Journal of Lie Theory Volume 13 (2003) 401–425 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Invariant Theory of a Class of Infinite-Dimensional Groups

Tuong Ton-That and Thai-Duong Tran

Communicated by K.-H. Neeb

Abstract. The representation theory of a class of infinite-dimensional groups which are inductive limits of inductive systems of linear algebraic groups leads to a new invariant theory. In this article, we develop a coherent and comprehensive invariant theory of inductive limits of groups acting on inverse limits of modules, rings, or algebras. In this context, the Fundamental Theorem of the Invariant Theory is proved, a notion of basis of the rings of invariants is introduced, and a generalization of Hilbert’s Finiteness Theorem is given. A generalization of some notions attached to the classical invariant theory such as Hilbert’s Nullstellensatz, the primeness condition of the ideals of invariants are also discussed. Many examples of invariants of the infinite-dimensional classical groups are given.

Key words and phrases. Invariant theory of inductive limits of groups acting on inverse limits of modules, rings, or algebras, Fundamental Theorem of Invariant Theory.

2000 Mathematics Subject Classification: Primary 13A50, Secondary 22E65, 13F20.

1. Introduction

In the preface to his book, The Classical Groups: Their Invariants and Representations, Hermann Weyl wrote “The notion of an algebraic invariant of an abstract group γ cannot be formulated until we have before us the concept of a representation of γ by linear transformations, or the equivalent concept of a “quantity of type 𝔄.” The problem of finding all representations or quantities of γ must therefore precede that of finding all algebraic invariants of γ.” His book has been and remains the most important work in the theory of representations of the classical groups and their invariants.

In recent years there is great interest, both in Physics and in Mathematics, in the theory of unitary representations of infinite-dimensional groups and their Lie algebras (see, e.g., [10], [9], [8] and the literature cited therein). One class of representations of infinite-dimensional groups is the class of tame representations of inductive limits of classical groups. They were studied thoroughly in the comprehensive and important work of Ol’shanskiĭ [13].

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

Research partially supported by a grant of the Obermann Center for Advanced Studies and by a Carver Scientific Research Initiative grant.

As in Weyl’s case with the classical groups, we also discovered a new type of invariants when we studied concrete realizations of irreducible tame representations of inductive limits of classical groups [22, 23]. One type of invariants that is extremely important in Physics is the Casimir invariants (see, e.g., [2]). Several of their generalizations to the case of infinite-dimensional groups may be found in [10], [14], [6], and [22]. However, to our knowledge, there is no systematic study of the invariant theory of inductive limits of groups acting on inverse limits of modules, rings, or algebras. In this article we develop a coherent and comprehensive theory of these invariants. To illustrate how they arise naturally from the representation theory of infinite-dimensional groups we shall consider the following typical examples.

Example 1.1. Set V_{k} = ℂ^{1×k} and let A_{k} = P (V_{k}) denote the algebra of polynomial functions on V_{k}. Set G_{k} = SO_{k}(ℂ) and G_{k}^{0} = SO_{k}(ℝ). Then G_{k} (resp. G_{k}^{0} ) acts on V_{k} by right multiplication, and this induces an action of G_{k} (resp. G_{k}^{0} ) on A_{k}. Then the ring of G_{k} (resp. G_{k}^{0} )-invariants is generated by the constants and p_{k}^{0} = ∑_{i=1}^{k} X_{i}^{2}, where (X_{1}, …, X_{k}) = X ∈ V_{k}, and the G_{k} - invariant differential operators are generated by △_{k} = p_{k}^{0}(D) = ∑_{i=1}^{k} ∂_{i}^{2}, where ∂_{i}^{2} = \frac{∂^{2}}{∂X_{i}^{2}} . If H_{k} (resp. H_{k}^{d}) denote the subspace of harmonic polynomials (resp. harmonic homogeneous of degree d), i.e., polynomials that are annihilated by △_{k}, then for k > 2 we have the “separation of variables” theorem

P^{(m)}(V_{k}) = ∑_{i=0,... ,[m/2]} ⊕(p_{k}^{0})^{(i)}H_{k}^{(m−2i)},  (1.1)

where P^{(m)}(V_{k}) denotes the subspace of all homogeneous polynomials of degree m ≥ 0, and [m/2] denotes the integral part of m/2. Moreover, each (p_{k}^{0})^{(i)}H_{k}^{(m−2i)} is an irreducible G_{k} (resp. G_{k}^{0} )-module of signature (\underbrace{m − 2i, 0, …, 0}_{[k/2]}). Now

observe that a polynomial in k variables (X_{1}, …, X_{k}) can be considered as a polynomial in l variables (X_{1}, …, X_{l}) for k ≤ l in the obvious sense. It follows that A_{k} can be embedded in A_{l} so that the inductive limit A of A_{k} can be considered as the algebra of polynomials in infinitely many variables; in the sense that an element of A is a polynomial in n variables, where n ranges over ℕ. Let G = ⋃_{k=1}^{∞} G_{k} (resp. G^{0} = ⋃_{k=1}^{∞} G_{k}^{0} ); then G acts on A in the following sense:

If g ∈ G then g ∈ G_{k}, for some k; if f ∈ A then f ∈ A_{l}, for some l. We may always assume that k ≤ l and g ∈ G_{l}, so that g · f is well-defined. Thus under the identification defined above, it makes sense to define H^{d} as the subspace of A which consists of all harmonic homogeneous polynomials of degree d. Then it was shown in [23] that H^{d} is an irreducible G-(resp. G^{0})-module. But now what are the G-invariants? It is easy to see that no elements of A as well as no polynomial differential operators can be G-invariant. Now observe that if we let p^{0} denote the formal sum ∑_{i=1}^{∞} X_{i}^{2} and X = (X_{1}, …, X_{k}, …) denote the formal infinite row matrix, then Xg, g ∈ G (i.e., g ∈ G_{k} for some k), equals ((X_{1}, …, X_{k})g, X_{k+1}, …), and it follows p^{0} is formally G-invariant. Set △ = ∑_{i=1}^{∞} ∂_{i}^{2} and let △ operate on A as follows:

If f ∈ A then f ∈ A_{k} for some k ∈ ℕ, and △f := △_{k}f.

Thus f ∈ A is harmonic if △f = 0, and H^{d} = {f ∈ A^{d} | △f = 0}.

This intuitive generalization of invariants can be rigorously formalized by defining p^{0} as an element of the inverse (or projective) limit A_{∞} of the algebras A_{k}. Then A_{∞} is an algebra over ℂ and one can define an action of G on A_{∞}. The subalgebra J of all elements of A_{∞} which are pointwise fixed by this action is called the algebra of G-invariants, and p^{0} ∈ J. It turns out that this can be done in a very general context.

It is also well-known that the ideal in A_{k} generated by p_{k}^{0} is prime if k > 2. It will be shown that the ring of G-invariants in A_{∞} is generated by the constants and by p^{0}, and the ideal in A_{∞} generated by p_{0} is prime.

Example 1.2. This example will be studied in great detail in Subsection 4.7., but since we want to use it to motivate the need to introduce a topology on A_{∞} in Section 2., we shall give a brief description below.

Let X_{k} = (x_{ij}) ∈ ℂ^{k×k} and let A_{k} denote the algebra of polynomial functions in the variables x_{ij}. Let G_{k} = GL_{k}(ℂ); then G_{k} operates on A_{k} via the co-adjoint representation. Set

T_{k}^{n} = Tr(X_{k}^{n}), 1 ≤ n ≤ k;

then the subalgebra of all G_{k} -invariants is generated by the constants and by the algebraically independent polynomials T_{k}^{1}, …, T_{k}^{k}. Let G denote the inductive limit of the G_{k} ’s and let A_{∞} denote the inverse limit of the A_{k} ’s. Let T^{n} denote the inverse limit of T_{k}^{n}; then it will be shown that {T^{n}; n ∈ ℕ} is an algebraically independent set of G-invariants. However, if we let ⟨T^{n}; n ∈ ℕ⟩ denote the subalgebra of A_{∞} generated by the T^{n}’s, and J denote the subalgebra of Ginvariants in A_{∞}, then we can only show that ⟨T^{n}; n ∈ ℕ⟩ is dense in J under the topology of inverse limits defined on A_{∞}. In general, we can give examples of ideals that are not closed in A_{∞} (see Example 2.12). Thus in order to have a notion of basis for the rings of invariants it is necessary to introduce a topology on A_{∞}.

It turns out that, in general, the topology introduced in Section 2. is the most natural and the only nontrivial that one can define on inverse limits of algebraic structures.

In the spirit of Hilbert’s Fourteenth Problem (see [12]) we shall also prove a sufficient condition for our rings of G-invariants to be finitely generated (Theorem 3.6). Some of the results in this article were presented in [20], [21] and [24].

2. Inverse limits of algebraic structures as topological spaces

Let 𝕀 be an infinite subset of the set of natural numbers ℕ. Let C be a category. Suppose for each i ∈ 𝕀 there is an object A_{i} ∈ C and whenever i ≤ j there is a morphism μ_{i}^{j} : A_{j} → A_{i} such that

(i) μ_{i}^{i} : A_{i} → A_{i} is the identity for every i ∈ 𝕀,

(ii) if i ≤ j ≤ k then μ_{i}^{k} = μ_{i}^{j} ∘ μ_{j}^{k}.

Then the family {A_{i}; μ_{i}^{j}} is called an inverse spectrum over the index set 𝕀 with connecting morphisms μ_{i}^{j}.

Form ∏_{i∈𝕀} A_{i} and let p_{i} denote its projection onto A_{i}. The subset

{a = (a_{i}) ∈ ∏_{i∈𝕀}A_{i} | a_{i} = p_{i}(a) = μ_{i}^{j} ∘ p_{j}(a) = μ_{i}^{j}(a_{j}), whenever i ≤ j}

is called the inverse (or projective) limit of the inverse spectrum {A_{i}; μ_{i}^{j}} and is denoted by A_{∞} (or lim_{←}A_{i}). The restriction p_{i}|_{A_{∞}} : A_{∞} → A_{i} is denoted by μ_{i}

and is called the i^{th} canonical map. The elements of A_{∞} are called threads.

In this article C can be either the category of modules, vector spaces, rings, or algebras over a field; then clearly if A_{∞} ≠ ∅ it belongs to the same category of the A_{i}. For example if each A_{i} is an algebra over the field 𝔽 then A_{∞} is an algebra over 𝔽 with the operations defined as follows:

For a = (a_{i}), b = (b_{i}) in A_{∞} and c in 𝔽,

(a + b)_{i} := (a_{i} + b_{i}), (ab)_{i} := a_{i}b_{i}, (ca)_{i} := ca_{i}.

These operations are well-defined since the connecting morphisms μ_{i}^{j} are algebra homomorphisms. It follows that the canonical maps are also morphisms. In general the inverse limit A_{∞} can be made into a topological space as follows:

Endow each A_{i} with the discrete topology. Then the Cartesian product ∏_{i∈𝕀} A_{i} has a nontrivial product topology. Since each mapping μ_{i}^{j} is clearly continuous it follows that the projection maps p_{i}, and hence, the canonical maps μ_{i} are continuous. It follows from Theorem 2.3, p. 428, of [5] that the sets {μ_{i}^{−1}(U) | all i ∈ 𝕀, all subsets U of A_{i}} form a topological basis for A_{∞}. We have the following refinement.

Lemma 2.1. The space A_{∞} equipped with the topology defined above satisfies the first axiom of countability with the sets {μ^{−1}(a_{i}) | all i ∈ 𝕀} forming a countable topological basis at each point a = (a_{i}) of A_{∞}. Moreover, μ_{j}^{−1}(a_{j}) ⊂ μ_{i}^{−1}(a_{i}) whenever i ≤ j.

