Weyl vertex algebras over fields of prime characteristic

Let ℋ(m;t) be the finite-dimensional odd Hamiltonian superalgebra over a field of prime characteristic. By determining ad-nilpotent elements in the even part, the natural filtration of ℋ (m;t) is proved to be invariant in the following sense: If ϕ: ℋ (m;t) → ℋ (m′t′) is an isomorphism then ϕ(ℋ(m;t)i) = ℋ (m′ t′) i for all i ≥ –1. Using the result, we complete the classification of odd Hamiltonian superalgebras. Finally, we determine the automorphism group of the restricted odd Hamiltonian superalgebra and give further properties.

A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic

Vertex superalgebras over fields of prime characteristic

Lattice vertex algebras over fields of prime characteristic

Twisted modules for affine vertex algebras over fields of prime characteristic

This paper considers centralizers of the Lie superalgebra~$\frak{sl}(0,n)$ over prime characteristic fields. Using homological methods, the centralizers of the even and odd parts of ~$\frak{sl}(0,n)$ in the generalized Witt Lie superalgebra are calculated and a summary of their structural properties is provided.

Let p be a prime number. Denote by C(xm ) and C(xn ) the companion matrices of the polynomials xm and xn of positive degree over the field . Let ρ and σ be non-zero elements of an extension field K of . The Jordan form of the Kronecker product of invertible Jordan block matrices over K is determined via an equivalent study of the nilpotent transformation S(m, n) of the vector space of m × n matrices X over defined by and represented by the Kronecker sum matrix . Using the p-adic expansions of m and n, an inductive method of constructing a Jordan basis for S(m, n) is described; this method is direct and based on classical formulae. The elementary divisors xL of S(m, n) and their multiplicities μ are specified in terms of these p-adic expansions, thus allowing computations in the representation algebra of a finite cyclic p-group to be carried out more readily than previously.

The maximal graded subalgebras for four families of Lie superalgebras of Cartan type over a field of prime characteristic are studied. All maximal graded subalgebras are described completely by a constructive method and their isomorphism classes, dimension formulas are found except for maximal irreducible graded subalgebras. The classification of maximal irreducible graded subalgebras is reduced to the classification of the maximal irreducible subalgebras for the classical Lie superalgebras 𝔤𝔩(m, n), 𝔰𝔩(m, n) and 𝔬𝔰𝔭(m, n).

In spite of many efforts over the past 50 years, the irreducible representations of the Lie algebra of a simple algebraic group over a field of prime characteristic are poorly understood. Recent work on quantum groups at a root of unity has provided new impetus for the subject. This article surveys what has been done and what remains to be done.

A representation of multidifferential polynomials in fields of prime characteristic

The finite-dimensional odd contact Lie superalgebras KO(n, n + 1, t ) over a field of prime characteristic are studied, where n is a positive integer and t is an n-tuple of non-negative integers. In particular, it is proven that KO(n, n + 1, t ) is simple and has no non-singular associative bilinear forms. Moreover, an explicit description of the derivation superalgebra of KO(n, n + 1, t ) is given, and as a consequence it is shown that the outer derivation superalgebra of KO(n, n + 1, t ) is Abelian of dimension . Communicated by K. Misra.

The outer derivation algebras of the Cartan-type Lie superalgebras W, S, H and K over a field of prime characteristic are determined completely in this article. In particular, they are metabelian Lie algebras.

Regular local rings essentially of finite type over fields of prime characteristic

Let G be a finite group having a normal Hall subgroup H, let K be a field, and let T be an irreducible (linear) K-representation of H of degree deg T whose character is invariant under the action of G. We say that T is extendible to G if there exists a K-representation S of G such that S(h) = T(h) for all hCH. In [5, Theorem 6] Gallagher proved that T is extendible if K is the field of complex numbers. The case when K is an arbitrary field of characteristic zero is treated by Isaacs in [7]. In this note we show that the arguments in Isaacs' paper can be extended to yield the following result:

In 1981, Herman and Yoccoz (1983 Generalizations of some theorems of small divisors to non Archimedean fields Geometric Dynamics (Lecture Notes in Mathematics) ed J Palis Jr, pp 408–47 (Berlin: Springer) Proc. Rio de Janeiro, 1981) proved that Siegel's linearization theorem (Siegel C L 1942 Ann. Math. 43 607–12) is true also for non-Archimedean fields. However, the condition in Siegel's theorem is usually not satisfied over fields of prime characteristic. We consider the following open problem from non-Archimedean dynamics. Given an analytic function f defined over a complete, non-trivial valued field of characteristic p > 0, does there exist a convergent power series solution to the Schröder functional equation (2) that conjugates f to its linear part near an indifferent fixed point? We will give both positive and negative answers to this question, one of the problems being the presence of small divisors. When small divisors are present this brings about a problem of a combinatorial nature, where the convergence of the conjugacy is determined in terms of the characteristic of the state space and the powers of the monomials of f, rather than in terms of the diophantine properties of the multiplier, as in the complex case. In the case that small divisors are present, we show that quadratic polynomials are analytically linearizable if p = 2. We find an explicit formula for the coefficients of the conjugacy, and applying a result of Benedetto (2003 Am. J. Math. 125 581–622), we find the exact size of the corresponding Siegel disc and show that there is an indifferent periodic point on the boundary. In the case p > 2 we give a sufficient condition for divergence of the conjugacy for quadratic maps as well as for a certain class of power series containing a quadratic term (corollary 2.1).