Grothendieck [7] showed that finite dimensional vector bundles over the projective line split into direct sums of line bundles (see also Andersen [1]). We begin this paper, after two preliminary sections, by proving this result (Theorem (3.5)) for induced bundles using representation theoretic techniques. The proof yields partial results in the infinite dimensional case and leads to the definit...

This article studies the Albanese kernel TF (E × E), for an elliptic curve E over a p-adic field F . The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from TF (E ×E)/p to the Galois cohomology group H(F, E[p] ⊗ E[p]), for E/F ordinary, without requiring that the ptorsion points are F -rational, or even that the Galois module E[p] is ...

Kazhdan-Lusztig polynomials Px,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. In particular, the value Px,w(1) is the dimension of the intersection cohomology sheaf of the Schubert variety Xw at the T -fixed point indexed by x [KL2]. It is also the multiplicity of a certain irreducible module in a Verma module [BeB...

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $${\mathcal {M}}(p)$$ , $$p\ge 2$$ . The first category consists all finite-length -modules with atypical composition factors, while the second is subcategory that induce to local triplet {W}}(p)$$ show every irreducible module has a projective cover in these categories, although...

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 ...

It is well-known that a ring R is semiperfect if and only if RR (or RR) is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (or RR) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie d...

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...