Proof. Let a = (a_{i}) ∈ V , where V is open in A_{∞}. Then there exist an i ∈ 𝕀 and a subset U_{i} of A_{i} such that a ∈ μ_{i}^{−1}(U_{i}) ⊂ V . This implies that a_{i} = μ_{i}(a) ∈ U_{i}, and therefore, μ_{i}^{−1}(a_{i}) ⊂ μ_{i}^{−1}(U_{i}) ⊂ V . Since the set {a_{i}} is open in A_{i}, μ_{i}^{−1}(a_{i}) is a basic open set in A_{∞} containing a. This shows that A_{∞} is first countable. Now let i ≤ j and let b ∈ μ_{j}^{−1}(a_{j}). Then b_{j} = μ_{j}(b) ∈ μ_{j}(μ_{j}^{−1}(a_{j})) = {a_{j}}, or b_{j} = a_{j}. This implies that b_{i} = μ_{i}^{j}(b_{j}) = μ_{i}^{j}(a_{j}) = a_{i}, and thus b ∈ μ_{i}^{−1}(a_{i}). This shows that μ_{j}^{−1}(a_{j}) ⊂ μ_{i}^{−1}(a_{i}).

Remark 2.2. In this article when we refer to the topological space A_{∞} we mean that A_{∞} is equipped with the topology defined by the topological basis {μ_{i}^{−1}(a_{i}) | all i ∈ 𝕀, and all a = (a_{i}) ∈ A_{∞}}, unless otherwise specified.

For S any subset of A_{∞} let S̄ denote the closure of S in A_{∞}. Lemma 2.1 implies the following

Lemma 2.3. Let S ⊂ A_{∞}. Then x ∈ S̄ if and only if there is a sequence {x^{n}} in S converging to x.

Proof. See [5, Theorem 6.2, p. 218].

Theorem 2.4. Let {x^{n}} be a sequence in A_{∞}. Then x^{n} → x if and only if for every i ∈ 𝕀 there exists a positive integer N_{i}, depending on i, such that x_{i}^{n} = x_{i} whenever n ≥ N_{i}.

Proof. By definition the sequence {x^{n}} converges to x if: “for every neighborhood U of x ∃N ∀ n ≥ N : x^{n} ∈ U ”. By Lemma 2.1 it is sufficient to consider the neighborhoods of x of the form μ_{i}^{−1}(x_{i}), ∀ i ∈ 𝕀. This means that x^{n} → x if and only if

“ ∀ i ∈ 𝕀 ∃N_{i} ∀ n ≥ N_{i} : x_{i}^{n} = x_{i}”.

Theorem 2.5. If A_{∞} belongs to the category C of modules, rings, etc., then the operations in A_{∞} are continuous.

Proof. For example, A_{∞} is an algebra over a field and the operation is the multiplication in A_{∞}. Let f : A_{∞} × A_{∞} → A_{∞} be the map defined by

f(a, b) = a · b, ∀ a, b ∈ A_{∞}.

Since A_{∞} is first countable it follows that A_{∞} × A_{∞} is first countable. It follows from Theorem 6.3, p. 218, of [5] that f is continuous at (a, b) if and only if f(a^{n}, b^{n}) → f(a, b) for each sequence (a^{n}, b^{n}) → (a, b). By Theorem 2.4

“a^{n} → a if and only if ∀ i ∈ 𝕀 ∃N_{i}^{a} ∀ n ≥ N_{i}^{a} : a_{i}^{n} = a_{i}”,
“b^{n} → b if and only if ∀ i ∈ 𝕀 ∃N_{i}^{b} ∀ n ≥ N_{i}^{b} : b_{i}^{n} = b_{i}”.

Thus ∀ i ∈ 𝕀 let N_{i} = max(N_{i}^{a}, N_{i}^{b}); then ∀ n ≥ N_{i} we have a_{i}^{n} = a_{i} and b_{i}^{n} = b_{i}. This implies that

“∀ i ∈ 𝕀 ∃N_{i} ∀ n ≥ N_{i} (a · b)_{i} = a_{i} · b_{i} = a_{i}^{n} · b_{i}^{n} = (a^{n} · b^{n})_{i}”

which implies that f(a^{n}, b^{n}) → f(a, b), or f is continuous at (a, b).

For each i ∈ 𝕀, let S_{i} ⊂ A_{i} and assume that μ_{i}^{j}(S_{j}) ⊂ S_{i} whenever i ≤ j. Then {S_{i}; μ_{i}^{j}|_{S_{j}}} is an inverse spectrum over 𝕀. Theorem 2.8, p. 423, of [5] implies that the inverse limit S_{∞} is homeomorphic to the subspace A_{∞} ∩ ∏_{i∈𝕀} S_{i}. In this article we shall identify S_{∞} with this subspace.

Theorem 2.6. Let S be any subset of A_{∞}, and let S_{i} = μ_{i}(S), all i ∈ 𝕀; then S_{∞} = S̄.

Proof. Let s ∈ S; then s_{i} = μ_{i}(s) ∈ S_{i}, ∀ i ∈ 𝕀, and μ_{i}^{j}(s_{j}) = μ_{i}^{j} ∘ μ^{j}(s) = μ_{i}(s) = s_{i}. Thus μ_{i}^{j}(S_{j}) ⊂ S_{i} and S ⊂ S_{∞}. Let us show that S_{∞} is closed in A_{∞}. Let s^{0} ∈ S̄_{∞}; then Lemma 2.3 implies that there exists a sequence {s^{n}} in S_{∞} converging to s^{0}. By Theorem 2.4 it follows that for every i ∈ 𝕀, there exists N_{i} such that s_{i}^{n} = s_{i}^{0} whenever n ≥ N_{i}. This implies that s_{i}^{0} ∈ S_{i} for every i ∈ 𝕀, and hence, s^{0} ∈ S_{∞}. Thus S_{∞} is closed, and it follows that S̄ ⊂ S_{∞}. Now let s ∈ S_{∞}; then by definition, for every i ∈ 𝕀, there exists an element s^{i} ∈ S such that s_{i} = s_{i}^{i}. Now the set {s^{i} | i ∈ 𝕀} is a sequence in S since 𝕀 is an infinite subset of ℕ. For any i, j ∈ 𝕀 such that j ≥ i; then s_{i}^{j} = μ_{i}^{j}(s_{j}^{j}) = μ_{i}^{j}(s_{j}) = s_{i}. It follows that, for every i ∈ 𝕀, there exists N_{i} = i such that s_{i}^{j} = s_{i} whenever j ≥ N_{i} = i. Theorem 2.4 implies that s^{j} → s, and thus s ∈ S̄. Therefore, S_{∞} ⊂ S̄, and hence S̄ = S_{∞}.

In the following theorems C is the category of (unital) rings but whenever it is appropriate the theorems remain valid if C is either the category of modules, vector spaces or algebras over a field 𝔽. The proofs of Theorems 2.7 and 2.8 and Corollary 2.9 are straightforward.

Theorem 2.7. Let {R_{i}; μ_{i}^{j} | i ∈ 𝕀} be an inverse spectrum in the category C of unital rings. Then R_{∞} is a unital ring and the following hold:

(i) If for all i ∈ 𝕀, S_{i} are subrings of R_{i} such that μ_{i}^{j}(S_{j}) ⊂ S_{i} whenever j ≥ i, then S_{∞} is a subring of R_{∞}.

(ii) If S is a subring of R_{∞} and S_{i} = μ_{i}(S), all i ∈ 𝕀, then each S_{i} is a subring of R_{i}. Moreover, S_{∞} is also a subring of R_{∞} such that S_{∞} = S̄.

Theorem 2.8. Let {R_{i}; μ_{i}^{j} | i ∈ 𝕀} be an inverse spectrum in the category C of commutative and unital rings. Then the following hold:

(i) If for all i ∈ 𝕀, I_{i} are ideals of R_{i} such that μ_{i}^{j}(I_{j}) ⊂ I_{i} whenever j ≥ i, then I_{∞} is an ideal of R_{∞}.

(ii) If I is an ideal of R_{∞}, if I_{i} = μ_{i}(I), and if the canonical homomorphisms μ_{i} : R_{∞} → R_{i} are surjective, then each I_{i} is an ideal of R_{i}. Moreover, I_{∞} is also an ideal of R_{∞} such that I_{∞} = Ī.

Let R be a unital commutative ring and let S ≠ ∅ be any subset of R. Let ⟨S⟩ denote the subring generated by S ; i.e., the smallest subring containing S. Similarly if S ≠ ∅ is a subset of R there exists a smallest ideal containing S. This ideal is called the ideal generated by S and is denoted by (S). The set S is then called a system of generators of this ideal. In fact an element of (S) can be written as ∑_{finite} r_{i}s_{i} where r_{i} ∈ R, and s_{i} ∈ S.

Corollary 2.9. Let S be any non-empty subset of R_{∞} and set ⟨S⟩_{i} = μ_{i}(⟨S⟩), (S)_{i} = μ_{i}((S)), all i ∈ 𝕀. Then the following hold:

(i) lim_{←} ⟨S⟩_{i} is the smallest closed subring of R_{∞} that contains S.

(ii) If the canonical homomorphisms μ_{i} are surjective, all i ∈ 𝕀, then lim_{←}(S)_{i} is the smallest closed ideal of R_{∞} that contains S.

A subset 𝕃 of the index set 𝕀 is called cofinal in 𝕀 if ∀ i ∈ 𝕀 ∃l ∈ 𝕃 : i ≤ l. Since 𝕀 ⊂ ℕ it is clear that 𝕃 ⊂ 𝕀 is cofinal in 𝕀 if and only if 𝕃 is an infinite subset of 𝕀.

Let {A_{i}; μ_{i}^{j}} be an inverse spectrum in a category C and let 𝕃 be cofinal in 𝕀. Then Theorem 2.7, p. 431, of [5] implies that lim_{←}A_{i∈𝕀} is homomorphic to lim_{←}A_{l∈𝕃}. Clearly both limits are in the category C and they are also isomorphic. So we may without loss of generality assume that lim_{←}A_{i∈𝕀} = lim_{←}A_{l∈𝕃}.

Theorem 2.10. If for every i ∈ 𝕀, R_{i} is an integral domain, then R_{∞} is an integral domain. If the connecting homomorphisms μ_{i} are surjective, all i ∈ 𝕀, then every principal ideal I in the integral domain R_{∞} is closed.

Proof. If a, b ∈ R_{∞} are such that a·b = 0 then a_{i} ·b_{i} = (a·b)_{i} = 0 for all i ∈ 𝕀. Since each R_{i} is an integral domain either a_{i} = 0 or b_{i} = 0. We may suppose without loss of generality that a_{i} = 0 for infinitely many indices i ∈ 𝕀. Since this set of indices is cofinal in 𝕀, Theorem 2.7 of [5] implies that a = 0. This implies that R_{∞} is an integral domain. Now let I be a principal ideal of the integral domain R_{∞} and let a be a generator of I. Since each μ_{i} is surjective, Theorem 2.8(ii) implies that each I_{i} = μ_{i}(I) is an ideal in R_{i}. For each s_{i} ∈ I_{i} there exists an s ∈ I such that μ_{i}(s) = s_{i}. Since I is a principal ideal there exists r ∈ R_{∞} such that s = ra. This implies that s_{i} = r_{i}a_{i}, and thus each I_{i} is a principal ideal in R_{i} with a_{i} as a generator. Let b ∈ I_{∞}; then b_{i} ∈ I_{i}, all i ∈ 𝕀. Therefore, for each i ∈ 𝕀 there exists r_{i} ∈ R_{i} such that b_{i} = r_{i}a_{i}. We have, for all j ≥ i,

r_{i}a_{i} = b_{i} = μ_{i}^{j}(b_{j}) = μ_{i}^{j}(r_{j}a_{j}) = μ_{i}^{j}(r_{j})μ_{i}^{j}(a_{j}), or  (2.1)
r_{i}a_{i} = μ_{i}^{j}(r_{j})a_{i}.

