On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

Modular Decomposition of the Orlik-Terao Algebra

Let ${\mathcal{A}}$ be a collection of n linear hyperplanes in ${\mathbb{k}^\ell}$ , where ${\mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${\mathcal{A}}$ is the subalgebra ${{\rm R}(\mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${\mathcal{A}}$ . It determines an irreducible subvariety ${Y (\mathcal{A})}$ of ${\mathbb{P}^{n-1}}$ . We show that a flat X of ${\mathcal{A}}$ is modular if and only if ${{\rm R}(\mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${\mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${\mathcal{A}}$ is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Diagram Groupoids and von Neumann Algebras

The main purpose of this paper is to study certain algebraic structures induced by directed graphs. We have studied graph groupoids, which are algebraic structures induced by given graphs. By defining a certain groupoid-homomorphism ?? on the graph groupoid ${\mathbb{G}}$ of a given graph G, we define the diagram of G by the image ${\delta(\mathbb{G})}$ of ??, equipped with the inherited binary operation on ${\mathbb{G}}$ . We study the fundamental properties of the diagram ${\delta(\mathbb{G})}$ , and compare them with those of ${\mathbb{G}}$ . Similar to Cho (Acta Appl Math 95:95?C134, 2007), we construct the groupoid von Neumann algebra ${\mathcal{M}_{G}=vN(\delta(\mathbb{G}))}$ , generated by ${\delta(\mathbb{G})}$ , and consider the operator algebraic properties of ${\mathcal{M}_{G}}$ . In particular, we show ${\mathcal{M}_{G}}$ is *-isomorphic to a von Neumann algebra generated by a family of idempotent operators and nilpotent operators, under suitable representations.     

Chromatic Gallai identities operating on Lovász number

If $G$ is a triangle-free graph, then two Gallai identities can be written as $\alpha (G)+\overline{\chi }(L(G))=|V(G)|=\alpha (L(G))+\overline{\chi }(G)$ , where $\alpha $ and $\overline{\chi }$ denote the stability number and the clique-partition number, and $L(G)$ is the line graph of  $G$ . We show that, surprisingly, both equalities can be preserved for any graph $G$ by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, according to any acyclic orientation of  $G$ . As a consequence, one obtains an operator $\Phi $ which associates to any graph parameter $\beta $ such that $\alpha (G) \le \beta (G) \le \overline{\chi }(G)$ for all graph $G$ , a graph parameter $\Phi _\beta $ such that $\alpha (G) \le \Phi _\beta (G) \le \overline{\chi }(G)$ for all graph $G$ . We prove that $\vartheta (G) \le \Phi _\vartheta (G)$ and that $\Phi _{\overline{\chi }_f}(G)\le \overline{\chi }_f(G)$ for all graph  $G$ , where $\vartheta $ is Lovász theta function and $\overline{\chi }_f$ is the fractional clique-partition number. Moreover, $\overline{\chi }_f(G) \le \Phi _\vartheta (G)$ for triangle-free $G$ . Comparing to the previous strengthenings $\Psi _\vartheta $ and $\vartheta ^{+ \triangle }$ of $\vartheta $ , numerical experiments show that $\Phi _\vartheta $ is a significant better lower bound for $\overline{\chi }$ than $\vartheta $ .     

On the Stretch Factor of Randomly Embedded Random Graphs

We consider a random graph $\mathcal{G}(n,p)$ whose vertex set $V,$ of cardinality $n,$ has been randomly embedded in the unit square and whose edges, which occur independently with probability $p,$ are given weight equal to the geometric distance between their end vertices. Then each pair $\{u,v\}$ of vertices has a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all $\{u,v\}\subseteq V.$ We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for $p$ not too close to 0 or 1, these bounds are the best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if $n(1-p)$ tends to 0, answering a question of O’Rourke.     

On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

A graph associated to a lattice

In this paper, we associate a simple graph to a lattice $\mathcal L $ , in which the vertex set is being the set of all elements of $\mathcal L $ , and two distinct vertices $x$ and $y$ are adjacent if $x\vee y\in S$ , when $S$ is a multiplicatively closed subset of $\mathcal L $ . We denote this graph by $\Gamma _S(\mathcal L )$ . We study some properties of $\Gamma _S(\mathcal L )$ . Moreover, we investigate the planarity of $\Gamma _S(\mathcal L )$ , whenever $S$ is a saturated multiplicatively closed subset of $\mathcal L $ .     

The \mathfrak {sl}_{3}-web algebra

In this paper we use Kuperberg’s $\mathfrak {sl}_3$ -webs and Khovanov’s $\mathfrak {sl}_3$ -foams to define a new algebra $K^S$ , which we call the $\mathfrak {sl}_3$ -web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$ -skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}$ , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.     

A construction of cylindric and polyadic algebras from atomic relation algebras

Given a simple atomic relation algebra ${\mathcal{A}}$ and a finite n ?? 3, we construct effectively an atomic n-dimensional polyadic equality-type algebra ${\mathcal{P}}$ such that for any subsignature L of the signature of ${\mathcal{P}}$ that contains the boolean operations and cylindrifications, the L-reduct of ${\mathcal{P}}$ is completely representable if and only if ${\mathcal{A}}$ is completely representable. If ${\mathcal{A}}$ is finite then so is ${\mathcal{P}}$ . It follows that there is no algorithm to determine whether a finite n-dimensional cylindric algebra, diagonal-free cylindric algebra, polyadic algebra, or polyadic equality algebra is representable (for diagonal-free algebras this was known). We also obtain a new proof that the classes of completely representable n-dimensional algebras of these types are non-elementary, a result that remains true for infinite dimensions if the diagonals are present, and also for infinite-dimensional diagonal-free cylindric algebras.     

