Tag Archives: hopf algebras

Homomorphisms from G_a to G_m

Let $k$ be a commutative ring. Let $\G_a$ be group functor $\G_a(R) = R$ and $\G_m$ be the group functor $\G_m(R) = R^\times$, both over the base ring $k$. What are the homomorphisms $\G_a\to \G_m$? In other words, what are the characters of $\G_a$? This depends on the ring, of course! The representing Hopf algebra […]

Highlights in Linear Algebraic Groups 7: Representations II

In the previous post, we saw that if $G\times X\to X$ is an algebraic group acting on a variety $X$ and $F\subseteq k[X]$ is a finite-dimensional subspace then there exists a finite dimensional subspace $E\subseteq k[X]$ with $E\supseteq F$ such that $E$ is invariant under translations. Recall that if […]

Highlights in Linear Algebraic Groups 1: Introduction

As in many mathematics departments, graduate students in McGill's Department of Mathematics and Statistics have to take a comprehensive examination comprising of two parts: a written part (Part A) and an oral part (Part B). The Part B exam is based on two topics related to the student's field of research. One of my Part […]