If I = {0} then obviously I is closed in R_{∞}. If I ≠ {0} then we may assume without loss of generality that a_{i} ≠ 0 for sufficiently large i. For such an i, Eq. (2.1) implies that μ_{i}^{j}(r_{j}) = r_{i} since I_{i} is an integral domain. Set r = (r_{i}); then since μ_{i}^{j}(r_{j}) = r_{i} whenever j ≥ i it follows that r ∈ R_{∞} and b = ra ∈ I. Thus I = I_{∞} and I is closed.

Theorem 2.11. For each i ∈ 𝕀 let I_{i} be an ideal of R_{i} such that μ_{i}^{j}(I_{j}) ⊂ I_{i} whenever j ≥ i. If I_{i} are prime for infinitely many i ∈ 𝕀 then I_{∞} is a prime ideal of R_{∞}.

Proof. Since the set 𝕃 of indices l ∈ 𝕀 for which I_{l} are prime is infinite lim_{←}I_{i∈𝕀} = lim_{←}I_{l∈𝕃} as remarked above. Thus we may assume without loss of generality that I_{i} are prime for all i ∈ 𝕀. Suppose a, b ∈ R_{∞} such that ab ∈ I_{∞}. Then by definition a_{i}b_{i} = (ab)_{i} ∈ I_{i}, ∀ i ∈ 𝕀. Since each I_{i} is prime either a_{i} ∈ I_{i} or b_{i} ∈ I_{i}. Suppose that there are infinitely many j ∈ 𝕀 such that a_{j} ∈ I_{j}. Then for each i ∈ 𝕀 there exists j ≥ i such that a_{j} ∈ I_{j}. Since μ_{i}^{j}(I_{j}) ⊂ I_{i} it follows that a_{i} = μ_{i}^{j}(a_{j}) ∈ I_{i}. Since i is arbitrary it follows that a = (a_{i}) ∈ I_{∞}. If there is only a finite number of j ∈ 𝕀 such that a_{j} ∈ I_{j} there must be infinitely many j ∈ 𝕀 such that b_{j} ∈ J_{j}, and the same argument as above shows that b ∈ I_{∞}. Thus, ab ∈ I_{∞} implies either a ∈ I_{∞} or b ∈ I_{∞}, and therefore I_{∞} is prime.

Example 2.12. We are giving below a class of examples which is typical of the category of objects that we will study in the remainder of this article.

Let R denote a commutative unital ring. Let k be a positive integer and let A_{k} denote the free commutative algebra R[X_{k}] ≡ R[(X_{ij})] of polynomials with respect to the indeterminates X_{ij}, where i is any integer ≥ 1 and 1 ≤ j ≤ k (see [3], Chapter 4, for polynomial algebras in general). Let (α)_{k} = (α_{11}, …, α_{1k}, α_{21}, …, α_{2k}, …) be a multi-index of integers ≥ 0 such that all but a finite number of the α_{ij} are nonzero. Set X_{k}^{(α)_{k}} = X_{11}^{α_{11}} … X_{ij}^{α_{ij}} . . .. Then the set {X_{k}^{(α)_{k}}} is a basis for the R-module A_{k} when (α)_{k} ranges over all multi-indices defined above. Set |(α)_{k}| = ∑_{i,j} α_{ij}. Then every polynomial p_{k} ∈ A_{k} can be written in exactly one way in the form

p_{k} = ∑_{|(α)_{k}|≥0} c_{(α)_{k}}X^{(α)_{k}}  (2.2)

where c_{(α)_{k}} ∈ R and the c_{(α)_{k}} are zero except for a finite number; the c_{(α)_{k}} are called the coefficients of p_{k}; the c_{(α)_{k}}X^{(α)_{k}} are called the terms of p_{k}. For l ≥ k every polynomial p_{l} = ∑ c_{(α)_{l}}X^{(α)_{l}} of A_{l} can be written uniquely in the form

p_{l} = ∑_{(α′)_{l}} c_{(α′)_{l}}X^{(α′)_{l}} + ∑_{(α′′)_{l}} c_{(α′′)_{l}}X^{(α′′)_{l}}  (2.3)

where in each (α′)_{l} all the integers α′_{ij} are zero whenever j > k, and in each (α′′)_{l} there must be an integer α′′_{ij} > 0 whenever k < j ≤ l. Identify each (α′)_{l} with an element (α)_{k} and define the map μ_{k}^{l} : A_{l} → A_{k} by

p_{k} = μ_{k}^{l}(p_{l}) = ∑_{(α′)_{l}} c_{(α′)_{l}}X^{(α′)_{l}} = ∑_{(α)_{k}} c_{(α)_{k}}X^{(α)_{k}}.  (2.4)

Using Eqs. (2.3) and (2.4) we can easily deduce that μ_{k}^{l} is an algebra homomorphism and we have μ_{k}^{m} = μ_{k}^{l} ∘ μ_{l}^{m} whenever k ≤ l ≤ m. In fact A_{k} can be considered as a subalgebra of A_{l} whenever k ≤ l. Thus all the connecting homomorphisms μ_{k}^{l} are surjective. These connecting homomorphisms are called truncation homomorphisms.

Let A_{∞} denote the inverse limit of the inverse spectrum {A_{k}; μ_{k}^{l}}. Then since every element p ∈ A_{k} can be considered as an element of A_{l} for l ≥ k we can identify p with a thread (p) in A_{∞} by defining p_{l} = p whenever l ≥ k. Since the set of all integers l ≥ k is cofinal in 𝕀 = ℕ the thread (p) is well-defined. It follows from Theorem 2.7 that A_{∞} is nonempty and each A_{k} is a subalgebra of A_{∞}. If R is an integral domain then Théorème 1, p. 10, of [3] implies that each A_{k} is an integral domain, and hence by Theorem 2.10, A_{∞} is an integral domain and every principal ideal I in A_{∞} is closed.

For a fixed integer n ≥ 1 let A_{n,k} denote the algebra R[(X_{ij})] for 1 ≤ i ≤ n and 1 ≤ j ≤ k; then obviously A_{n,k} is a subalgebra of A_{k} such that μ_{k}^{l}(A_{n,l}) = A_{n,k} whenever l ≥ k. Set A_{n,∞} = lim_{k}_{←}A_{n,k}; then Theorem 2.7 implies that A_{n,∞} is a subalgebra of A_{∞}.

For each i ≥ 1 define p_{k}^{i} ∈ A_{k} by p_{k}^{i} = ∑_{j=1}^{k} X_{ij}^{2} . Consider the thread f^{i} = (p_{k}^{i}) in A_{∞}. As remarked above for each k, p_{k}^{i} can be considered as an element of A_{∞} so that if we set f^{i,k} = p_{k}^{i} then the set {f^{i,k} | k ∈ ℕ} is a sequence in A_{∞}. This sequence has a particular property in that its k^{th} term f^{i,k} is a stationary thread at k. We claim that lim_{k→∞}f^{i,k} = f^{i}. Indeed, for every k there

exists N_{k} = k such that f_{k}^{i,l} = μ_{k}^{l}(p_{l}^{i}) = p_{k}^{i} = f_{k}^{i} whenever l ≥ k. Theorem 2.4 implies that lim_{k→∞}f^{i,k} = f^{i}. In general, if S^{k} = ∑_{i=1}^{k} g^{i}, k ∈ ℕ, is a convergent

sequence in A_{∞}, then we write its limit as ∑_{i=1}^{∞} g^{i}. Similarly, if P^{k} = ∏_{i=1}^{k} g^{i} is a convergent sequence in A_{∞}, then we write its limit as ∏_{i=1}^{∞} g^{i}. Thus with this convention we have f^{i} = ∑_{j=1}^{∞} X_{ij}^{2} , all i ∈ ℕ.

We claim that the ideal I generated by the set {f^{i} | i ∈ ℕ} is not closed in A_{∞}. Indeed, let S^{n} = ∑_{i=1}^{n} X_{ii}f^{i}; then {S^{n} | n ∈ ℕ} is a sequence in I such that

S_{k}^{n} = μ_{k}(S^{n}) = ∑_{i=1}^{n} μ_{k}(X_{ii})μ_{k}(f^{i}) = \{ \begin{matrix}∑_{i=1}^{n} X_{ii}f_{k}^{i}, k > n, \\ ∑_{i=1}^{k} X_{ii}f_{k}^{i}, k ≤ n. \end{matrix}  (2.5)

Set S = ∑_{i=1}^{∞} X_{ii}f^{i} and let us show that S ∈ A_{∞} and lim_{n→∞}S^{n} = S. First

consider {(∑_{i=1}^{k} X_{ii}f_{k}^{i})_{k} | k ∈ ℕ}. Then μ_{k}^{l}(∑_{i=1}^{l} X_{ii}f_{l}^{i}) = ∑_{i=1}^{k} X_{ii}f_{k}^{i} whenever l ≥ k. Thus ((∑_{i=1}^{k} X_{ii}f_{k}^{i})_{k}) is a thread in A_{∞} and S = ((∑_{i=1}^{k} X_{ii}f_{k}^{i})_{k}). Now we have from Eq. (2.5) “ ∀ k ∈ ℕ ∃N_{k} = k ∀ n ≥ k : S_{k}^{n} = S_{k} ”, which means that

lim_{n→∞}S^{n} = S ∈ Ī.

Now let us show that S ∉ I. A general element in I is of the form g = ∑_{i=1}^{m} h^{i}f^{i} where h^{i} ∈ A_{∞}, 1 ≤ i ≤ m. Then g_{k} = μ_{k}(g) = ∑_{i=1}^{m} μ_{k}(h^{i})μ_{k}(f^{i}) = ∑_{i=1}^{m} h_{k}^{i}f_{k}^{i} where the h_{k}^{i} belong to A_{k}, 1 ≤ i ≤ m. Thus each h_{k}^{i} is a polynomial in the indeterminates X_{rs}, 1 ≤ r ≤ n_{i}, 1 ≤ s ≤ k. Let n = max{n_{i} | 1 ≤ i ≤ m}; then clearly n is independent of k. Now choose k > n; then S_{k} = ∑_{i=1}^{k} X_{ii}f_{k}^{i}, and S cannot be an element g in I since the term X_{kk}f_{k}^{i} of S_{k} does not occur in g_{k}.

3. Invariant theory of inductive limits of groups acting on inverse limits of rings

Let 𝕀 be an infinite subset of the set of natural numbers ℕ. Let C be a category. Let {Y_{i} | i ∈ 𝕀} be a family of objects in the category C. Suppose for each pair of indices i, j satisfying i ≤ j there is a morphism λ_{ij} : Y_{i} → Y_{j} such that

(i) λ_{ii} : Y_{i} → Y_{i} is the identity for every i ∈ 𝕀,

(ii) if i ≤ j ≤ k then λ_{ik} = λ_{jk} ∘ λ_{ij}.

Then the family {Y_{i}; λ_{ij}} is called a direct (or inductive) system with index set 𝕀 and connecting morphisms λ_{ij}.

The image of y_{i} ∈ Y_{i} under any connecting morphism is called a successor of y_{i}. Let Y = ⋃_{i∈𝕀} Y_{i} and call two elements y_{i} ∈ Y_{i} and y_{j} ∈ Y_{j} in Y equivalent whenever they have a common successor in the spectrum. This relation, R, is obviously an equivalence relation in Y . The quotient Y/R is called the inductive (or direct) limit of the spectrum, and is denoted by Y^{∞} (or lim_{→}Y_{i}). Let p : ⋃_{i∈𝕀} Y_{i} → Y^{∞} be the projection; its restriction p|Y_{i} is denoted by λ_{i} and is called the canonical morphism of Y_{i} into Y^{∞}. In general, Y^{∞} may not have the same algebraic structure as the Y_{i}, but in many instances it does. For example, if {G_{i}; λ_{ij}} is an inductive system of groups, the inductive limit of the operations on G_{i} defines on lim_{→}G_{i} a group structure. Similar results hold for inductive limits of rings, modules, algebras, or Hilbert spaces; for details see [4, p. 139].

