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$ denotes the sign of the permutation $ \sigma$. The computation of the determinant is much more easily done via the ‘minor expansion method’, but this just reduces to the above formula. Now, typically we would never use the above formula to calculate the determinant, but that doesn’t mean it isn’t useful!

The Lie Algebra of an Algebraic Group

In fact, computating the Lie algebra of $ \mathrm{SL}_n$ over a fixed commutative ring $ k$ happens to be a quick and interesting application of the above determinant formula. First, let us recall the definition of the Lie algebra, or at least one of the many equivalent definitions. Consider the algebra of dual numbers, $ k[\tau]/(\tau^2)$. Each element in this algebra can be written uniquely as $ a + b\tau$: this gives a $ k$-algebra map $ C: k[\tau]/(\tau^2)\to k$ by sending $ a + b\tau$ to $ a$.
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