Algebraic Groups

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Algebraic Groups

We know many examples of mathematical objects acting on other objects. Algebraically, the most important instance is perhaps a group acting on a vector space; geometrically, there are Lie group actions. Since algebraic geometry endows algebraic objects with geometric meaning, the action of an algebraic group naturally integrates these two perspectives. In this post, for convenience we set \(\mathbb{k}=\mathbb{C}\) throughout.

First, the following definition is obvious.

Definition 1 An algebraic group \(G\) is an algebraic variety satisfying the following conditions:

1. \(G\) carries a group structure.
2. Multiplication \(m: G \times G \to G\) and inverse \(i: G \to G\) are both morphisms of varieties.

As with Lie groups, the most important examples are usually matrix groups.

Example 2 The most basic examples are as follows:

1. The general linear group \(\GL(n, \mathbb{C}) = \{A \in M_{n \times n}(\mathbb{C}) \mid \det A \ne 0\}\) has the structure of an algebraic group as an open subvariety of \(\mathbb{C}^{n\times n}\).
2. The special linear group \(\SL(n, \mathbb{C}) = \{A \in \GL(n, \mathbb{C}) \mid \det A = 1\}\) is an algebraic group as a closed subvariety of \(\GL(n, \mathbb{C})\).
3. The two abelian groups \(\mathbb{G}_a = \mathbb{C}\) (addition) and \(\mathbb{G}_m = \mathbb{C}^\ast\) (multiplication) are both one-dimensional algebraic groups.

Among algebraic groups, those that play a particularly important role are, of course, affine algebraic groups.

Definition 3 An algebraic group \(G\) is called an affine algebraic group if \(G\) is an affine variety.

Actions of Algebraic Groups

We now define how an algebraic group acts on an algebraic variety.

Definition 4 An action of an algebraic group \(G\) on an algebraic variety \(X\) is a morphism

\[\alpha: G \times X \to X\]

satisfying the following conditions:

1. \(\alpha(e, x) = x\) for all \(x \in X\) (identity)
2. \(\alpha(g, \alpha(h, x)) = \alpha(gh, x)\) for all \(g, h \in G\), \(x \in X\) (associativity)

In principle, an algebraic group action is built by examining how affine algebraic groups act on affine varieties and then gluing these local pictures together. To look more closely at this case, consider an affine algebraic group \(G = \Spec(A)\) acting on an affine variety \(X = \Spec(B)\). Since \(\Spec\) is a contravariant functor, the action \(G \times X \to X\) translates into a structure on the coordinate ring. Specifically, it yields the following data.

Definition 5 For an affine algebraic group \(G = \Spec(A)\) and an affine variety \(X = \Spec(B)\), a comodule structure corresponding to an action of \(G\) on \(X\) is an algebra homomorphism

\[\rho: B \to B \otimes_\mathbb{C} A\]

satisfying the following conditions:

1. (Coassociativity) \((\rho \otimes \id_A) \circ \rho = (\id_B \otimes \Delta) \circ \rho\)
2. (Counit) \((\id_B \otimes \epsilon) \circ \rho = \id_B\)

Here \(\Delta: A \to A \otimes A\) is the comultiplication induced from the multiplication on \(G\), and \(\epsilon: A \to \mathbb{C}\) is the counit induced from the identity element.

Representations of Algebraic Groups

Just as with Lie groups, algebraic groups can be better understood through representation theory.

Definition 6 A representation of an algebraic group \(G\) is a finite-dimensional vector space \(V\) together with a group homomorphism

\[\rho: G \to \GL(V)\]

such that the induced map \(G \times V \to V\) is a morphism.

Then the following definition is also the same as for Lie groups.

Definition 7 The character \(\chi_\rho: G \to \mathbb{C}\) of a representation \(\rho: G \to \GL(V)\) is defined by

\[\chi_\rho(g) = \tr(\rho(g))\]

Moreover, as the following proposition shows, representations of algebraic groups simultaneously carry both algebraic and geometric natures.

Proposition 8 A representation \(\rho: G \to \GL(V)\) of an affine algebraic group \(G = \Spec(A)\) is in one-to-one correspondence with a comodule structure \(V \to V \otimes_\mathbb{C} A\) on \(V\).

Proof

Given a representation \(\rho: G \to \GL(V)\), this induces a map \(G \times V \to V\), and by the contravariance of \(\Spec\) we obtain \(V^\ast \to V^\ast \otimes A\).

Conversely, given a comodule structure \(V \to V \otimes A\), for each \(g \in G\) we obtain \(V \to V \otimes \mathbb{C} \cong V\) via the evaluation map \(\operatorname{ev}_g: A \to \mathbb{C}\), and this defines a representation.

Algebraic Tori and Weight Decomposition

Among algebraic groups, one of the objects we encounter most frequently is a torus \(T\). We have already examined the one-dimensional torus. (Example 2)

Definition 9 An algebraic torus is an algebraic group isomorphic to a finite direct sum of copies of \(\mathbb{G}_m = \mathbb{C}^\ast\). That is, there exists \(n \ge 1\) such that

\[T \cong (\mathbb{C}^\ast)^n\]

In the theory of Lie groups, we saw that a torus decomposes into one-dimensional representations and that the information about each of these is encoded in its characters. Let us carry out the same program here.