Now assume that for each k ∈ 𝕀 we have a linear subgroup G_{k} of GL_{k}(ℂ) such that G_{k} is naturally embedded (as a subgroup) in G_{l}, k < l; then we can define the inductive limit G^{∞} = ⋃_{k∈𝕀} G_{k}, and the connecting morphisms λ_{kl} are just the embedding isomorphisms of G_{k} into G_{l}.

Let A_{∞} be the inverse limit of an inverse spectrum {A_{i}; μ_{i}^{j}} of a category of objects considered in Section 2.. Suppose that each A_{k} is acted on by the group G_{k}.

Lemma 3.1. Assume that the homomorphisms μ_{j}^{k} and λ_{jk} satisfy the following condition

g · (μ_{j}^{k}(a_{k})) = μ_{j}^{k}(λ_{jk}(g) · a_{k}),  (3.1)

for all g ∈ G_{j}, a_{k} ∈ A_{k} and k ≥ j. Then there is a well-defined action of G^{∞} on A_{∞} given by

\{ \begin{matrix}(g · a)_{k} := g · a_{k}, \\ (g · a)_{n} := λ_{kn}(g) · a_{n}, & if n ≥ k, \\ (g · a)_{j} := μ_{j}^{k}(g · a_{k}), & if j ≤ k, \\ ∀ g ∈ G_{k}, ∀ a = (a_{k}) ∈ A_{∞}. \end{matrix}  (3.2)

Proof. First, let us prove that g · a ∈ A_{∞} whenever g ∈ G_{k} and a ∈ A_{∞}. For this we need to show that (g · a)_{i} = μ_{i}^{l}((g · a)_{l}) whenever l ≥ i.

If l ≥ i ≥ k then by Eq. (3.1) we have

μ_{i}^{l}((g · a)_{l}) = μ_{i}^{l}(λ_{kl}(g) · a_{l}) = μ_{i}^{l}(λ_{il}(λ_{ki}(g)) · a_{l})
= λ_{ki}(g) · (μ_{i}^{l}(a_{l})) = λ_{ki}(g) · a_{i} = (g · a)_{i}.

If l ≥ k > i then by definition we have

μ_{i}^{l}((g · a)_{l}) = μ_{i}^{l}(λ_{kl}(g) · a_{l}) = μ_{i}^{k}(μ_{k}^{l}(λ_{kl}(g) · a_{l}))
= μ_{i}^{k}(g · μ_{k}^{l}(a_{l})) = μ_{i}^{k}(g · a_{k}) = (g · a)_{i}.

If k > l ≥ i then by definition we have

μ_{i}^{l}((g · a)_{l}) = μ_{i}^{l}(μ_{l}^{k}(g · a_{k})) = μ_{i}^{k}(g · a_{k}) = (g · a)_{i}.

Now let us show that Eq. (3.2) defines an action of G^{∞} on A_{∞}. Let g_{1} ∈ G_{i} and g_{2} ∈ G_{k}. If i < k we may identify g_{1} with λ_{ik}(g_{1}), if k < i we may identify g_{2} with λ_{ki}(g_{2}). So we may assume without loss of generality that g_{1}, g_{2} ∈ G_{k}. We must show that

(g_{1}g_{2}) · a = g_{1} · (g_{2} · a), for all a = (a_{k}) ∈ A_{∞}.

For n ≥ k we have

((g_{1}g_{2}) · a)_{n} = λ_{kn}(g_{1}g_{2}) · a_{n} = (λ_{kn}(g_{1})λ_{kn}(g_{2})) · a_{n}
= λ_{kn}(g_{1}) · (λ_{kn}(g_{2}) · a_{n}) = λ_{kn}(g_{1}) · (g_{2} · a)_{n} = (g_{1} · (g_{2} · a))_{n}.

For j ≤ k we have

((g_{1}g_{2}) · a)_{j} = μ_{j}^{k}((g_{1}g_{2}) · a_{k}) = μ_{j}^{k}(g_{1} · (g_{2} · a_{k}))
= μ_{j}^{k}(g_{1} · (g_{2} · a)_{k}) = (g_{1} · (g_{2} · a))_{j}.

Let e_{i} denote the identity element of G_{i} for all i ∈ 𝕀. Then the unique element e ∈ G^{∞} such that e = λ_{i}(e_{i}) for all i ∈ 𝕀 is obviously the identity of G^{∞}, and we can easily verify that e · a = a, ∀ a ∈ A_{∞}.

Definition. An element a_{k} ∈ A_{k} is said to be G_{k} -invariant if g_{k} · a_{k} = a_{k} for all g_{k} ∈ G_{k}. An element a = (a_{k}) ∈ A_{∞} is said to be G^{∞}-invariant if g · a = a for all g ∈ G^{∞}.

The proofs of Lemmas 3.2 and 3.3 are straightforward.

Lemma 3.2. An element a = (a_{k}) ∈ A_{∞} is G^{∞}-invariant if and only if each a_{k} is G_{k} -invariant.

Lemma 3.3. If x ∈ A_{k} is G_{k} -invariant then μ_{j}^{k}(x) is G_{j} -invariant for all j ≤ k.

Now let 𝔽 = ℝ or ℂ and consider the free commutative algebra 𝔽[X_{k}] = 𝔽[(X_{ij})], i ≥ 1, 1 ≤ j ≤ k, of polynomials as described in Example 2.12. For every p ∈ 𝔽[(X_{ij})] let p̃ denote the polynomial function obtained by substituting X_{ij} by x_{ij} ∈ 𝔽. Since 𝔽 is an infinite field the mapping p → p̃ of 𝔽[X_{k}] onto 𝔽[x_{k}] is an algebra isomorphism (cf. [3, Proposition 9, p. 27]). Thus we can identify 𝔽[X_{k}] with 𝔽[x_{k}] = A_{k} and continue to call elements of A_{k} polynomials for the sake of brevity. Let μ_{k}^{l} be the truncation homomorphisms described in Example 2.12. Let A_{∞} denote the inverse limit of the inverse spectrum {A_{k}; μ_{k}^{l}}. Let {G_{k}; λ_{kl}} be an inductive system of groups such that each G_{k} acts on A_{k}. Then it can be easily verified that condition (3.1) is satisfied, and thus the action of G^{∞} on A_{∞} given in Lemma 3.1 is well-defined. We have now the Fundamental Theorem of the Invariant Theory of inductive limits of groups acting on inverse limits of polynomial algebras. Since the action of each G_{k} on A_{k} is such that g · (p + q) = gp + g · q, g · (cp) = c(g · p), and g · (pq) = (g · p)(g · q) for all g ∈ G_{k}, p, q ∈ A_{k}, and c ∈ 𝔽, it follows that the action of G^{∞} on A_{∞} has the same algebraic structure (see [4, Section 6, p. 140]). This implies immediately that the subset of all G^{∞}-invariants in A_{∞} is a subalgebra of A_{∞}.

Theorem 3.4. For each k ∈ 𝕀 let J_{k} denote the subalgebra of G_{k} -invariants in A_{k}. Let J denote the subalgebra of G^{∞}-invariants in A_{∞}. Then J_{∞} = lim_{←}J_{k} = J, and hence, J is closed in A_{∞}.

Proof. For each k ∈ 𝕀, Theorem 2.7(ii) implies that μ_{k}(J) is a subalgebra of A_{k}. Lemma 3.2 implies that μ_{k}(J) ⊂ J_{k} for all k ∈ 𝕀. Lemma 3.3 implies that μ_{k}^{l}(J_{l}) ⊂ J_{k} whenever l ≥ k. Now Theorem 2.7(i) implies that J_{∞} is a subalgebra of A_{∞}, and Theorem 2.7(ii) implies that lim_{←}μ_{k}(J) is also a subalgebra of A_{∞}. Obviously we have lim_{←}μ_{k}(J) ⊂ J_{∞}. Lemma 3.2 implies that J_{∞} ⊂ J. Theorem

2.7(ii) implies that lim_{←}μ_{k}(J) = J̄. Thus, finally we have the chain of inclusions.

J ⊂ J̄ = lim_{←}μ_{k}(J) ⊂ J_{∞} ⊂ J.  (3.3)

Then the Theorem now follows immediately from Eq. (3.3).

In the Invariant Theory of the Classical Groups the subalgebra of invariants is generated by an algebraically independent set of polynomials. We shall generalize this result by introducing a notion of algebraic basis for an inverse limit of polynomial algebras.

Definition. 1. A family {f^{α}}_{α∈Λ} of elements in A_{∞} is said to be algebraically independent if the relation p({f^{α}}) = 0, where p is a polynomial in 𝔽[{X^{α}}]_{α∈Λ} where X^{α} is an indeterminate, implies p = 0. The family is said to be algebraically dependent if it is not algebraically independent.

It is clear from the definition of a polynomial that a family is algebraically independent if and only if every finite subfamily of this family is algebraically independent.

2. A family \{f^{α}\}_{α∈Λ} of elements in A_{∞} is said to generate A_{∞} if \overline{⟨\{f^{α}\}_{α∈Λ}⟩} = A_{∞}, where ⟨{f^{α}}_{α∈Λ}⟩ denotes the subalgebra generated by the f^{α}, and the bar denotes the closure in the topology of inverse limits defined in Section 2.

3. An algebraically independent family of elements in A_{∞} that generates A_{∞} is called an inverse limit basis of A_{∞}.

4. For the standard definition of an algebraically independent family of polynomials see [3, p. 95].

Theorem 3.5. Let {f^{α}}_{α∈Λ} be a family of elements in A_{∞}. If for every finite subset of indices {α_{1}, …, α_{n}} ⊂ Λ there exists an integer k ∈ 𝕀, possibly depending on n, such that the subset of polynomials {f_{k}^{α_{1}}, …, f_{k}^{α_{n}}} is algebraically independent in A_{k}, then {f^{α}}_{α∈Λ} is algebraically independent in A_{∞}.

Proof. Suppose p({f^{α}}) = 0, p ∈ 𝔽[{X^{α}}]_{α∈Λ}; then p(f^{α_{1}}, …, f^{α_{n}}) = 0 for some finite subset of indices {α_{1}, …, α_{n}}. By hypothesis there exists an integer k such that {f_{k}^{α_{1}}, …, f_{k}^{α_{n}}} is algebraically independent in A_{k}. Since the canonical map μ_{k} : A_{∞} → A_{k} is an algebra homomorphism it follows that p(f_{k}^{α_{1}}, …, f_{k}^{α_{n}}) = 0. Hence p = 0 and the theorem is proved.

Theorem 3.6. Let {f^{α}}_{α∈Λ} be a family of elements in A_{∞}. If there exists k_{0} ∈ 𝕀 such that the family of polynomials {f_{k_{0}}^{α}}_{α∈Λ} is algebraically independent in A_{k_{0}} then {f^{α}}_{α∈Λ} is also algebraically independent in A_{∞} and ⟨{f^{α}}_{α∈Λ}⟩ is closed in A_{∞}.

Proof. The fact that {f^{α}}_{α∈Λ} is algebraically independent follows immediately from Theorem 3.5. By Lemma 2.3 to prove that ⟨{f^{α}}_{α∈Λ}⟩ is closed we suppose that φ is the limit of a sequence {φ^{n}} in ⟨{f^{α}}_{α∈Λ}⟩ and verify that φ ∈ ⟨{f^{α}}_{α∈Λ}⟩.

