Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ? on 𝔤 is said to preserve solvability in both directions if ?, as well as ?^?1, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ? on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.     

Higher Indicators for the Doubles of Some Totally Orthogonal Groups

We investigate the indicators for certain groups of the form ? _k  ? D _l and their doubles, where D _l is the dihedral group of order 2l. We subsequently obtain an infinite family of totally orthogonal, completely real groups which are generated by involutions, and whose doubles admit modules with second indicator of ?1. This provides us with answers to several questions concerning the doubles of totally orthogonal finite groups.     

The Limiting Shape of the Generic Initial System of a Complete Intersection

Consider a complete intersection I of type (d ₁,…, d _r ) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(I ⁿ )} _n obtained by taking the reverse lexicographic generic initial ideals of the powers of I and describe its asymptotic behavior. This behavior is nicely captured by the limiting shape which is shown to depend only on the type of the complete intersection.     

Why the square root function is not linear

Six proofs are given for the fact that for each integer n?2, the nth root function, viewed as a function from the set of non-negative real numbers to itself, is not linear. If p is a prime number, then ?/p? is characterized, up to isomorphism, as the only integral domain D of characteristic p such that D admits a pth root function D→D which is linear. The first part of this note could find classroom use in courses on precalculus or calculus; the second part, in courses on abstract algebra.     

A Note on the Radical of a Module Category

We characterize the finiteness of the representation type of an artin algebra in terms of the behavior of the projective covers and the injective envelopes of the simple modules with respect to the infinite radical of the module category. In case the algebra is representation-finite, we show that the nilpotency of the radical of the module category is the maximal depth of the composites of these maps, which is independent from the maximal length of the indecomposable modules.     

Classification of Involutions on Incidence Algebras

For the case when a finite partially ordered set X has an element that is comparable with all the elements of X, we present necessary and sufficient conditions for two involutions on the incidence algebra of X (over a field of characteristic different from 2) to be equivalent via inner automorphism. Furthermore, we use this classification to get the general classification of involutions.