# 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 […]