Category Archives: group-theory

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 […]

Guess The Algebraic Group

Suppose one day you run into the following algebraic group, defined on $\mathbb{Z}$-algebras $R$ by $$G(R) = \left\{ \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{11} & -a_{14} & -a_{13} \\ a_{13} & -a_{14} & a_{11} & -a_{12} \\ a_{14} & -a_{13} & -a_{12} & a_{11} \end{pmatrix} \in\mathrm{GL}_4(R) \right\}$$ Can you […]

Determinants, Permutations and the Lie Algebra of SL(n)

Here is an old classic from linear algebra: given an $ n\times n$ matrix $ A = (a_{ij})$, the determinant of $ A$ can be calculated using the permuation formula for the determinant: $ \det(A) = \sum_{\sigma\in S_n} (-1)^\sigma a_{1\sigma(1)}\cdots a_{n\sigma(n)}$. Here $ S_n$ denotes the permutation group on $ n$ symbols and $ (-1)^\sigma$ […]

2 Dimensional Connected Algebraic Groups are Solvable

Conventions: $ G$ is an algebraic group over an algebraically closed field $ k$ and we identify $ G$ with $ G(k)$. Consider the algebraic groups $ \mathbb{G}_a$ and $ \mathbb{G}_m$. They are the only one-dimensional connected groups and they are both solvable. What about two-dimensional connected groups? It turns out that if $ \mathrm{dim} […]

Visualising the Real Orthogonal Group

In the post Can You See in Four Dimensions?, we saw some ways of visualising functions plotted in four-dimensional 'space' in various ways. Of course, we used colour and time for two dimensions because it is a bit difficult to plot in four actual spatial dimensions! Here is another example: the orthogonal group $ \mathrm{O}_2(\mathbb{R})$ […]

Highlights in Linear Algebraic Groups 14: Singular Tori

From Highlights 12 and Highlights 13, we have gained quite a bit of information on connected reductive groups $ G$ of semisimple rank 1. Recall, this means that $ G/R(G)$ has rank 1 where $ R(G)$ is the radical of $ G$, which is in turn the connected component of the unique maximal normal solvable […]

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 […]

Wild Spectral Sequences Ep. 3: Cohomological Dimension

Last time in Wild Spectral Squences 2, we saw how to prove the five lemma using a spectral sequence. Today, we'll see a very simple application of spectral sequences to the concept of cohomological dimension in group cohomology. We will define the well-known concept of cohomological dimension of a group $ G$, and then show […]

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 […]