \overline{\mathscr {R}(A^*)}\).     

On the peripheral spectrum of positive elements

Let A be an ordered Banach algebra with a unit \(\mathbf{e}\) and a cone \(A^+\). An element p of A is said to be an order idempotent if \(p^2 = p\) and \(0 \le p\le \mathbf{e}\). An element \(a\in A^+\) is said to be irreducible if the relation \((\mathbf{e}-p)ap = 0\), where p is an order idempotent, implies \(p = 0\) or \(p = \mathbf{e}\). For an arbitrary element a of A the peripheral spectrum \(\sigma _\mathrm{per}(a)\) of a is the set \(\sigma _\mathrm{per}(a) = \{\lambda \in \sigma (a):|\lambda | = r(a)\}\), where \(\sigma (a)\) is the spectrum of a and r(a) is the spectral radius of a. We investigate properties of the peripheral spectrum of an irreducible element a. Conditions under which \(\sigma _\mathrm{per}(a)\) contains or coincides with \(r(a)H_m\), where \(H_m\) is the group of all \(m^\mathrm{th}\) roots of unity, and the spectrum \(\sigma (a)\) is invariant under rotation by the angle \(\frac{2\pi }{m}\) for some \(m\in {\mathbb N}\), are given. The correlation between these results and the existence of a cyclic form of a is considered. The conditions under which a is primitive, i.e., \(\sigma _\mathrm{per}(a) = \{r(a)\}\), are studied. The necessary assumptions on the algebra A which imply the validity of these results, are discussed. In particular, the Lotz–Schaefer axiom is introduced and finite-rank elements of A are defined. Other approaches to the notions of irreducibility and primitivity are discussed. Conditions under which the inequalities \(0 \le b < a\) imply \(r(b) < r(a)\) are studied. The closedness of the center \(A_\mathbf{e}\), i.e., of the order ideal generated by \(\mathbf{e}\) in A, is proved.     

A geometric approach to alternating <Emphasis Type="Italic">k</Emphasis>-linear forms

Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.     

On Isomorphisms of Marušič–Scapellato Graphs

Maru?i?–Scapellato graphs are vertex-transitive graphs of order \(m(2^k + 1)\), where m divides \(2^k - 1\), whose automorphism group contains an imprimitive subgroup that is a quasiprimitive representation of \(\mathrm{SL}(2,2^k)\) of degree \(m(2^k + 1)\). We show that any two Maru?i?–Scapellato graphs of order pq, where p is a Fermat prime, and q is a prime divisor of \(p - 2\), are isomorphic if and only if they are isomorphic by a natural isomorphism derived from an automorphism of \(\mathrm{SL}(2,2^k)\). This work is a contribution towards the full characterization of vertex-transitive graphs of order a product of two distinct primes.