Free subalgebras of division algebras over uncountable fields

We study the existence of free subalgebras in division algebras, and prove the following general result: if $A$ is a noetherian domain which is countably generated over an uncountable algebraically closed field $k$ of characteristic $0$ , then either the quotient division algebra of $A$ contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if $k$ is an uncountable algebraically closed field and $A$ is a finitely generated $k$ -algebra that is a domain of GK-dimension strictly less than $3$ , then either $A$ satisfies a polynomial identity, or the quotient division algebra of $A$ contains a free $k$ -algebra on two generators.     

Global injectivity conditions for planar maps

We prove that a planar $C^1$ -smooth map $f:D\longrightarrow \mathbb{R }^{2n}$ , where $D\subseteq \mathbb{R }^{2n}$ is a convex open set, is injective if $\mathbb{R }\cap \mathrm{Spec}(df)_z=\emptyset $ for all $z\in D$ . We continue by showing that the triangulability of the differentials $(df)_z$ , $z\in D$ , ensure the global injectivity as well.     

Geometric rigidity for sequences of W^{2,2} conformal immersions

We analyse sequences of discs conformally immersed in $ \mathbb{R }^ n$ with energy $ \int _{ D} |A_k |^ 2 \le \gamma _n$ , where $ \gamma _n = 8\pi $ if $ n=3$ and $ \gamma _n = 4 \pi $ when $n\ge 4$ . We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper’s minimal surface if $ n=3$ or Chen’s minimal graph if $ n \ge 4$ . In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313–340, 2012; Rivière, Adv Calculus Variations 6(1), 1–31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then $\liminf _ {k\rightarrow \infty } \mathcal W ( f_k )\ge 8\pi $ . We apply the above analysis to show that in $ \mathbb{R }^3$ if the sequence diverges so that $ \lim _{ k \rightarrow \infty } \mathcal W (f_k) =8\pi $ then there exists a sequence of Möbius transforms $ \sigma _{k}$ such that $ \sigma _k\circ f _k$ converges weakly to a catenoid.     

Graphs, partitions and classes of groups

If ${\mathfrak X}$ is a class of groups, Delizia et?al. (Bull Austral Math Soc 75:313–320, 2007) call a group G ${\mathfrak X}$ -transitive (or an ${\mathfrak XT}$ -group) if whenever ${\langle a,b\rangle}$ and ${\langle b,c\rangle}$ are in ${\mathfrak X} \langle a,c\rangle$ is also in ${\mathfrak X}$ ( ${a,b,c\in G}$ ). The structure of ${\mathfrak XT}$ -groups has been investigated for a number of classes of groups, by Delizia, Moravec and Nicotera and others. A graph can be associated with a group in many ways. Delizia, Moravec and Nicotera introduce a graph which is a generalisation of the commuting graph of a group, but do not make use of the graph. We will use the properties of the graph to investigate further classes of groups and to obtain more detailed structural information.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Extending structures for Lie algebras

Let $\mathfrak{g }$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g }$ as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on $E$ such that $\mathfrak{g }$ is a Lie subalgebra of $E$ . A general product, called the unified product, is introduced as a tool for our approach. Let $V$ be a complement of $\mathfrak{g }$ in $E$ : the unified product $\mathfrak{g } \,\natural \, V$ is associated to a system $(\triangleleft , \, \triangleright , \, f, \{-, \, -\})$ consisting of two actions $\triangleleft $ and $\triangleright $ , a generalized cocycle $f$ and a twisted Jacobi bracket $\{-, \, -\}$ on $V$ . There exists a Lie algebra structure $[-,-]$ on $E$ containing $\mathfrak{g }$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $(E, [-,-]) \cong \mathfrak{g } \,\natural \, V$ . All such Lie algebra structures on $E$ are classified by two cohomological type objects which are explicitly constructed. The first one $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ will classify all Lie algebra structures on $E$ up to an isomorphism that stabilizes $\mathfrak{g }$ while the second object $\mathcal{H }^{2} (V, \mathfrak{g })$ provides the classification from the view point of the extension problem. Several examples that compute both classifying objects $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ and $\mathcal{H }^{2} (V, \mathfrak{g })$ are worked out in detail in the case of flag extending structures.     

Representation of Nelson algebras by rough sets determined by quasiorders

In this paper, we show that every quasiorder R induces a Nelson algebra ${{\mathbb R}{\mathbb S}}$ such that the underlying rough set lattice RS is algebraic. We note that ${{\mathbb R}{\mathbb S}}$ is a three-valued ?ukasiewicz algebra if and only if R is an equivalence. Our main result says that if ${{\mathbb A}}$ is a Nelson algebra defined on an algebraic lattice, then there exists a set U and a quasiorder R on U such that ${{\mathbb A} \cong {\mathbb R}{\mathbb S}}$ .     

Circular Coloring of Planar Digraphs

Let D be a digraph. The circular chromatic number ${\chi_c(D)}$ and chromatic number ${\chi(D)}$ of D were proposed recently by Bokal et?al. Let ${\vec{\chi_c}(G)={\rm max}\{\chi_c(D)| D\, {\rm is\, an\, orientation\, of} G\}}$ . Let G be a planar graph and n?≥ 2. We prove that if the girth of G is at least ${\frac{10n-5}{3},}$ then ${\vec{\chi_c}(G)\leq \frac{n}{n-1}}$ . We also study the circular chromatic number of some special planar digraphs.