Definition 10 A character \(\rchi: T \to \mathbb{C}^\ast\) of a torus \(T = (\mathbb{C}^\ast)^n\) is defined as follows. For each coordinate \(t_i \in \mathbb{C}^\ast\),

\[\rchi(t_1, \ldots, t_n) = t_1^{a_1} \cdots t_n^{a_n}\]

where \(a_i \in \mathbb{Z}\) and \(a = (a_1, \ldots, a_n) \in \mathbb{Z}^n\).

The product of two characters \(\rchi^a\) and \(\rchi^b\) is defined by

\[\rchi^a \cdot \rchi^b = \rchi^{a+b}\]

This corresponds to addition in \(\mathbb{Z}^n\)

\[a + b = (a_1 + b_1, \ldots, a_n + b_n)\]

The set of characters \(X^\ast(T) = \{\rchi^a \mid a \in \mathbb{Z}^n\}\) forms an abelian group under this product; it is called the character group of the torus \(T\).

It is not difficult to show that \(\rchi: T \to \mathbb{C}^\ast\) is a group homomorphism.

Proposition 11 The coordinate ring of a torus \(T = (\mathbb{C}^\ast)^n\) is isomorphic to the polynomial ring in the characters \(\mathbb{C}[\rchi_1^{\pm 1}, \ldots, \rchi_n^{\pm 1}]\). Here \(\rchi_i(t_1, \ldots, t_n) = t_i\) is the character corresponding to the \(i\)-th coordinate.

Proof

Since the torus \(T = (\mathbb{C}^\ast)^n\) is the direct product of \(n\) copies of \(\mathbb{C}^\ast\), its coordinate ring is the tensor product of the coordinate rings of each factor. We know that the coordinate ring of \(\mathbb{C}^\ast = \mathbb{C} \setminus \{0\}\) is \(\mathbb{C}[x, x^{-1}]\). (§Schemes, ⁋Example 10)

Hence the coordinate ring of \(T\) is

\[\mathbb{C}[x_1, x_1^{-1}] \otimes \cdots \otimes \mathbb{C}[x_n, x_n^{-1}] \cong \mathbb{C}[x_1, x_1^{-1}, \ldots, x_n, x_n^{-1}]\]

Each \(x_i\) corresponds to the \(i\)-th coordinate function \(\rchi_i: T \to \mathbb{C}^\ast\), defined by \(\rchi_i(t_1, \ldots, t_n) = t_i\). This is exactly a character of the torus, and its inverse \(\rchi_i^{-1}\) is likewise a character.

Therefore the coordinate ring of \(T\) coincides with the polynomial ring generated by all the characters.

When a torus \(T\) acts on an affine variety \(X = \Spec(A)\), the coordinate ring \(A\) decomposes into weight spaces.

Proposition 12 When a torus \(T\) acts on an affine variety \(X = \Spec(A)\), the coordinate ring \(A\) decomposes into weight spaces:

\[A = \bigoplus_{\rchi \in X^\ast(T)} A_\rchi\]

Here \(X^\ast(T)\) is the character group of the torus, and each weight space \(A_\rchi\) is defined by

\[A_\rchi = \{f \in A \mid t \cdot f = \rchi(t) f \text{ for all } t \in T\}\]

Proof

Given a torus action \(T \times X \to X\), we can define an action on the coordinate ring \(A\) by \(t \cdot f = f(t)\). This action is linear, and each element \(f\) of the coordinate ring can be decomposed into eigenvectors with eigenvalues. Since these eigenvalues coincide with the characters of the torus, \(A\) decomposes as a direct sum of weight spaces.

Quotient Varieties

On an algebraic variety with a given action, we often wish to construct a quotient variety. However, such objects are well defined primarily in algebraic settings; even for topological spaces, the simplest geometric objects, this is not always well defined. The same holds in the world of algebraic varieties.

Definition 13 When an algebraic group \(G\) acts on an algebraic variety \(X\), we define the following.

• Orbit: \(G \cdot x = \{g \cdot x \mid g \in G\} \subseteq X\)
• Stabilizer: \(G_x = \{g \in G \mid g \cdot x = x\} \le G\)
• Fixed point set: \(X^G = \{x \in X \mid g \cdot x = x \text{ for all } g \in G\}\)

To construct a quotient, we must examine \(G\)-invariant functions.

Definition 14 When an affine algebraic group \(G\) acts on an affine variety \(X = \Spec(A)\), the invariant ring \(A^G\) with respect to the corresponding comodule structure \(\rho: A \to A \otimes_\mathbb{C} \mathbb{C}[G]\) is defined by

\[A^G = \{a \in A \mid \rho(a) = a \otimes 1\}\]

This is a subalgebra of \(A\).

The elements of the invariant ring capture the symmetries that are unchanged by the group action. Regarding elements of the coordinate ring \(A\) as functions on \(\Spec A\), the elements of \(A^G\) become constant on each orbit. Therefore the space \(\Spec(A^G)\) constructed from them can be regarded as a good approximation to the orbit space \(X/G\). The difficulty, however, is that \(A^G\) need not behave well algebraically.

