Category Archives: group-theory

## The Torsion Subgroup of an Abelian Group

Let $A$ be an abelian group. We call an element $a\in A$ torsion if there exists a natural number $n$ such that $na = 0$. The set of all torsion elements $T(A)$ of $A$ form a subgroup of $A$, and we can think of $T$ as an endofunctor on the category of abelian groups. Here […]

## 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 $(-1)^\sigma$ […]