Category Archives: group-theory

Highlights in Linear Algebraic Groups 10: G/B is Projective

In Highlights 9 of this series, we showed that for an algebraic group $ G$ and a closed subgroup $ H\subseteq G$, we can always choose a representation $ G\to\rm{GL}(V)$ with a line $ L\subseteq V$ whose stabiliser is $ H$. In turn, this allows us to identify the quotient $ G/H$ with the orbit […]

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 […]

When Are Discrete Subgroups Closed?

Let $ H\subseteq G$ be a subgroup of a topological group $ G$ (henceforth abbreviated "group"). If the induced topology on $ H$ is discrete, then we say that $ H$ is a discrete subgroup of $ G$. A commonplace example is the subgroup $ \mathbb{Z}\subseteq \mathbb{R}$: the integers are normal subgroup of the real […]

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 3: Lie Algebras II

In Linear Algebraic Groups 2, we defined the Lie algebra of an algebraic group $ G$ to be the Lie algebra of all left-invariant derivations $ D:k[G]\to k[G]$ where $ k[G]$ is the representing algebra of $ G$. However, we were left trying to figure out exactly how a morphism $ \varphi:G\to H$ determines a […]

Dihedral Groups and Automorphisms, Part 2

Welcome back readers! In the last post, Dihedral Groups and Automorphisms, Part 1 we introduced the dihedral group. To briefly recap, the dihedral group $ D_n$ of order $ 2n$ for $ n\geq 3$ is the symmetry group of the regular Euclidean $ n$-gon. Any dihedral group is generated by a reflection and a certain […]

Dihedral Groups and Automorphisms, Part 1

Welcome to the second two-post series on AZC! The evil secret plot of this series is to make group cohomology seem interesting for those who have not seen much group cohomology. To do this, we will dissect the dihedral group, which most math majors have probably seen as undergrads. However, to be thorough, the first […]