Volume 8 (2008), Number 4. Abstracts

V. Batyrev and F. Haddad. On the Geometry of SL(2)-Equivariant Flips [PDF]

In this paper, we show that any 3-dimensional normal affine quasihomogeneous SL(2)-variety can be described as a categorical quotient of a 4-dimensional affine hypersurface. Moreover, we show that the Cox ring of an arbitrary 3-dimensional normal affine quasihomogeneous SL(2)-variety has a unique defining equation. This allows us to construct SL(2)-equivariant flips by different GIT-quotients of hypersurfaces. Using the theory of spherical varieties, we describe SL(2)-flips by means of 2-dimensional colored cones.

Keywords. Geometric invariant theory, categorical quotient, Mori theory.

2000 Mathematics Subject Classification. Primary: 14L24, 14L30; Secondary 14E30.

M. Brion. Local Structure of Algebraic Monoids [PDF]

We describe the local structure of an irreducible algebraic monoid M at an idempotent element e. When e is minimal, we show that M is an induced variety over the kernel MeM (a homogeneous space) with fibre the two-sided stabilizer Me (a connected affine monoid having a zero element and a dense unit group). This yields the irreducibility of stabilizers and centralizers of idempotents when M is normal, and criteria for normality and smoothness of an arbitrary monoid M. Also, we show that M is an induced variety over an abelian variety, with fiber a connected affine monoid having a dense unit group.

Keywords. Algebraic monoid, idempotent, local structure, induced variety.

2000 Mathematics Subject Classification. 14L10, 14L30, 14M17, 20M20.

C. De Concini, S. Kannan, and A. Maffei. The Quotient of a Complete Symmetric Variety [PDF]

We study the quotient of a completion of a symmetric variety G/H under the action of H. We prove that this is isomorphic to the closure of the image of an isotropic torus under the action of the restricted Weyl group. In the case the completion is smooth and toroidal we describe the set of semistable points.

Keywords. Symmetric varieties, compactification of symmetric varieties, geometric invariant theory, Chevalley theorem.

2000 Mathematics Subject Classification. 14L30, 14L24, 14M17.

V. Gichev. Invariant Function Algebras on Compact Commutative Homogeneous Spaces [PDF]

Let M be a commutative homogeneous space of a compact Lie group G and A be a closed G-invariant subalgebra of the Banach algebra C(M). A function algebra is called antisymmetric if it does not contain nonconstant real functions. By the main result of this paper, A is antisymmetric if and only if the invariant probability measure on M is multiplicative on A. This implies, for example, the following theorem: if GC acts transitively on a Stein manifold M, vM, and the compact orbit M=Gv is a commutative homogeneous space, then M is a real form of M.

Keywords. Invariant function algebra, commutative homogeneous space, maximal ideal space.

2000 Mathematics Subject Classification. Primary: 46J10; Secondary: 14R20, 32E20, 46J20.

J. Hausen. Cox Rings and Combinatorics II [PDF]

We study varieties with a finitely generated Cox ring. In the first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow-ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combinatorially minimal one.

Keywords. Cox ring, total coordinate ring, divisors, modifications.

2000 Mathematics Subject Classification. 14C20, 14M25.

V. Kac, P. Möseneder Frajria, and P. Papi. On the Kernel of the Affine Dirac Operator [PDF]

Let g be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form (·,·), σ an elliptic automorphism of g leaving the form (·,·) invariant, and a a σ-invariant subalgebra of g, such that the restriction of the form (·,·) to a is non-degenerate. Let \widehatL(g,σ) and \widehatL(a,σ) be the associated twisted affine Lie algebras and Fσ(p) the σ-twisted Clifford module over \widehatL(a,σ), associated to the orthocomplement p of a in g. Under suitable hypotheses on σ and a, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight \widehatL(g,σ)-module and Fσ(p), into irreducible \widehatL(a,σ)-submodules.

As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.

Keywords. Affine algebra, Dirac operator, Lie algebra automorphism.

2000 Mathematics Subject Classification. 17B67.

V. Nikulin. On Ground Fields of Arithmetic Hyperbolic Reflection Groups. II [PDF]

This paper continues two our papers that appeared in 2007. Using our methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 4 are defined, and good explicit bounds of their degrees (over Q) are obtained. This extends the results of our previous paper where it was done in dimensions at least 6. These results could be important for the further classification of these groups.

Keywords. Groups generated by reflections, arithmetic groups, hyperbolic groups.

2000 Mathematics Subject Classification. 20F55, 51F15, 22E40.

A. Vershik and A. Sergeev. A New Approach to the Representation Theory of the Symmetric Groups, IV. Z2-Graded Groups and Algebras; Projective Representations of the Group Sn [PDF]

We start with definitions of the general notions of the theory of Z2-graded algebras. Then we consider theory of inductive families of Z2-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers [VO], [OV] to representation theory of symmetric groups. The main example is the theory of the projective representations of symmetric groups.

Keywords. Chains of Z2-graded algebras, Gelfand–Tsetlin superalgebras, Young formulas.

2000 Mathematics Subject Classification. Primary: 20C07; Secondary: 17A70.

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