Solvability Of Algebraic Groups When Dimension Is At Most 2

Jul 14, 2025 by stackunigon 60 views

Exploring the Solvability of Algebraic Groups with Dimension at Most 2

In the realm of algebraic groups, a fundamental question arises concerning the relationship between the dimension of a connected algebraic group and its solvability. Specifically, we delve into proving that if the dimension of a connected algebraic group G is less than or equal to 2, then G is solvable. This exploration will take us through various cases based on the dimension of G, leveraging key concepts and theorems from the theory of algebraic groups.

Introduction to Solvable Algebraic Groups

Before diving into the proof, let's establish a clear understanding of solvable algebraic groups. In the context of group theory, a group G is termed solvable if there exists a chain of subgroups:

G = G₀ ▹ G₁ ▹ ... ▹ Gₙ = {e}

where each Gᵢ is a normal subgroup of Gᵢ₋₁, and the quotient group Gᵢ₋₁/ Gᵢ is abelian for all i. This definition extends naturally to algebraic groups, where the subgroups are required to be closed in the Zariski topology.

The solvability of an algebraic group is a crucial property that has significant implications for its structure and representation theory. Solvable groups exhibit a hierarchical structure that allows for a systematic analysis of their subgroups and homomorphisms. Understanding the conditions under which an algebraic group is solvable is thus a central theme in the study of these groups.

Case 1: Dimension of G Equals 0

The simplest case to consider is when the dimension of G is 0. Recall that the dimension of an algebraic group is the dimension of its underlying algebraic variety. If dim(G) = 0, it implies that G is a finite group. A finite algebraic group is necessarily solvable. To see this, consider the derived series of G, defined iteratively as D₀G = G and Dᵢ₊₁G = [DᵢG, DᵢG], where [A, B] denotes the commutator subgroup generated by elements of the form aba⁻¹b⁻¹, with a in A and b in B. Since G is finite, the derived series must eventually stabilize, i.e., there exists an n such that DₙG = Dₙ₊₁G. If DₙG is trivial, then G is solvable. Otherwise, DₙG is a perfect group (equal to its own commutator subgroup), which must be trivial since G is finite. Therefore, if dim(G) = 0, then G is solvable.

Case 2: Dimension of G Equals 1

Now, let's consider the case where dim(G) = 1. This scenario is more intricate and requires a deeper understanding of the structure of one-dimensional algebraic groups. A fundamental result in the theory of algebraic groups states that a connected algebraic group of dimension 1 is isomorphic either to the additive group Ga or the multiplicative group Gm. Here, Ga is the algebraic group corresponding to the affine line with addition as the group operation, and Gm is the algebraic group corresponding to the non-zero elements of the field with multiplication as the group operation.

Both Ga and Gm are abelian groups, and any abelian group is solvable. To verify this, consider the series G₀ = G and G₁ = {e}, where e is the identity element. Since G is abelian, the quotient group G/{e} is isomorphic to G, which is abelian. Thus, the group G is solvable. Consequently, if dim(G) = 1, then G is solvable, as it is isomorphic to either Ga or Gm, both of which are abelian and therefore solvable.

Case 3: Dimension of G Equals 2

The most challenging and interesting case is when dim(G) = 2. To prove that G is solvable in this case, we invoke the Borel Fixed Point Theorem. This powerful theorem states that if a solvable algebraic group G acts morphically on a complete algebraic variety X, then there exists a point in X that is fixed by the action of G. The Borel Fixed Point Theorem is a cornerstone in the theory of algebraic groups and provides a crucial link between solvability and group actions.

Let B be a Borel subgroup of G. A Borel subgroup is a maximal connected solvable subgroup. The existence of Borel subgroups is a fundamental result in the theory of algebraic groups. The quotient space G/B is a complete variety known as the flag variety. Since G acts on itself by left multiplication, it also acts on G/B. This action is a morphism, and since G/B is a complete variety, we can apply the Borel Fixed Point Theorem if G is solvable.

However, we are trying to prove that G is solvable. Instead, we consider a Borel subgroup B of G. If dim(G) = 2, then the dimension of any Borel subgroup B must be less than or equal to 2. If dim(B) = 2, then B = G, and G is solvable by definition. If dim(B) = 1, then B is isomorphic to either Ga or Gm, both of which are solvable. If dim(B) = 0, then B is trivial, which is also solvable.

The key idea is to consider the adjoint representation of G on its Lie algebra Lie(G). The Lie algebra Lie(G) is a vector space that captures the infinitesimal structure of the algebraic group G. The adjoint representation is a homomorphism Ad: G → GL(Lie(G)), where GL(Lie(G)) is the general linear group of Lie(G). The image of the adjoint representation, denoted Ad(G), is an algebraic subgroup of GL(Lie(G)).

If dim(G) = 2, then dim(Lie(G)) = 2. The adjoint representation gives us a homomorphism from G to GL₂(K), where K is the underlying field. The kernel of this homomorphism, denoted Ker(Ad), is the center of G, which is a closed normal subgroup. The image Ad(G) is an algebraic subgroup of GL₂(K), and its dimension is at most 2.

Now, consider the derived subgroup D(G) of G. The derived subgroup is the subgroup generated by all commutators aba⁻¹b⁻¹, where a, b ∈ G. The derived subgroup measures the non-commutativity of G. If D(G) is a proper subgroup of G, then its dimension is less than 2. If dim(D(G)) = 0 or 1, then D(G) is solvable, as we have already shown. If D(G) is solvable, then G is solvable.

If D(G) = G, then G is perfect. However, in the case where dim(G) = 2, a connected algebraic group cannot be perfect unless it is semisimple. If G is semisimple, then it is a product of simple algebraic groups. The only simple algebraic groups of dimension at most 2 are isomorphic to SL₂(K) or PSL₂(K), which have dimension 3. Thus, G cannot be semisimple, and D(G) must be a proper subgroup.

Therefore, if dim(G) = 2, G must be solvable. This completes the proof for the case where the dimension of G is 2.

Conclusion

In summary, we have demonstrated that if the dimension of a connected algebraic group G is less than or equal to 2, then G is solvable. This result was established by considering three cases: dim(G) = 0, dim(G) = 1, and dim(G) = 2. In each case, we utilized fundamental concepts and theorems from the theory of algebraic groups, such as the solvability of finite groups, the structure of one-dimensional algebraic groups, and the Borel Fixed Point Theorem, to arrive at the desired conclusion. This exploration underscores the deep connections between the dimension and solvability of algebraic groups, providing valuable insights into their structure and properties.