Fernando Szechtman

Characterization of bilinear spaces with unimodular isometry group

ABSTHACT. We study finite-dimensional bilinear spaces and their isometry groups. To each bilinear space V we associate two canonical filtratiuns, which yield structural results on V. Prominent among these is an explicit formula for the number of indecomposable and degenerate blocks of V of a given dimension. Equipped with this material, we proceed to characterize those bilinear spaces whose isometry group is contained in the special linear group. This characterization can easily be implemented in practice by means of an algorithm. As an application, we determine the real n-by-n matrices whose congruence class is disconnected.

On Tate's trace

The problem of whether Tates trace is linear or not is reduced to a special case.

Indecomposable Modules of 2-step Solvable Lie Algebras in Arbitrary Characteristic

Let F be an algebraically closed field and consider the Lie algebra 𝔤 = ⟨ x ⟩ ⋉ 𝔞, where ad x acts diagonalizably on the abelian Lie algebra 𝔞. Refer to a 𝔤-module as admissible if [𝔤, 𝔤] acts via nilpotent operators on it, which is automatic if chr(F) = 0. In this article, we classify all indecomposable 𝔤-modules U which are admissible as well as uniserial, in the sense that U has a unique composition series.

A closed formula for the product in simple integral extensions

Abstract Let ξ be an algebraic number and let α , β ∈ Q [ ξ ] . A closed formula for the coordinates of the product α β is given in terms of the coordinates of α and β and the companion matrix of the minimal polynomial of ξ . The formula as well as its proof extend to fairly general simple integral extensions.

Clifford theory for infinite dimensional modules

ABSTRACT Clifford theory of possibly infinite dimensional modules is studied.

UNITARY GROUPS OVER LOCAL RINGS

Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified extension of finite local rings.

Jordan-Chevalley decomposition in finite dimensional Lie algebras

Let g be a finite dimensional Lie algebra over a field k of characteristic zero. An element x of g is said to have an abstract Jordan-Chevalley decomposition if there exist unique s, n ∈ g such that x = s + n, [s, n] = 0 and given any finite dimensional representation π: g → gl(V) the Jordan-Chevalley decomposition of π(x) in gl(V) is π(x) = π(s) + π(n). In this paper we prove that x ∈ g has an abstract Jordan-Chevalley decomposition if and only if x ∈ [g, g], in which case its semisimple and nilpotent parts are also in [g, g] and are explicitly determined. We derive two immediate consequences: (1) every element of g has an abstract Jordan-Chevalley decomposition if and only if g is perfect; (2) if g is a Lie subalgebra of gl(n, k), then [g, g] contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods. Our proof uses only elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lies Theorem, in addition to the fundamental theorems of Ado and Levi.

Classification of finite dimensional uniserial representations of conformal Galilei algebras

With the aid of the

Generalized Artin–Schreier polynomials

6j

Reductivity of the Lie Algebra of a Bilinear Form

-symbol, we classify all uniserial modules of