Tag Archives: algebraic group

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} […]

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 […]

Highlights in Linear Algebraic Groups 4: Lie Algebras III

Last time in this series, we saw the definition of the Lie algebra of a linear algebraic group $ G$ over a arbitrary field $ k$ as the set of differentiations $ f:k[G]\to k$; these are the $ k$-linear maps satisfying $ f(ab) = \epsilon(a)f(b) + f(a)\epsilon(b)$ where $ \epsilon:k[G]\to k$ is the counit morphism […]

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 […]