By Theorem 2.4, φ^{n} → φ if and only if for every i ∈ 𝕀 there exists a positive integer N_{i}, depending on i, such that φ_{i}^{n} = φ_{i} whenever n ≥ N_{i}. In particular, for i = k_{0} there exists N_{k_{0}} such that φ_{k_{0}}^{n} = φ_{k_{0}} whenever n ≥ N_{k_{0}} . Thus for i ≥ k_{0} we can choose N_{i} ≥ N_{k_{0}} . Therefore for n ≥ N_{i} we have

φ_{i} = φ_{i}^{n} = p_{n}({f_{i}^{α}}) = (p_{n}({f^{α}}))_{i},

where p_{n} is a polynomial depending on n. Since μ_{k_{0}}^{i} is an algebraic homomorphism, φ_{k_{0}} = φ_{k_{0}}^{n} = μ_{k_{0}}^{i}(φ_{i}^{n}) = p_{n}({f_{k_{0}}^{α}}). The fact that {f_{k_{0}}^{α}}_{α∈Λ} is algebraically independent implies that all the polynomials p_{n} are the same for sufficiently large n. Let p denote such a polynomial. Then we have φ_{i} = (p({f^{α}}))_{i} for all i ∈ 𝕀. This means that φ ∈ ⟨{f^{α}}_{α∈Λ}⟩, and this achieves the proof of the theorem.

Remark 3.7. Suppose {f^{α}}_{α∈Λ} is a family of elements in A_{∞} such that {f_{i}^{α}}_{α∈Λ} is an algebraic basis for the polynomial algebra A_{i} for all i ≥ k_{0} for some k_{0} ∈ 𝕀, i.e., the family {f_{i}^{α}}_{α∈Λ} is algebraically independent in A_{i} and ⟨{f_{i}^{α}}_{α∈Λ}⟩ = A_{i}. Then Theorem 3.6 implies that {f^{α}}_{α∈Λ} is also an algebraic basis for A_{∞}. Thus in this case the notion of algebraic basis and inverse limit basis for A_{∞} coincide, and the notion of (inverse limit) basis does indeed generalize the notion of algebraic basis.

Corollary 3.8. We preserve the notations of Theorem 3.4. Suppose {j^{α}}_{α∈Λ} is a family of elements in J such that {j_{k}^{α}}_{α∈Λ} is an algebraic basis for J_{k} for all k ≥ k_{0}. Then {j^{α}}_{α∈Λ} is an algebraic basis for J.

Proof. By Theorem 3.4, J = J_{∞} and Theorem 3.6 implies that {j^{α}}_{α∈Λ} is an algebraic basis for J_{∞}. Thus the corollary is proved.

Example 3.9. Let A_{k} be the algebra of polynomials in k variables x_{1}, …, x_{k} in 𝔽. Let {A_{k}; μ_{k}^{l}} denote the inverse spectrum with connecting homomorphisms μ_{k}^{l} : A_{l} → A_{k}, l ≥ k, l, k ∈ ℕ. The μ_{k}^{l} are truncation homomorphisms, which in this case can be defined simply by setting μ_{k}^{l}(x_{j}) = x_{j} for 1 ≤ j ≤ k and μ_{k}^{l}(x_{j}) = 0 for k < j ≤ l, and by extending algebraically to all polynomials in A_{l}. Let A_{∞} denote the inverse limit of the inverse spectrum {A_{k}; μ_{k}^{l}}. Then the set {x^{(α)}}_{(α)∈Λ}, where x^{(α)} = x^{α_{1}} ⋯ x^{α_{k}}, and (α) = (α_{1}, …, α_{k}) is a multi-index, forms an inverse limit basis for A_{∞} when (α) ranges over all multi-indices, and k = 1, 2, …, etc.

Now let G_{k} ⊂ GL_{k}(ℂ) be a reductive algebraic group, let V_{k} be a complex vector space of dimension k on which G_{k} acts linearly. Let ℂ[V_{k}] denote the ring of all polynomial functions on V_{k}. Let ℂ[V_{k}]^{G_{k}} denote the subring of all G_{k} - invariant polynomial functions. Then we have the following Hilbert’s Finiteness Theorem: There exist s algebraically independent G-invariants p_{1}, …, p_{s} such that ℂ[V_{k}]^{G_{k}} = ℂ[p_{1}, …, p_{s}]. (See [7] and [16]). Set A_{k} = ℂ[V_{k}] and J_{k} = ℂ[V_{k}]^{G_{k}}. We preserve the notation of Theorem 3.4 and assume in addition that each G_{k}, k ∈ 𝕀, is a reductive linear algebraic group. Then by Hilbert’s Finiteness Theorem there exists a set of algebraically independent polynomials {f_{k}^{α}}_{α∈Λ_{k}} that generates J_{k}, where the index set Λ_{k} is a finite subset of ℕ. It follows that, for all pairs (l, k) such that l ≥ k, we may assume that Λ_{k} ⊂ Λ_{l}. Set Λ = ⋃_{k∈𝕀} Λ_{k}. In general if the V_{k} are infinite-dimensional then the J_{k} may not be finitely generated but we still have Λ_{k} ⊂ Λ_{l} for l ≥ k.

Theorem 3.10. For each k ∈ 𝕀 let {f_{k}^{α}}_{α∈Λ_{k}} be a set of generators for J_{k}. If f^{α} = lim_{←}f_{k}^{α} then the family {f^{α}}_{α∈Λ} generates J. In particular, if {f_{k}^{α}}_{α∈Λ_{k}} is an algebraic basis for J_{k} then {f^{α}}_{α∈Λ} is an inverse limit basis for J.

Proof. Let J′ = ⟨{f^{α}}; α ∈ Λ⟩; then by assumption μ_{k}(J′) = J_{k}, for all k ∈ 𝕀. By Theorem 2.6, J̄′ = J_{∞}. By Theorem 3.4, J_{∞} = J. Therefore J̄′ = J. Now if in addition the sets {f_{k}^{α}}_{α∈Λ_{k}} are algebraically independent, then by assumption every finite subset of indices {α_{1}, …, α_{m}} of Λ is contained in Λ_{k} for some k ∈ 𝕀; therefore, the set {f_{k}^{α_{1}}, …, f_{k}^{α_{m}}} is algebraically independent. Theorem 3.5 implies that the set {f^{α}; α ∈ Λ} is algebraically independent. Thus by definition {f^{α}; α ∈ Λ} is an inverse limit basis for J.

Remark 3.11. In some examples in Section 4., for each k ∈ 𝕀 the set of generators {f_{k}^{α}}_{α∈Λ_{k}} are not algebraically independent yet the set {f^{α}}_{α∈Λ} can be shown to be algebraically independent.

We conclude this section by generalizing to the case of inverse limits of polynomial algebras one of the most fundamental theorems in the theory of polynomial algebras, namely, Hilbert’s Nullstellensatz. This theorem plays an important role in the theory of algebraic invariants.

Let I be a proper ideal in the polynomial ring in k complex variables ℂ[Z] = ℂ[z_{1}, …, z_{k}]. Then the algebraic variety of I is defined as the set

V (I) = {Z ∈ ℂ^{k} | p(Z) = 0, ∀ p ∈ I}.

If V is a subset of ℂ^{k} then the ideal of V is the ideal in ℂ[Z] defined by

I(V) = {p ∈ ℂ[Z] | p(Z) = 0, ∀ Z ∈ V }.

The nilradical of I is defined as the set

\sqrt{I} = \{p ∈ ℂ[Z] | p^{n} ∈ I for some integer n ≥ 1\}.

Then it can be easily shown that \sqrt{I} is the intersection of all prime ideals in ℂ[Z] which contain I. Then Hilbert’s Nullstellensatz can be simply stated as I(V (I)) = \sqrt{I} (See [7, p. 142]).

For each k ∈ ℕ let A_{k} = ℂ[Z]_{k}. For l ≥ k let μ_{k}^{l} : A_{l} → A_{k} denote the truncation homomorphism. Let A_{∞} = lim_{←}A_{k}. Let i_{kl} : ℂ^{k} → ℂ^{l} denote the

embedding defined by i_{kl}(Z) = i_{kl}(z_{1}, …, z_{k}) = (\underbrace{z_{1}, …, z_{k}, 0, …, 0})_{l}, ∀ Z ∈ ℂ^{k}.

Let ℂ^{∞} = ⋃_{k∈ℕ} ℂ^{k} denote the inductive limit of the spectrum {ℂ^{k}; i_{kl}}. Then it is easy to show that

p(i_{kl}(Z)) = [μ_{k}^{l}(p)](Z), ∀ p ∈ A_{l}, ∀ Z ∈ ℂ^{k}, ∀ k, l ∈ ℕ, l ≥ k.  (3.4)

Then for every f = (f_{k}) ∈ A_{∞} and every Z ∈ ℂ^{∞}, i.e., Z ∈ ℂ^{k} for some k ∈ ℕ, we define

f(Z) := f_{k}(Z).  (3.5)

Eq. (3.4) implies that Eq. (3.5) is well-defined, i.e., the complex number f(Z) is independent of k. This allows us to define the following concept: For any subset V of ℂ^{∞} let I(V) = {f ∈ A_{∞} | f(Z) = 0, ∀ Z ∈ V }.

Lemma 3.12. For any V ⊂ ℂ^{∞} the set I(V) is an ideal in A_{∞}. Moreover I(V) is closed.

Proof. If f, f′ ∈ I(V), h ∈ A_{∞} and c ∈ ℂ, then (f + cf′)(Z) = f(Z) + cf′(Z) = 0, ∀ Z ∈ ℂ^{∞}, and (hf)(Z) = h(Z)f(Z) = 0. Thus I(V) is an ideal in A_{∞}.

Suppose a sequence {f^{n}} ⊂ I(V) converges for f ∈ A_{∞}. Then by Theorem 2.4 we have

“ ∀ k ∈ ℕ, ∃ N_{k} ∀ n ≥ N_{k} : f_{k}^{n} = f_{k} ”.

This implies that for every k, f_{k}(Z) = f_{k}^{n}(Z) = 0 for all Z ∈ V^{k}, where V^{k} = V ∩ ℂ^{k}. This means that f(Z) = 0, ∀ Z ∈ V , and thus f ∈ I(V).

Now let I be an ideal in A_{∞} and set V (I) = {Z ∈ ℂ^{∞} | f(Z) = 0, ∀ f ∈ I}. If f = (f_{l}) ∈ I then Z ∈ V (I) if Z ∈ ℂ^{k} for some k ∈ ℕ, and f(Z) = f_{k}(Z) = 0. Choose k to be the smallest integer such that Z ∈ ℂ^{k}\ℂ^{k−1} and define V^{k}(I) = {Z ∈ ℂ^{k} | f_{k}(Z) = 0, ∀ f ∈ I}. Then V^{k}(I) may be empty but we always have V^{k}(I) ⊂ V^{l}(I), ∀ l ≥ k, and V (I) = ⋃_{k∈ℕ} V^{k}(I). Indeed, Z ∈ V (I) if and only Z ∈ V^{k}(I) for some k ∈ ℕ. And if l ≥ k then Eq. (3.3) implies that Z ∈ V^{l}(I). Thus if V (I) is not empty then there exists a smallest integer k_{0} such that V^{k}(I) = ∅ for k < k_{0} and V^{k}(I) ⊃ V^{k_{0}}(I) ≠ ∅ for k ≥ k_{0}. Thus in any case we can write V (I) = ⋃_{k∈ℕ} V^{k}(I).

Theorem 3.13. (The Nullstellensatz for inverse limits of polynomial rings) Let I be an ideal in A_{∞} and let I_{k} = μ_{k}(I), ∀ k ∈ ℕ. Then I(V (I)) = lim_{←}(\sqrt{I_{k}}).

