Tag Archives: algebraic groups

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 […]

From Rational Canonical Form to The Kostant Section

Suppose we have a $2\times 2$ matrix $$ M = \begin{pmatrix} x_{11} & x_{12}\\ x_{21} & x_{22} \end{pmatrix} $$ with entries in a field $F$. The characteristic polynomial of this matrix is $p(t) := {\rm det}(tI_2 – M) = t^2 – (x_{11} + x_{22})t + x_{11}x_{22} – x_{21}x_{12}$. One might ask: how can we produce […]

Finite Normal Subgroups Of Connected Groups Are Central

The previous series on algebraic groups is over. Actually, I barely got to the root system and root datum of a reductive group, but I found that the whole slew of material was getting too complex to organise on this blog, which I feel is better for more self-contained posts. Instead, I have begun to […]

Highlights in Linear Algebraic Groups 12: Radical, Reductive

To analyse the structure of a group G you will need the radical and a torus T. The group of Weyl may also may also suit to prevent the scattering of many a root. Functors are nice including the one of Lie Parabolics bring in the ge-o-metry! The theory of weights may seem oh so […]

Highlights in Linear Algebraic Groups 11: Semisimple Rank 1

In order to understand the structure of reductive groups, we will first look at some “base cases” of groups that are quite small. These are the groups of so-called semisimple rank 1, which by definition are the algebraic groups $ G$ such that $ G/R(G)$ has rank 1, where $ R(G)$ is the connected component […]

Highlights in Linear Algebraic Groups 9: Quotients as Varieties

Conventions: an algebraic group here is a linear algebraic group over a fixed algebraically closed field $ k$. In Highlights 6 and Highlights 7 in this series on algebraic groups, we saw that given any finite dimensional $ k$-subspace $ F\subseteq k[G]$, we can find a finite dimensional subspace $ E\supseteq F$ such that $ […]

Highlights in Linear Algebraic Groups 8: Borel Subgroups I

Borel subgroups are an important type of subgroup that will allow us to gain insight into the mysterious structure of algebraic groups. We shall look at the definition and some basic examples in this post. As usual, algebraic group means some linear algebraic group defined over an algebraically closed field $ k$. A Borel subgroup […]

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 6: Representations I

Soon it will be time to explore some aspects of root systems and structure theory for reductive groups. Our goal is to understand everything in the classical setting over an algebraically closed field, and then explore reductive groups over arbitrary base schemes. Before we do this, I will give a few examples for some of […]

Highlights in Linear Algebraic Groups 2: Lie Algebras

I’ve decided to start this series with a few posts on the Lie algebra of an algebraic group. This seems to me the first real technical aspect of the classical theory that arises in Humphreys’ book. We shall loosely follow this book as a guide, but we shall also deviate and look at more scheme-theoretic […]