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A construction of hyperbolic coxeter groups
We give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct e.g. new groups having some interesting asphericity properties.
New representations of matroids and generalizations
We extend the notion of matroid representations by matrices over fields by considering new representations of matroids by matrices over finite semirings, more precisely over the boolean and the superboolean semirings. This idea of representations is naturally generalized to include hereditary collections (also known as abstract simplicial complexes). We show that a matroid that can be directly decomposed as matroids, each of which is representable over a field, has a boolean representation, and more generally that any arbitrary hereditary collection is superboolean-representable.
The Weyl group of the Cuntz algebra
The Weyl group of the Cuntz algebra On is investigated. This is (isomorphic to) the group of polynomial automorphisms λu of On, namely those induced by unitaries u that can be written as finite sums of words in the canonical generating isometries Si and their adjoints. A necessary and sufficient algorithmic combinatorial condition is found for deciding when a polynomial endomorphism λu restricts to an automorphism of the canonical diagonal MASA. Some steps towards a general criterion for invertibility of λu on the whole of On are also taken. A condition for verifying invertibility of a certain subclass of polynomial endomorphisms is given. First examples of polynomial automorphisms of On not inner related to permutative ones are exhibited, for every n≥2. In particular, the image of the Weyl group in the outer automorphism group of On is strictly larger than the image of the reduced Weyl group analyzed in previous papers. Results about the action of the Weyl group on the spectrum of the diagonal are also included.
Infeasibility certificates for linear matrix inequalities
Farkas' lemma is a fundamental result from linear programming providing linear certi cates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry. More precisely, we show that a linear matrix inequality L(x)⪰0 is infeasible if and only if −1 lies in the quadratic module associated to L. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certi cate, we use a more involved algebraic certificate motivated by the real radical and Prestel's theory of semiorderings. Completely different methods, namely complete positivity from operator algebras, are employed to consider linear matrix inequality domination.
Locally conformally Kähler manifolds admitting a holomorphic conformal flow
Abstract A manifold M is locally conformally Kähler (LCK) if it admits a Kähler covering ˜M with monodromy acting by holomorphic homotheties. Let M be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on ˜M . We show that M admits an automorphic potential, and the monodromy group of its conformal weight bundle is Z.
Localized endomorphisms of graph algebras
Endomorphisms of graph C*-algebras are investigated. A combinatorial ap- proach to analysis of permutative endomorphisms is developed. Then invertibility criteria for localized endomorphisms are given. Furthermore, proper endomor- phisms which restrict to automorphisms of the canonical diagonal MASA are analyzed. The Weyl group and the restricted Weyl group of a graph C*-algebra are introduced and investigated. Criteria of outerness for automorphisms in the restricted Weyl group are found.
A series of algebras generalizing the octonions and Hurwitz-Radon identity
We study non-associative twisted group algebras over (Z2)n with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: (a21+...+a2N)(b21+...+b2ρ(N))=c21+...+c2N, where ck are bilinear functions of the ai and bj and where ρ(N) is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop.
Legendrian knots in Lens spaces
In this note, we first classify all topological torus knots lying on the Heegaard torus in Lens spaces, and then we classify Legendrian representatives of torus knots. We show that all Legendrian torus knots in universally tight contact structures on Lens spaces are determined up to contactomorphism by their knot type, rational Thurston-Bennequin invariant and rational rotation number.
Braid equivalences and the L-moves
In this survey paper we present the L–moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3–manifolds and in handlebodies, as well as for virtual knots, for flat virtuals, for welded knots and for singular knots. The L–moves are local and they provide a uniform ground for formulating and proving braid equivalence theorems for any diagrammatic setting where the notion of braid and diagrammatic isotopy is defined, the statements being first geometric and then algebraic.