Proof. The proof of the theorem consists of the following logically equivalent statements: Let f = (f_{k}); then

“f ∈ I(V (I))” ⟺ “f(Z) = 0, ∀ Z ∈ V (I)”

⟺ “f(Z) = 0, ∀ Z ∈ V^{k}(I), ∀ k ∈ ℕ” ⟺ “f_{k}(Z) = 0, ∀ Z ∈ V^{k}(I_{k}), ∀k ∈ ℕ”

⟺ “f_{k} ∈ \sqrt{I_{k}}, ∀ k ∈ ℕ” (by the classical form of the Nullstellensatz)

⟺ f ∈ lim_{←}(\sqrt{I_{k}}).

Corollary 3.14. In Theorem 3.13 suppose in addition that I is closed and that each I_{k} is radical. Then I(V (I)) = I.

Proof. We have I(V (I)) = lim_{←}(\sqrt{I_{k}}) = lim_{←}(I_{k}) = I_{∞} = Ī = I.

4. Invariant theory of the infinite-dimensional classical groups

In this section we apply the results of Sections 2. and 3. to the invariant theory of inductive limits of the classical groups and the symmetric groups as they act on inverse limits of polynomials in many variables. Our basic reference is [26]. As remarked by H. Weyl, the results are valid in any field of characteristic zero, but since in all our examples the underlying field is ℂ, we shall restrict ourselves to this case. Also, as noted by H. Weyl, “nothing of algebraic import is lost by unitary restriction” (for Weyl’s “unitarian trick” see [25, Lemma 4.11.13, p. 349]), and by the principle of permanence of the identities we shall consider only the case of the complex classical groups acting on polynomial algebras over ℂ. Also, all theorems in this section are consequences of results in previous sections, especially of Corollary 3.8 and Theorem 3.10. Thus we shall only give a detailed proof for the case O^{∞} (ℂ).

4.1. Invariant theory of the orthogonal group O^{∞} (ℂ).

For k ∈ ℕ let O_{k} (ℂ) = {g ∈ GL_{k} (ℂ) | g^{−1} = g^{t}} and set G_{k} = O_{k} (ℂ). For k ≤ l define the connecting isomorphism λ_{kl} of G_{k} into G_{l} by

λ_{kl} (g) = \begin{aligned}& & ^{k} & ^{l−k} \\ k\{ & ( & g & 0 & ) \\ l−k\{ & & 0 & 1 \end{aligned} , for all g ∈ G_{k}.  (4.1)

Let O^{∞} (ℂ) = G^{∞} = ⋃_{k∈ℕ} G_{k} denote the inductive limit of the inductive system {G_{k} ; λ_{kl}}. Let X_{k} denote the matrix of indeterminates X_{ij}, i ≥ 1, 1 ≤ j ≤ k. Let X_{k}^{i} denote the i^{th} row of X_{k}. We shall consider both cases when 1 ≤ i ≤ n and when the index set {i} is unbounded. As in Example 2.12 we let A_{k} = ℂ [X_{k}] and μ_{k}^{l}: A_{l} → A_{k}, k, l ∈ ℕ, l ≥ k, denote the truncation homomorphisms. Let A_{∞} denote the inverse limit of the inverse spectrum {A_{k} ; μ_{k}^{l}}.

For a fixed k, G_{k} acts on A_{k} by right translation, i.e.,

(g, p) ⟶ g · p, where (g · p) (X_{k}) := p (X_{k}g) , g ∈ G_{k}, p ∈ A_{k}.  (4.2)

Then according to [26, Theorem (2.9A), p. 53], the subalgebra J_{k} of G_{k} -invariants in A_{k} is generated by 1 and the homogeneous quadratic polynomials

f_{k}^{i_{1}i_{2}} = (X_{k}^{i_{1}}, X_{k}^{i_{2}}) = ∑_{j=1}^{k} X_{i_{1}j}X_{i_{2}j},  ∀ i_{1}, i_{2} ≥ 1.  (4.3)

If 1 ≤ i_{1}, i_{2} ≤ n ≤ k, then the f_{k}^{i_{1}i_{2}} are algebraically independent. Otherwise, all relations between them are algebraic consequences of relations of the following type (see [26, Theorem (2.17.A), p. 75]):

| \begin{matrix}(X_{k}^{i_{1}}, X_{k}^{j_{1}}) & … & (X_{k}^{i_{1}}, X_{k}^{j_{k+1}}) \\ ⋮ & & ⋮ \\ (X_{k}^{i_{k+1}}, X_{k}^{j_{1}}) & … & (X_{k}^{i_{k+1}}, X_{k}^{j_{k+1}}) \end{matrix} | = 0.  (4.4)

Example 4.1. Suppose that 1 ≤ i ≤ 3, k = 2, and X_{3} = [ \begin{matrix}X \\ Y \\ Z \end{matrix} ], where

X, Y, Z are two-dimensional row vectors. Then J_{k} is generated by 1 and

(X, X) , (X, Y) , (X, Z) , (Y, Y) , (Y, Z) , (Z, Z)

and the relations between these generators are algebraic consequences of the relation

| \begin{matrix}(X, X) & (X, Y) & (X, Z) \\ (Y, X) & (Y, Y) & (Y, Z) \\ (Z, X) & (Z, Y) & (Z, Z) \end{matrix} | = 0.

Let us verify that there is a well-defined action of G^{∞} on A_{∞} given by Eq. (3.2) of Lemma 3.1; i.e., we must verify Eq. (3.1) for g ∈ G_{k}, p ∈ A_{l}, and l ≥ k. We have

[g · (μ_{k}^{l} (p))] (X_{k}) = [μ_{k}^{l} (p)] (X_{k}g) .

Since p ∈ A_{l}, p is a polynomial in the indeterminates X_{ij}, for 1 ≤ i ≤ m and 1 ≤ j ≤ l. By identifying the indeterminates X_{ij} with the variables X_{ij} ∈ ℂ we can write

p (X_{l}) = p (X_{k} X_{l−k}) ,

where X_{l} is an m × l matrix, X_{k} is an m × k matrix, and X_{l−k} is an m × (l − k) matrix. And μ_{k}^{l} (p) (X_{k}) is just p (X_{k} 0). Therefore,

μ_{k}^{l} (p) (X_{k}g) = p (X_{k}g 0) .

On the other hand,

μ_{k}^{l} (λ_{kl} (g) · p) (X_{k}) = [λ_{kl} (g) · p] (X_{k} 0)

= p ([X_{k} 0] ( \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} )) = p (X_{k}g 0) .

Since these relations hold for all X_{k} ∈ ℂ^{m×k} and g ∈ G_{k} we have

μ_{k}^{l} (λ_{kl} (g) · p) = g · (μ_{k}^{l} (p)) , ∀ g ∈ G_{k}, p ∈ A_{l}, l ≥ k,

which is exactly the relation (3.1).

Let

f^{i_{1}i_{2}} = ∑_{j=1}^{∞} X_{i_{1}j}X_{i_{2}j},  ∀ i_{1}, i_{2} ≥ 1.  (4.5)

Then we have the following Fundamental Theorem of Invariants for O^{∞} (ℂ) acting on A_{∞}.

Theorem 4.2. (i) The set {1, f^{i_{1}i_{2}} | i_{1}, i_{2} ≥ 1} forms an inverse limit basis for the subalgebra J of all O^{∞} (ℂ)-invariants in A_{∞}.

(ii) If 1 ≤ i_{1}, i_{2} ≤ n, then 1 and the \frac{1}{2}n (n + 1) formal sums f^{i_{1}i_{2}} form an algebraic basis for the subalgebra J of all O^{∞}-invariants in A_{∞}.

Proof. Part (i) follows from Theorem 3.10. Part (ii) follows from Corollary 3.8.

4.2. Invariant theory of the special orthogonal group SO^{∞} (ℂ).

For k ∈ ℕ let G_{k} = SO_{k}(ℂ) = {g ∈ O_{k}(ℂ) | det(g) = 1}. Then in general the subalgebra J_{k} of G_{k} -invariants is generated by 1, the homogeneous quadratic polynomials f_{i_{1}i_{2}}^{k} as defined by Eq. 4.3, and the determinants

[(X_{k}^{i_{1}}, …, X_{k}^{i_{k}})] = det [ X_{k}^{i_{1}} ; ⋮ ; X_{k}^{i_{k}} ] ,  (4.6)

where we again identify the indeterminate X_{ij} with the variable X_{ij} ∈ ℂ (see [26, pp. 41–53]). But since for l > k, μ_{k}^{l}([(X_{l}^{i_{1}}, …, X_{l}^{i_{l}})]) = 0, it follows from Theorem 3.10 and Corollary 3.8 that the Fundamental Theorem of Invariants for SO^{∞}(ℂ) acting on A_{∞} has exactly the same form as Theorem 4.2.

Theorem 4.3. (The Nullstellensatz for the homogeneous ideal of invariants of O^{∞}(ℂ) and SO^{∞}(ℂ)). In Theorem 4.2(ii) let I = (f^{i_{1}i_{2}}) denote the homogeneous ideal in A_{∞} generated by the \frac{1}{2}n(n + 1) elements f^{i_{1}i_{2}} , 1 ≤ i_{1}, i_{2} ≤ n. If I_{k} denotes the ideal in A_{k} generated by the polynomials f_{k}^{i_{1}i_{2}} , 1 ≤ i_{1}, i_{2} ≤ n, and I_{∞} = lim_{←}I_{k} then we have I(V (I)) = I_{∞}.

Proof. Since obviously the canonical homomorphisms μ_{k} : A_{∞} → A_{k} are surjective Theorem 2.8(ii) implies that μ_{k}(I) is an ideal in A_{k}. Clearly, μ_{k}(I) ⊂ I_{k}, and since μ_{k}(f^{i_{1}i_{2}}) = f_{k}^{i_{1}i_{2}} it follows that μ_{k}(I) = I_{k}. By [17, Theorem 2.5, p. 11] the ideals I_{k} are prime for k > 2n. Hence by Theorem 2.11, I_{∞} is a prime ideal of A_{∞}, and \sqrt{I_{∞}} = I_{∞}. Therefore, by Theorem 3.13, I(V (I)) = I_{∞}.

4.3. Invariant theory of the symplectic group Sp^{∞}(ℂ).

The symplectic group G_{k} = Sp_{2k}(ℂ) is defined as the group of all linear transformations which leave a skew-symmetric bilinear form invariant. For our purpose we choose the skew-symmetric bilinear form

[x, y] = (x_{1}y_{2} − x_{2}y_{1}) + (x_{3}y_{4} − x_{4}y_{3}) + ⋯ + (x_{2k−1}y_{2k} − x_{2k}y_{2k−1})  (4.7)

for all x, y ∈ ℂ^{1×2k}.

If we let S_{k} denote the k × k block diagonal matrix

[ \begin{matrix}( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & & ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & & & & & . & . & . \\ & & & & & & & & & ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \end{matrix} ]  (4.8)

then Eq. (4.7) can be written as [x, y] = xS_{k}y^{t}. Then G_{k} can be defined as

G_{k} = {g ∈ ℂ^{2k×2k} | gS_{k}g^{t} = S_{k}}.  (4.9)

It can be easily verified that if g ∈ G_{k} then automatically det(g) = 1. For k ≤ l define the connecting isomorphism λ_{kl} of G_{k} into G_{l} by

λ_{kl} (g) = \begin{aligned}& & ^{2k} & ^{2(l−k)} \\ 2k\{ & ( & g & 0 & ) \\ 2(l−k)\{ & & 0 & 1 \end{aligned} ,  for all g ∈ G_{k}.  (4.10)

Let Sp^{∞}(ℂ) = G^{∞} = ⋃_{k∈ℕ} G_{k} denote the inductive limit of the inductive system {G_{k}; λ_{kl}}. Let X_{k} denote the matrix of indeterminates X_{ij}, i ≥ 1, 1 ≤ j ≤ 2k.

