Monomial Hopf algebras over fields of positive characteristic

In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on Cd(n) are completely classified. Moreover, we construct a Hopf algebras filtration on Cd(n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.     

Cycles on curves over global fields of positive characteristic

Let be a global field of positive characteristic, and let be a smooth projective curve. We study the zero-dimensional cycle group and the one-dimensional cycle group , addressing the conjecture that is torsion and is finitely generated. The main idea is to use Abhyankar's Theorem on resolution of singularities to relate the study of these cycle groups to that of the -groups of a certain smooth projective surface over a finite field.

     

Complexity of multiplication in commutative group algebras over fields of characteristic 0

Let rkA denote the bilinear complexity (also known as rank) of a finite-dimension associative algebra A. Algebras of minimal rank are widely studied from the point of view of bilinear complexity. These are the algebras A for which the Alder-Strassen inequality is satisfied as an equality, i.e., rkA = 2dimA ? t, where t is the number of maximum two-sided ideals in A. It is proved in this work that an arbitrary commutative group algebra over a field of characteristic 0 is an algebra of minimal rank. The structure and precise values of the bilinear complexity of commutative group algebras over a field of rational numbers are obtained.     

The cohomology of the Heisenberg Lie algebras over fields of finite characteristic

We give explicit formulas for the cohomology of the Heisenberg Lie algebras over fields of finite characteristic. We use this to show that in characteristic two, unlike all other cases, the Betti numbers are unimodal.

     

Grassmann algebras over finite fields

We study an analogue of a problem of procesi about matrices [9, page 185(e)]: are there non-trivial polynomials over Z which become identities over Z _p - for grassmann algebras E? when 1 ? E, we show that such polynomials do not exist, but when 1 ?,E such polynomials exist - also for matrices over E. these results are deduced from a careful study of the various codimensions of these algebras.     

Indicators of Hopf algebras in positive characteristic

The notion of n-th indicator for a finite-dimensional Hopf algebra was introduced by Kashina, Montgomery, and Ng as invariance of the tensor category of its representations. In this short note, we show that the indicators appearing in positive characteristic all share the same sequence pattern if we assume the coradical of the Hopf algebra is a local Hopf subalgebra.     

On vanishing coefficients of algebraic power series over fields of positive characteristic

Let K be a field of characteristic p>0 and let f(t ₁,…,t _d ) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K(t ₁,…,t _d ). We prove a generalization of both Derksen’s recent analogue of the Skolem–Mahler–Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices (n ₁,…,n _d )∈? ^d for which the coefficient of \(t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\) in f(t ₁,…,t _d ) is zero is a p-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to S-unit equations and more generally to the Mordell–Lang Theorem over fields of positive characteristic.     

Graded algebras

We study the growth in the number of dimensions d_n of the homogeneous component of a graded algebra with a finite number of defining relations and generators for the Poincaré series d_nxⁿ. It is proved that if the defining relations are words, the Poincaré series is a rational function. In the general case inequalities are proved linking the number of dimensions d_n with the number of generators defining relations and their degree.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 197–204, August, 1972.     

Involutions of reductive Lie algebras in positive characteristic

Let G be a reductive group over a field k of characteristic ≠2, let , let θ be an involutive automorphism of G and let be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G^θ on is well understood, since the well-known paper of Kostant and Rallis [B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753–809]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic.Among other results, we prove that the variety of nilpotent elements of has a dense open orbit, and that the same is true for every fibre of the quotient map . However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of , extending a result of Sekiguchi for . Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants .