Narrow Results

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four-dimensional projective space associated to the root system D₅. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces.

We depict the weight diagrams (alias, crystal graphs) of basic and adjoint representations of complex simple Lie algebras/algebraic groups and describe some of their uses.

A construction of the Virasoro algebra in terms of free massless two-dimensional boson fields is studied. The ansatz for the Virasoro field contains the most general unitary scaling dimension 2 expression built from vertex operators. The ansatz leads in a natural way to a concept of a quasi root systems. This is a new notion generalizing the notion of a root system in the theory of Lie algebras. We introduce a definition of a quasi root systems and provide an extensive list of examples. Explicit solutions of the ansatz are presented for a range of quasi root systems.

In this paper, we consider the classification of singularities [P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction. I, II, III, Proc. Camb. Philos. Soc.30 (1934) 453–491] and real structures [C. T. C. Wall, Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math.1987(375/376) (1987) 47–66, ISSN 0075-4102] of weak Del Pezzo surfaces from an algorithmic point of view. It is well-known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.

We study bosonic–fermionic constructions of the root spaces of the orthosymplectic Lie superalgebras. We also present explicitly bosonic–fermionic realizations of the supertrace forms for the orthosymplectic Lie superalgebras. As by-products, we obtain bosonic–fermionic realizations of both the general linear Lie superalgebras and vertex representations of the affine orthosymplectic Lie superalgebras.

We construct the basic representation of the double affine Hecke algebra at critical level $q=1$ associated to an irreducible reduced affine root system $R$ with a reduced gradient root system. For $R$ of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights.

Let $G$ be a non-compact classical semisimple Lie group and let $G/V$ be the adjoint orbit with respect to a fixed element in $G$. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on $G/V$ purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type $A_{\ell }$, $B_{\ell }$, $C_{\ell }$, $D_{\ell }$, where $\ell$ denotes the rank of the Lie algebra.

Let L be a simple Lie algebra with irreducible root system Φ having roots of different length, F be a field of characteristic different from 2, G=L(F) be a Chevalley group of type L over F. Denote by Φ^l the set of all long roots in Φ. Set G^l=<x_r(t); r∈Φ^l, t∈ F>. It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of G^l in G when L is not of type G₂.

This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics.