Let g be a finite-dimensional semisimple Lie algebra and (· , ·) its Killing form, σ an elliptic automorphism of g, and a a σ-invariant reductive subalgebra of g, such that the restriction of the form (· , ·) to a is non-degenerate. Let L̂(g, σ) and L̂(a, σ) be the associated twisted affine Lie algebras and F σ(p) the σ-twisted Clifford module over L̂(a, σ), associated to the orthocomplement p of ...

We prove that if G is a non-uniform lattice in a rank-one semisimple Lie group = Isom(H2 R ), then G is quasi-isometrically co-Hopf. This means that every quasi-isometric embedding G → G is coarsely surjective and thus is a quasi-isometry.

We prove that the lowest nonzero piece in the Hodge filtration of a mixed Hodge module is always weakly positive in the sense of Viehweg.

Let G be a complex semisimple algebraic group with Lie algebra g. The goal of this note is to show that combining some ideas of Gunnells and Sommers [Math. Res. Lett. 10 (2–3) (2003) 363–373] and Vinberg and Popov [Invariant Theory, in: Algebraic Geometry IV, in: Encyclopaedia Math. Sci., Vol. 55, Springer, Berlin, 1994, pp.123–284] yields a geometric description of the characteristic of a nilp...

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/H ⊂ P(V ) is a spherical orbit, set X = G/H ⊂ P(V ) its closure. Then we describe the orbits of X and those of its normalization e X in terms of spherical systems and we give necessary and sufficient conditions so that the normalization e ...

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a “reflection” defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig’s automorphisms of quantized Kac-Moody algebras to the nilpotent part. As a direct application we co...

The present paper can be considered as a natural extension the article [Ar7]. Fix root data (Y,X, . . . ) of the finite type (I, ·) and a positive integer number l. In [Ar7] we obtained a nice description for semiinfinite cohomology of the trivial module C over the small quantum group ul corresponding to the root data (Y,X, . . . ) in terms of local cohomology of the structure sheaf on the nilp...

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/H ⊂ P(V ) is a spherical orbit, set X = G/H ⊂ P(V ) its closure. Then we describe the orbits of X and of its normalization e X in terms of spherical systems and we give necessary and sufficient conditions so that the normalization e X → X ...

We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U(sl2), Uq(sl2) and the enveloping algebra of the 3-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions and pr...

Let G be a semisimple connected linear algebraic group over C, and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. For any wonderful variety X we give a necessary and sufficient condition in terms of its set of spherical roots, which are certain characters of B (a fixed Borel subgroup of G) associated to the G...