Special Linear Group Scheme SL_n

Definition (functor of points)

For a commutative ring \(R\), define \[ \mathrm{SL}_n(R) = \{A \in M_n(R) : \det A = 1\}. \] This rule \(R \mapsto \mathrm{SL}_n(R)\) is a representable functor, represented by the affine group scheme \[ \mathrm{SL}_n = \operatorname{Spec}\bigl(\mathbb{Z}[x_{ij}]_{1\le i,j\le n}/(\det(x_{ij})-1)\bigr). \]

Coordinate Hopf algebra

The coordinate ring is \[ \mathcal{O}(\mathrm{SL}_n) = \mathbb{Z}[x_{ij}]/(\det - 1). \] The Hopf algebra structure is inherited from \(\mathcal{O}(\mathrm{GL}_n)\):

• Comultiplication \(\Delta(x_{ij}) = \sum_k x_{ik} \otimes x_{kj}\) (matrix multiplication).
• Counit \(\varepsilon(x_{ij}) = \delta_{ij}\) (identity matrix).
• Antipode \(S(x_{ij}) = (-1)^{i+j} M_{ji}\), where \(M_{ji}\) is the \((j,i)\)-minor (inverse via Cramer's rule; the determinant constraint makes this polynomial, not requiring inversion).

As kernel of the determinant

\(\mathrm{SL}_n\) is the kernel of the determinant character \[ \det : \mathrm{GL}_n \longrightarrow \mathbb{G}_m,\quad A \mapsto \det A. \] That is, \(\mathrm{SL}_n = \ker(\det)\) in the category of group schemes. This is a closed immersion of group schemes, making \(\mathrm{SL}_n\) a normal closed subgroup scheme of \(\mathrm{GL}_n\).

Structure

• Dimension: \(n^2 - 1\) (removing one degree of freedom from \(GL_n\)).
• Smoothness: \(\mathrm{SL}_n\) is smooth over \(\mathbb{Z}\) (the determinant map is surjective on tangent spaces; equivalently, the Jacobian criterion holds). In particular, it is a smooth group scheme over any base.
• Simply connected: \(\mathrm{SL}_n\) is the simply connected cover in its isogeny class (root datum \(A_{n-1}\) with weight lattice as character group).
• Quotient: The quotient \(\mathrm{GL}_n / \mathrm{SL}_n \cong \mathbb{G}_m\) via the determinant.
• Centre: \(Z(\mathrm{SL}_n) = \mu_n\) (the \(n\)-th roots of unity), embedded as scalar matrices \(\zeta I_n\).

Small cases

• \(\mathrm{SL}_1 = \{1\}\) (trivial group).
• \(\mathrm{SL}_2\): the simply connected cover of \(\mathrm{PGL}_2\); root datum \(A_1\); over \(\mathbb{C}\), isomorphic to \(\mathrm{SU}(2)\) as a real Lie group.
• \(\mathrm{SL}_n/\mu_n \cong \mathrm{PGL}_n\) (the adjoint form of type \(A_{n-1}\)).

Lie algebra

\[ \operatorname{Lie}(\mathrm{SL}_n) = \mathfrak{sl}_n = \{X \in M_n : \operatorname{tr}(X) = 0\}. \] Dimension \(n^2-1\); the adjoint representation on \(\mathfrak{sl}_n\) is the adjoint representation of the Lie algebra.

Root datum

\(\mathrm{SL}_n\) with maximal torus \(T = \{\mathrm{diag}(t_1,\dots,t_n) : \prod t_i = 1\}\) has \[ X^*(T) = \{(a_1,\dots,a_n)\in\mathbb{Z}^n : \sum a_i = 0\},\quad \Phi = \{\varepsilon_i - \varepsilon_j : i\ne j\}. \] This is the root datum of type \(A_{n-1}\), simply connected form.