Semisimple Groups
A semisimple algebraic group is a connected reductive group with trivial geometric radical (equivalently, with finite centre). The semisimple groups are the "purest" reductive groups — those that decompose as almost-products of simple factors.
Definition
Let \(G\) be a connected linear algebraic group over a field \(k\), and let \(R(G_{\bar k})\) denote the geometric radical (maximal connected solvable normal subgroup of \(G_{\bar k}\)).
\(G\) is semisimple if \(R(G_{\bar k}) = 1\).
Equivalently: \(G\) is reductive with finite centre (in good characteristic).
Almost-direct product decomposition
Theorem. A connected semisimple algebraic group \(G\) over an algebraically closed field is the almost-direct product of finitely many simple factors: \[ G \;=\; G_1 \cdot G_2 \cdots G_k, \] with each \(G_i\) simple (no normal connected proper subgroups), and the \(G_i\) commuting; their intersections are finite central subgroups.
This is the algebraic counterpart of "compact Lie group decomposes as a quotient of a product of simply-connected simple compact groups by a finite central subgroup."
Classification
Connected simple algebraic groups over \(\bar k\) (good characteristic) are classified by their Dynkin diagram:
\[ A_n, \quad B_n, \quad C_n, \quad D_n, \quad E_6, \quad E_7, \quad E_8, \quad F_4, \quad G_2. \]
Within each Dynkin type, the possible groups form a isogeny class parametrised by the lattice gap \(Q^\vee \subseteq X_* \subseteq P^\vee\):
- Simply connected form: \(X_* = Q^\vee\), trivial fundamental group, e.g.\ \(\mathrm{SL}_n, \mathrm{Spin}_n, \mathrm{Sp}_{2n}\).
- Adjoint form: \(X_* = P^\vee\), trivial centre, e.g.\ \(\mathrm{PGL}_n, \mathrm{PSO}_n, \mathrm{PSp}_{2n}\).
- Intermediate forms exist for types where \(|P^\vee/Q^\vee| > 1\).
Examples
| Group | Type | Form | Centre | \(\pi_1\) |
|---|---|---|---|---|
| \(\mathrm{SL}_n\) | \(A_{n-1}\) | sc | \(\mu_n\) | 0 |
| \(\mathrm{PGL}_n\) | \(A_{n-1}\) | ad | 1 | \(\mathbb{Z}/n\) |
| \(\mathrm{Spin}_{2n+1}\) | \(B_n\) | sc | \(\mu_2\) | 0 |
| \(\mathrm{SO}_{2n+1}\) | \(B_n\) | ad | 1 | \(\mathbb{Z}/2\) |
| \(\mathrm{Sp}_{2n}\) | \(C_n\) | sc | \(\mu_2\) | 0 |
| \(\mathrm{PSp}_{2n}\) | \(C_n\) | ad | 1 | \(\mathbb{Z}/2\) |
| \(\mathrm{Spin}_{2n}\) | \(D_n\) | sc | finite | 0 |
| \(\mathrm{SO}_{2n}\) | \(D_n\) | intermediate | \(\mu_2\) | \(\mathbb{Z}/2\) |
| \(G_2\) | \(G_2\) | sc = ad | 1 | 0 |
| \(F_4\) | \(F_4\) | sc = ad | 1 | 0 |
| \(E_6\) sc | \(E_6\) | sc | \(\mu_3\) | 0 |
| \(E_7\) sc | \(E_7\) | sc | \(\mu_2\) | 0 |
| \(E_8\) | \(E_8\) | sc = ad | 1 | 0 |
Universal cover (sc cover)
For any connected semisimple group \(G\), there is a canonical simply connected cover: \[ 1 \;\to\; \mu \;\to\; G^{\mathrm{sc}} \;\to\; G \;\to\; 1, \] where \(\mu = \ker(G^{\mathrm{sc}} \to G)\) is a finite central subgroup of multiplicative type, and \(G^{\mathrm{sc}}\) is simply connected (\(\pi_1 = 0\)).
Dually: the adjoint quotient \(G^{\mathrm{ad}} := G/Z(G)\), with \(\pi_1(G^{\mathrm{ad}}) = Z(G^{\mathrm{sc}})\) of size \(|P^\vee / Q^\vee|\).
Relation to reductive groups
For any connected reductive \(G\): \[ G \;=\; Z(G)^\circ \cdot G_{\mathrm{der}}, \] where \(G_{\mathrm{der}}\) is the derived group, semisimple, and \(Z(G)^\circ\) is the connected centre (a torus). The intersection \(Z(G)^\circ \cap G_{\mathrm{der}}\) is a finite group of multiplicative type.
So semisimple groups are exactly the derived groups of reductive groups, modulo the central torus.
Anisotropic semisimple groups
\(G\) semisimple is anisotropic if \(G\) has no proper parabolic \(k\)-subgroup (equivalently, no \(k\)-rational maximal split torus of positive dimension).
Over \(\mathbb{R}\): \(G\) anisotropic iff \(G(\mathbb{R})\) is compact.
Over local fields: anisotropic groups are exactly the inner forms of \(\mathrm{SL}_1(D)\) for \(D\) a central simple division algebra.
Role in Langlands
- Langlands functoriality: most conjectures are stated for connected reductive groups, but the deepest non-trivial cases are semisimple.
- Discrete series representations: parametrised by regular characters of compact-modulo-centre maximal tori; the existence of such tori requires the group to be of equal rank with its maximal compact subgroup (a constraint that is automatic for semisimple groups in many cases).
- Galois representations: for \(G = \mathrm{SL}_n\), the Langlands correspondence gives the deepest known instances of the Fontaine-Mazur conjecture.
- Geometric Langlands: for semisimple \(G\), \(\mathrm{Bun}_G\) is connected, and the Langlands functoriality has a cleaner shape.
Mathlib
Mathlib.LinearAlgebra.RootSystem.IsCrystallographic etc.\ provides
the abstract Dynkin/root-system infrastructure; explicit semisimple
algebraic groups are not yet there.
References
- Borel, Linear Algebraic Groups (1991), §14.
- Springer, Linear Algebraic Groups (1998), §8.
- Humphreys, Linear Algebraic Groups (1975), §27.
- Tits, Classification of algebraic semisimple groups (Proc.\ AMS Symp.\ 1966).