Let X_{k}^{i} denote the i^{th} row of X_{k}. Let Y_{k} denote the matrix of indeterminates Y_{ij}, i ≥ 1, 1 ≤ j ≤ 2k. Let Y_{k}^{i} denote the i^{th} row of Y_{k}. Let A_{k} = ℂ[X_{k}, Y_{k}] and μ_{k}^{l} : A_{l} → A_{k}, l ≥ k, denote the truncation homomorphisms. Let A_{∞} denote the inverse limit of the inverse spectrum {A_{k}; μ_{k}^{l}}. For a fixed k, G_{k} acts on A_{k} via

\{ \begin{matrix}(g, p) ⟶ g · p, where \\ (g · p)(X_{k}, Y_{k}) := p(X_{k}g, Y_{k}g^{✓}), \\ g ∈ G_{k}, p ∈ A_{k}, g^{✓} = (g^{−1})^{t}. \end{matrix}  (4.11)

Then according to [26, Theorem (6.1.A), p. 167] the subalgebra J_{k} of G_{k} -invariants in A_{k} is generated by 1 and by the homogeneous quadratic polynomials

\{ \begin{matrix}f_{k}^{i_{1}i_{2}} = [X_{k}^{i_{1}}, X_{k}^{i_{2}}] = X_{k}^{i_{1}}S_{k}(X_{k}^{i_{2}})^{t}, & i_{2} > i_{1} ≥ 1, \\ φ_{k}^{j_{1}j_{2}} = [Y_{k}^{j_{1}}, Y_{k}^{j_{2}}] = Y_{k}^{j_{1}}S_{k}(Y_{k}^{j_{2}})^{t}, & j_{2} > j_{1} ≥ 1, \\ h_{k}^{i_{1}j_{1}} = (X_{k}^{i_{1}}, Y_{k}^{j_{1}}) = X_{k}^{i_{1}}(Y_{k}^{j_{1}})^{t}, & i_{1}, j_{1} ≥ 1. \end{matrix}  (4.12)

These generators are not algebraically independent. All relations between them are algebraic consequences of relations of the type (4.4) for the h_{k}^{ij} ’s and of the types J_{1} ≡ 0, …, J_{k} ≡ 0 for the f_{k}^{i_{s}j_{s}} ’s and φ_{k}^{i_{s}j_{s}} ’s (see [26, Theorem (6.1.B), p. 168] for the definition of J_{1} ≡ 0, …, J_{k} ≡ 0).

Similarly to the case G^{∞} = O^{∞}(ℂ) we can easily verify that there is a well-defined action of G^{∞} = Sp^{∞}(ℂ) on A_{∞}.

Let X = lim_{k→∞}X_{k}, Y = lim_{k→∞}Y_{k}, and S = lim_{k→∞}S_{k}. Set

\{ \begin{matrix}f^{i_{1}i_{2}} = X^{i_{1}}S(X^{i_{2}})^{t}, & i_{2} > i_{1} ≥ 1, \\ φ^{j_{1}j_{2}} = Y^{j_{1}}S(Y^{j_{2}})^{t}, & j_{2} > j_{1} ≥ 1, \\ h^{i_{1}j_{1}} = X^{i_{1}}(Y^{j_{1}})^{t}, & i_{1}, j_{1} ≥ 1. \end{matrix}  (4.13)

Then we have the following Fundamental Theorem of Invariants of Sp^{∞}(ℂ) acting on A_{∞}.

Theorem 4.4. (i) The set {1, f^{i_{1}i_{2}}, φ^{j_{1}j_{2}}, h^{i_{1}j_{1}}} forms an inverse limit basis for the subalgebra J of all Sp^{∞}(ℂ)-invariants in A_{∞}.

(ii) If both X and Y have only a finite number of rows then the set {1, f^{i_{1}i_{2}}, φ^{j_{1}j_{2}}, h^{i_{1}j_{1}}} forms an algebraic basis for the subalgebra J of all Sp^{∞}(ℂ)-invariants in A_{∞}.

Proof. Part (i) follows from Theorems 3.5 and 3.10. Part (ii) follows from Corollary 3.8.

If X_{k} is an n × 2k matrix for all k ∈ ℕ, and A_{k} = ℂ[X_{k}], then the subalgebra J_{k} of G_{k} -invariant polynomials is generated by the set {1, f_{k}^{i_{1}i_{2}} | 1 ≤ i_{1} < i_{2} ≤ n}. Let I_{k} = (f_{k}^{i_{1}i_{2}}) denote the homogeneous ideal in A_{k} generated by the \frac{1}{2}n(n − 1) homogeneous quadratic polynomials f_{k}^{i_{1}i_{2}} , 1 ≤ i_{1} < i_{2} ≤ n. For k > n, the ideals I_{k} are shown to be prime in [18, Theorem 1.5, p. 269]. Hence by Theorem 2.11, I_{∞} = lim_{←}I_{k} is a prime ideal in A_{∞}, and we have Theorem 4.5. (The Nullstellensatz for the homogeneous ideal of invariants of Sp^{∞}(ℂ)). Let X_{k} denote an n × 2k matrix of indeterminates, let A_{k} = ℂ[X_{k}], and A_{∞} = lim_{←}A_{k}. Let I_{k} = (f_{k}^{i_{1}i_{2}}), and I_{∞} = lim_{←}I_{k}. Let I denote the ideal in

A_{∞} generated by the \frac{1}{2}n(n − 1) invariants f^{i_{1}i_{2}} , 1 ≤ i_{1} < i_{2} ≤ n. Then we have I(V (I)) = I_{∞}.

4.4. Invariant theory of the general linear group GL^{∞}(ℂ).

Let X_{k} (resp. Y_{k}) denote the matrix of indeterminates X_{ij} (resp. Y_{ik}), 1 ≤ i, 1 ≤ j ≤ k. Let X_{k}^{i} (resp. Y_{k}^{i}) denote the i^{th} row of X_{k} (resp. Y_{k}). Let A_{k} = ℂ[X_{k}, Y_{k}]. Set G_{k} = GL_{k}(ℂ); then the action of G_{k} on A_{k} is also given by Eq. (4.11). Then according to [26, Theorem (2.6.A), p. 45] the subalgebra of G_{k} - invariants in A_{k} is generated by 1 and by the homogeneous quadratic polynomials

\{ \begin{matrix}h_{k}^{i_{1}i_{2}} & = (X_{k}^{i_{1}}, Y_{k}^{i_{2}}) = X_{k}^{i_{1}}(Y_{k}^{i_{2}})^{t}, \\ & = ∑_{j=1}^{k} X_{i_{1}j}Y_{i_{2}j},  i_{1}, i_{2} ≥ 1. \end{matrix}  (4.14)

If 1 ≤ i_{1}, i_{2} ≤ n ≤ k then they are algebraically independent. Otherwise, all relations between them are algebraic consequences of relations of the following type:

| \begin{matrix}(X_{k}^{i_{1}}, Y_{k}^{j_{1}}) & ⋯ & (X_{k}^{i_{1}}, Y_{k}^{j_{k+1}}) \\ ⋮ & & ⋮ \\ (X_{k}^{i_{k+1}}, Y_{k}^{j_{1}}) & ⋯ & (X_{k}^{i_{k+1}}, Y_{k}^{j_{k+1}}) \end{matrix} | .  (4.15)

As in the case of Sp^{∞}(ℂ), we let GL^{∞}(ℂ) = G^{∞} = ⋃_{k∈ℕ} G_{k} denote the inductive limit of the inductive system {G_{k}; λ_{kl}}. Then there is a well-defined action of G^{∞} on A_{∞}. Set

h^{i_{1}i_{2}} = (X^{i_{1}}, Y^{i_{2}}) = X^{i_{1}}(Y^{i_{2}})^{t}, i_{1}, i_{2} ≥ 1.  (4.16)

Then we have the Fundamental Theorem of Invariants of GL^{∞}(ℂ) acting on A_{∞}.

Theorem 4.6. (i) The set {1, h^{i_{1}i_{2}}} forms an inverse limit basis for the subalgebra J of all GL^{∞}(ℂ)-invariants in A_{∞}.

(ii) If both X and Y have only a finite number of rows then the set {1, h^{i_{1}i_{2}}} forms an algebraic basis for J.

Proof. Same as in Theorem 4.4.

If X_{k} is a p × k matrix and Y_{k} is a q × k matrix for all k ∈ ℕ, let I_{k} = (h_{k}^{i_{1}i_{2}}) (resp. I = (h^{i_{1}i_{2}})) denote the ideal in A_{k} (resp. A_{∞}) generated by the pq invariants h_{k}^{i_{1}i_{2}} (resp. h^{i_{1}i_{2}} ), 1 ≤ i_{1} ≤ p, 1 ≤ i_{2} ≤ q. For k > max(p, q) the ideals I_{k} are shown to be prime in [19, Theorem 5.1, p. 213]. Hence by Theorem 2.11, I_{∞} = lim_{←}I_{k} is prime and we have the Nullstellensatz for GL^{∞}(ℂ), I(V (I)) = I_{∞}.

4.5. Invariant theory of the special linear group SL^{∞}(ℂ).

The setup is exactly the same as the general linear group GL^{∞}(ℂ) except that, in addition to the h_{k}^{i_{1}i_{2}} ’s in Eq. (4.14), the generators of the G_{k} -invariants are

[(X_{k}^{i_{1}}, …, X_{k}^{i_{k}})] = det [ X_{k}^{i_{1}} ; ⋮ ; X_{k}^{i_{k}} ] , and [(Y_{k}^{j_{1}}, …, Y_{k}^{j_{k}})] = det [ Y_{k}^{j_{1}} ; ⋮ ; Y_{k}^{j_{k}} ] . (4.17)

However, the images of the determinants under the truncation homomorphisms μ_{k}^{l}, l > k, are all zero. The Fundamental Theorem of Invariants and the Nullstellensatz of SL^{∞}(ℂ) acting on A_{∞} are exactly the same as those of GL^{∞}(ℂ).

Before studying the co-adjoint action of GL^{∞}(ℂ) we shall discuss the invariant theory of the infinite symmetric group since there is an intimate relation between the invariant theory of S_{k} and that of the co-adjoint action of GL_{k}(ℂ) (see, e.g., [16, Theorem 1.5.7, p. 10]).

4.6. Invariant theory of the infinite symmetric group S^{∞}.

Let R be any commutative ring with unit. Let A_{k} = R[X_{k}] denote the polynomial ring in k variables (x_{1}, …, x_{k}) = X_{k}. Let S_{k} denote the symmetric group of all permutations of the set {1, …, k}. Then S_{k} acts on A_{k} via

\{ \begin{matrix}(σ, p) ⟶ σ · p, σ ∈ S_{k}, p ∈ A_{k}, where \\ (σ · p)(x_{1}, …, x_{k}) = p(x_{σ^{−1}(1)}, …, x_{σ^{−1}(k)}). \end{matrix}  (4.18)

Then the subring J_{k} of S_{k} -invariant polynomials has an algebraic basis of the form

S_{k}^{0} = 1, ; S_{k}^{1} = ∑_{1≤i≤k} x_{i}, ; S_{k}^{2} = ∑_{1≤i<j≤k} x_{i}x_{j}, ; ⋮ ; S_{k}^{3} = ∑_{1≤i_{1}<i_{2}<i_{3}≤k} x_{i_{1}}x_{i_{2}}x_{i_{3}}, ; ⋮ ; S_{k}^{k} = x_{1}x_{2} ⋯ x_{k}.  (4.19)

The polynomials S_{k}^{1}, …, S_{k}^{k} are called the elementary symmetric functions (see [26, p. 37] or [1, Theorem 3.4, p. 548]). Set

T_{k}^{n} = ∑_{1≤i≤k} x_{i}^{n}, 1 ≤ n ≤ k;  (4.20)

then it can be shown [26, pp. 38–39] that the set {T_{k}^{n}; 0 ≤ n ≤ k} also forms an algebraic basis of J_{k}. In fact, there is a recursive formula expressing {T_{k}^{n}} in terms of {S_{k}^{n}}, and vice versa [26, p. 39].

