Given a smooth non-hyperelliptic curve C of genus 3 and a maximal isotropic subgroup (w.r.t. the Weil pairing) L ⊂ Jac(C)[2], there exists a smooth curve C′ s.t. Jac(C′) = Jac(C)/L. This construction is symmetric. i.e. if we start with C′ and the dual flag on it, we get C. A previous less explicit approach was taken by Donagi and Livné (see [DL]). The advantage of our construction is that it is...

In their fundamental studies of the finite dimensional representations of associative algebras, Artin and Procesi proved that the primitive ideals corresponding to irreducible n-dimensional representations (for fixed n, over an algebraically closed field) can be homeomorphically parameterized by a locally closed subset of the maximal spectrum of a suitably chosen affine commutative trace ring. ...

Every order invariant subset of Q[-r,r] has a maximal square whose sections at the x ∈ {0,...,r} agree below 0. Every universal property of the f::Q[-n,n] → Q[n,n] with no constants, has a 1-maximal regressive solution with f(0,...,n-1) ≡ f(1,...,n). We prove such purely order theoretic statements in the rationals within ZFC augmented with a standard large cardinal hypothesis, and show that ZFC...

Let G be a reductive algebraic group and H its reductive subgroup. Fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ B. The Cartan space aG,G/H is, by definition, the subspace of Lie(T )∗ generated by the weights of B-semiinvariant rational functions on G/H . We compute the spaces aG,G/H.

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay a foundation for this study. We formulate and study notions of quasi module and S-quasi module for a G1-vertex algebra V , where S is a quantum Yang-Baxter operator on V . We also formulate and study a notion of qu...

Let K be a global eld, V an innnite proper subset of the set of all primes of K, and S a nite subset of V. Denote the maximal Galois extension of K in which each p 2 S totally splits by K tot;S. Let M be an algebraic extension of K. A data for an (S; V)-Skolem density problem for M consists of a nite subset T of V containing S, polynomials f 1 ; : : : ; f m 2 ~ KX 1 ; : : : ; X n ] satisfying j...

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundation for this study. For any vector space W , we study what we call quasi compatible subsets of Hom(W,W ((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncom...

0.1. Let G be a connected semisimple algebraic group over an algebraically closed field. Let B,B− be two opposed Borel subgroups of G with unipotent radicals U,U− and let T = B ∩ B−, a maximal torus of G. Let NT be the normalizer of T in G and let W = NT/T be the Weyl group of T , a finite Coxeter group with length function l. For w ∈ W let ẇ be a representative of w in NT . The following resul...

We determine all nite groups G which admit a subgroup K of index p a ; p a prime, under the assumption that G has an irreducible and faithful GF (p)-module of dimension at most a. As an application to the theory of permutation groups we determine the maximal transitive subgroups of the primitive aane permutation groups.

It is proved that each group of order 32 that has a maximal subgroup isomorphic to C2×C2×C2×C2 or C4×C4 is determined by its endomorphism semigroup in the class of all groups.