Special linear group

Description

Let $F$ be a field and let $n \ge 1$. The special linear group is the set $$SL_n(F) = \{A \in GL_n(F) : \det A = 1\}$$ of $n \times n$ matrices over $F$ with determinant $1$, with matrix multiplication. It is a subgroup of the general linear group $GL_n(F)$.

Remarks

It is the kernel of the group homomorphism $\det \colon GL_n(F) \to F^\times$, so it is a normal subgroup. The quotient $GL_n(F) / SL_n(F)$ is isomorphic to $F^\times$. Over $\R$, its elements are the linear maps that keep volume and orientation. $SL_2(\Z)$ is the analogous group of integer matrices with determinant $1$.

Explanations

L1 human

The special linear group collects the matrices with determinant $1$. In the real case, these maps keep the volume of every region, and they do not flip orientation. Shears, such as $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$, are typical elements.