Definition 15 An affine algebraic group \(G\) is called reductive if every finite-dimensional representation of \(G\) is completely reducible. That is, any representation \(V\) decomposes as a direct sum of irreducible representations.

If \(G\) is a reductive group, then by the Hilbert basis theorem and Nagata’s theorem one can show that the invariant ring \(A^G\) is always finitely generated, and hence \(\Spec(A^G)\) is a well-defined affine variety.

Example 16 Most algebraic groups we deal with are reductive.

1. The algebraic torus \((\mathbb{C}^\ast)^n\) is reductive.
2. The general linear group \(\GL(n, \mathbb{C})\) is reductive.
3. The special linear group \(\SL(n, \mathbb{C})\) is reductive.
4. The orthogonal group and unitary group \(\operatorname{O}(n)\), \(\operatorname{U}(n)\), etc., are also reductive.

By contrast, \(\mathbb{G}_a = \mathbb{C}\) is not reductive.

The content regarding the finite generation of \(A^G\) mentioned after Definition 15 is usually treated in Geometric Invariant Theory (GIT), and through this theory we can define the quotient for a reductive group action.

Definition 17 When a reductive group \(G\) acts on an affine variety \(X = \Spec(A)\), the GIT quotient \(X /\!/ G\) is defined by

\[X /\!/ G = \Spec(A^G)\]

As discussed above, the GIT quotient \(X /\!/ G\) is geometrically a good approximation to the orbit space \(X/G\); this means that the GIT quotient satisfies the universal property expected of a quotient.

The discussion so far has concerned affine varieties; the situation is more complicated for projective varieties. For example, suppose \(\mathbb{C}^\ast\) acts on \(\mathbb{P}^1 = \{[x:y]\}\) by

\[t \cdot [x:y] = [tx:y]\]

In the homogeneous coordinate ring \(\mathbb{C}[x,y]\), if \(t\) acts by scaling \(x\) by \(t\) and \(y\) by \(1\), then in the degree \(d\) component \(\mathbb{C}_d[x,y]\) the monomial \(x^a y^{d-a}\) is acted on by \(t\) as \(t^a\). That is, each monomial carries a different weight.

In this case the only \(G\)-invariant subspace of \(\mathbb{C}_d[x,y]\) is spanned by \(y^d\), and \(A_d^G = \mathbb{C} \cdot y^d\) is well defined. However, for an action on a general projective variety, if the action does not extend to all of \(\mathbb{P}^n\), the grading and the action may fail to be compatible and \(A_d^G\) may not be well defined.

Definition 18 A linearization of an action of a reductive group \(G\) on a projective variety \(X\) is an extension of the action to an action of \(G\) on \(\mathbb{P}^n\) satisfying the following:

1. \(X \subseteq \mathbb{P}^n\) is \(G\)-invariant
2. The \(G\)-action on \(\mathbb{P}^n\) is linear (that is, it lifts to \(\GL(n+1, \mathbb{C})\))

Linearization resolves this problem. A linear action preserves each degree \(d\) component \(A_d\) of the homogeneous coordinate ring, so \(G\) acts on \(A_d\) in a well-behaved manner and we can form \(A_d^G\). In the example above, the action is already linear, \(A_d^G = \mathbb{C} \cdot y^d\), and we can construct the quotient as

\[\mathbb{P}^1 /\!/ \mathbb{C}^\ast = \Proj\left(\bigoplus_{d \ge 0} \mathbb{C} \cdot y^d\right) = \{[1]\} \cong \mathrm{pt}\]

Once a linearization is given, we can define stable and semistable points.

Definition 19 For a \(G\)-action on \(X \subseteq \mathbb{P}^n\) with a given linearization:

• A point \(x \in X\) is semistable if there exists a non-constant \(G\)-invariant homogeneous polynomial \(f\) such that \(f(x) \ne 0\).
• A point \(x \in X\) is stable if it satisfies the following conditions:
  1. \(x\) is semistable
  2. \(G_x\) is a finite group
  3. The orbit \(G \cdot x\) is closed

We denote the set of semistable points by \(X^{\mathrm{ss}}\) and the set of stable points by \(X^{\mathrm{s}}\).

Definition 20 The GIT quotient for a \(G\)-action on a projective variety \(X\) with a given linearization is defined by

\[X /\!/ G = \Proj\left(\bigoplus_{d \ge 0} A_d^G\right)\]

Here \(A_d\) is the degree \(d\) component of the homogeneous coordinate ring of \(X\).

Just as in the affine case discussed above, the GIT quotient of a projective variety is also a sufficiently good approximation. Moreover, in this case \(X/\!/G\) can be regarded as a compactification of the orbit space of semistable points.

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References

[Spr] T. A. Springer, Linear Algebraic Groups, Birkhäuser, 1998.
[Hum] J. E. Humphreys, Linear Algebraic Groups, Springer, 1975.
[Mil] J. S. Milne, Algebraic Groups, Cambridge University Press, 2017.
[MFK] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Springer, 1994.