We can embed S_{k} into S_{l} for k < l by defining λ_{kl} : S_{k} → S_{l} as follows:

λ_{kl}(σ)(i) = \{ \begin{matrix}σ(i), & 1 ≤ i ≤ k, \\ i, & k < i ≤ l, \end{matrix}  ∀ σ ∈ S_{k}.  (4.21)

Let S^{∞} = ⋃_{k=1}^{∞} S_{k} be the inductive limit of the inductive system {S_{k}; λ_{kl}}. Let A_{∞} denote the inverse limit of the inverse spectrum {A_{k}; μ_{k}^{l}}, where

the connecting homomorphisms μ_{k}^{l} : A_{l} → A_{k}, l > k, are the truncation homomorphisms. In fact, μ_{k}^{l} can be simply defined by setting

μ_{k}^{l}(x_{i}) = \{ \begin{matrix}x_{i}, & 1 ≤ i ≤ k, \\ 0, & k < i ≤ l. \end{matrix}

Then it is straightforward to verify that

σ · μ_{k}^{l}(p) = μ_{k}^{l}(λ_{kl}(σ) · p).

Set

S^{n} = ∑_{i_{1}<i_{2}<···<i_{n}} x_{i_{1}}x_{i_{2}} ⋯ x_{i_{n}}, T^{n} = ∑_{i∈ℕ}x_{i}^{n}, n = 1, 2, …;  (4.22)

then Theorem 3.10 implies the following

Theorem 4.7. Each of the sets {1, S^{n}; n ∈ ℕ} and {1, T^{n}; n ∈ ℕ} forms an inverse limit basis for the subring J of all S^{∞}-invariants in the ring A_{∞}.

See [11, Theorem (3.2), p. 3] for another proof that {S^{n}} is algebraically independent.

4.7. Invariant theory of the co-adjoint action of GL^{∞}(ℂ).

Let X_{k} = (x_{ij}) ∈ ℂ^{k×k}, and let A_{k} = ℂ[X_{k}] denote the algebra of all polynomial functions in the variables x_{ij}, 1 ≤ i, j ≤ k. Set G_{k} = GL_{k}(ℂ); then the adjoint representation of G_{k} on ℂ^{k×k} is defined by

g · X_{k} = gX_{k}g^{−1},  ∀ g ∈ G_{k}, X ∈ ℂ^{k×k}.  (4.23)

The co-adjoint representation of G_{k} on A_{k} is defined by

\{ \begin{matrix}(g, p) ⟶ g · p, where (g · p)(X_{k}) = p(g^{−1} · X_{k}) = p(g^{−1}X_{k}g), \\ ∀ g ∈ G_{k}, p ∈ A_{k}. \end{matrix}  (4.24)

If X_{k} ∈ ℂ^{k×k} let χ_{X_{k}}(t) = det(I − tX_{k}) denote the characteristic polynomial of X_{k} in the indeterminate t. Then we have

χ_{X_{k}}(t) = det(I − tX_{k})  (4.25)
= 1 − S^{1}(X_{k})t + S^{2}(X_{k})t^{2} + ⋯ + (−1)^{k}S_{k}^{k}(X_{k})t^{k},

and it can be shown that

S_{k}^{n}(X_{k}) = ∑ det [ \begin{matrix}X_{i_{1}i_{1}} & ⋯ & X_{i_{1}i_{n}} \\ ⋮ & & ⋮ \\ X_{i_{n}i_{1}} & ⋯ & X_{i_{n}i_{n}} \end{matrix} ] ≡ ∑ △_{i_{1}···i_{n}}^{i_{1}···i_{n}}(X_{k}),  (4.26)

where the sum is over all n-shuffles (i_{1}, …, i_{n}), i_{1} < i_{2} < ⋯ < i_{n}, 1 ≤ n ≤ k. If Y_{k} ∈ ℂ^{k×k} let Tr(Y) = ∑_{i=1}^{k} Y_{ii} denote the trace of Y and define

T_{k}^{n} = Tr(X_{k}^{n}),  1 ≤ n ≤ k.  (4.27)

Let J_{k} denote the subalgebra of all G_{k} -invariant polynomials. Then we have the following fundamental theorem for the theory of G_{k} -invariant polynomials.

Theorem 4.8. The algebra J_{k} is generated by the constants and the algebraically independent polynomials S_{k}^{1}, …, S_{k}^{k}. The same statement holds for T_{k}^{1}, …, T_{k}^{k}. Moreover, the following recursive formula also holds:

(−1)^{n}(n + 1)S_{k}^{n+1} = ∑_{i+j=n}(−1)^{i}S_{k}^{i}T_{k}^{j+1},  n = 0, 1, …, k − 1.  (4.28)

Proof. See [16, Theorem 1.5.7, p. 10] and [26, p. 39].

As in Subsection 4.4., it is easy to verify that there is a well-defined action of G^{∞} = GL^{∞}(ℂ) on A_{∞}.

Let S^{n} = lim_{←}S_{k}^{n} and T^{n} = lim_{←}T^{n}. Let X = lim X_{k} denote the inductive limit of X_{k}. Then S^{n} and T^{n} can be symbolically represented as

S^{n} = ∑_{i_{1}<i_{2}<···<i_{n}} △_{i_{1}···i_{n}}^{i_{1}···i_{n}}(X),  T^{n} = Tr(X^{n}),  n ∈ ℕ.  (4.29)

Then Theorem 3.10 implies the following

Theorem 4.9. Each of the sets {1, S^{n}; n ∈ ℕ} and {1, T^{n}; n ∈ ℕ} forms an inverse limit basis for the subalgebra J of all G^{∞}-invariants in the algebra A_{∞}.

Remark 4.10. Although they are both bases for J, for practical applications the basis {1, T^{n}; n ∈ ℕ} is more suitable because of its simpler form (see [22, Remark 3.12]).

5. Conclusion

In this article we have developed a coherent and comprehensive invariant theory of inductive limits of groups acting on inverse limits of the categories of modules, rings, or algebras. On one hand, we have succeeded in generalizing Hilbert’s finiteness theorem to many cases in our context. On the other hand, there are many cases in which the bases of the rings of invariants are not finitely generated, and in these cases the notion of an algebraic basis is not adequate. This led us to introduce the notion of inverse limit basis which naturally involves a topology on inverse limits. This is illustrated by examples in Subsections 4.6. and 4.7.. Also, the example in Subsection 4.7. can be generalized to the case of the inductive limit G^{∞} of the chain {G_{k}}, where each G_{k} is a semisimple connected complex analytic group acting on the algebra of polynomial functions on its Lie algebra via the co-adjoint representation. Then the Chevalley restriction theorem (see, e.g., [25, Theorem 4.9.2, p. 335]) gives us a procedure to find a basis for the ring of G_{k} -invariants. This was achieved in a more general context by Procesi in [15]. We hope to generalize these invariants to our context in a future work.

Acknowledgement: The authors wish to thank the technical staff of the Department of Mathematics of the University of Iowa, Ms. Cymie Wehr and Mr. Brian Treadway, for their impeccable job and for their patience in preparing the manuscript.

References

[1]  Artin, M., “Algebra,” Prentice-Hall, Englewood Cliffs, NJ, 1991.

[2] Barut, A., and R. Raczka, “Theory of Group Representations and Applications, ” Polish Scientific Publishers, Warsaw, 1977, 2nd ed., World Scientific Publishing, Singapore, 1986.

[3] Bourbaki, N., “Éléments de Mathématique: Algèbre,” Chapitres 4 et 5, Actualités Sci. Indust., vol. 1102, Hermann, Paris, 1959.

[4] —, “Éléments de Mathématique: Algèbre,” Chapitre 2, Actualités Sci. Indust., vol. 1236, Hermann, Paris, 1962.

[5]  Dugundji, J., “Topology,” Allyn and Bacon, Boston, 1978.

[6] Gould, M., and N. Stoilova, Casimir invariants and characteristic identities for gl(∞), J. Math. Phys. 38 (1997), 4783–4793.

[7] Hilbert, D., “Theory of Algebraic Invariants,” Cambridge University Press, Cambridge, 1993.

[8] Ismagilov, R., “Representations of Infinite-Dimensional Groups,” Transl. Math. Monographs, vol. 152, American Mathematical Society, Providence, 1996.

[9] Kac, V., “Infinite-Dimensional Lie Algebras,” Cambridge University Press, Cambridge, 1990.

[10] Kass, S., ed., “Infinite-Dimensional Lie Algebras and their Applications”, World Scientific Publishing, Teaneck, NJ, 1988.

[11] Macdonald, I. G., “Symmetric Functions and Orthogonal Polynomials, ” Univ. Lecture Ser., vol. 12, American Mathematical Society, Providence, 1998.

[12] Mumford, D., Hilbert’s fourteenth problem–the finite generation of subrings such as rings of invariants, pp. 431–444, in: “Mathematical Developments Arising from Hilbert Problems” (F. E. Browder, ed.), Proc. Sympos. Pure Math., vol. 28, American Mathematical Society, Providence, 1976.

[13] Ol’shanskiĭ, G. I., The method of holomorphic extensions in the theory of unitary representations of infinite-dimensional classical groups, Functional Anal. Appl. 22 (1988), 273–285.

[14] —, Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians, “Topics in Representation Theory” (A. A. Kirillov, ed.), Adv. Soviet Math., vol. 2, American Mathematical Society, Providence, 1991.

[15] Procesi, C., The invariant theory of n×n matrices, Adv. Math. 19 (1976), 306–381.

[16] Springer, T. A., “Invariant Theory, ” Lecture Notes in Math., vol. 585, Springer-Verlag, Berlin-New York, 1977.

[17] Ton-That, T., Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc. 216 (1976), 1–46.

[18] —, Symplectic Stiefel harmonics and holomorphic representations of symplectic groups, Trans. Amer. Math. Soc. 232 (1977), 265–277.

[19] —, Sur la décomposition des produits tensoriels des représentations irréductibles de GL(k, C), J. Math. Pures Appl. (9) 56 (1977), 251–261; Dual representations and invariant theory, pp. 205–221, “Representation Theory and Harmonic Analysis” (T. Ton-That, et al., eds.), Contemp. Math., vol. 191, American Mathematical Society, Providence, 1995.

[20] —, Invariant theory for tame representations of infinitedimensional classical groups, Special Session in Invariant Theory, 924 AMS Meeting, Montréal, Sept. 26–28, 1997, Abstract 924-22-210.

[21] —, Invariant theory of some infinite-dimensional Lie groups, p. 110, International Congress of Mathematicians, Berlin, 1998, Short Communications, 1998.

[22] —, Poincaré-Birkhoff-Witt theorems and generalized Casimir invariants for some infinite-dimensional Lie groups: I , J. Phys. A 32 (1999), 5975– 5991.

[23] —, Reciprocity theorems for holomorphic representations of some infinitedimensional groups, Helv. Phys. Acta 72 (1999), 221–249.

[24] Tran, T. D., “Invariant Theory for Infinite–Dimensional Classical Groups”, Ph.D. Thesis, The University of Iowa, December 1999.

[25] Varadarajan, V. S., “Lie Groups, Lie Algebras, and Their Representations,” 2nd ed., Springer-Verlag, Berlin-New York, 1984.

[26] Weyl, H., “The Classical Groups: Their Invariants and Representations,” 2nd ed., Princeton University Press, Princeton, NJ, 1946.

Tuong Ton-That Department of Mathematics University of Iowa Iowa City, IA 52242 USA tonthat@math.uiowa.edu Thai-Duong Tran Department of Computer Science MCS #560 Southwest Texas University 601 University Drive San Marcos, TX 78666 USA tt10@swt.edu

Received February 4, 2002 and in final form December 19, 2002