Tag Archives: lie algebra

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$ […]

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 […]