Category Archives: algebraic-geometry

Highlights in Linear Algebraic Groups 13: Centralisers of Tori

In Highlights 12, we used some of the equivalent conditions for a connected algebraic group $ G$ over a field $ k=\overline{k}$ to have semisimple rank 1 in the study of reductive groups (these are the groups whose unipotent radical $ R(G)_u$ is trivial). Precisely, we showed that such a $ G$ must have a […]


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 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 […]


Montreal Spring '13 Conferences: Number Theory and Algebra

This is a fairly recent picture of the McGill campus: However, soon the spell of unbearable heat will dawn on the city and there will be plenty of fun things to do outside. Despite the hot sun we shouldn't neglect the indoor activities, such as the many awesome conferences and workshops that will be going […]


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 […]


An Example Using Chevalley Restriction

Here is a classic problem of geometric invariant theory: let $ G$ be a reductive linear algebraic group such as $ \mathrm{GL}_n$ and let $ \mathfrak{g}$ be its Lie algebra. Determine the invariant functions $ k[\mathfrak{g}]^G$, where $ G$ acts on $ \mathfrak{g}$ via the adjoint action. This problem is motivated by the search for […]