c 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 ofbasisof the rings of invariants is introduced, and a generalization ofHilbert’s Finiteness Theoremis given. A generalization of some notions attached to the classical invariant theory such asHilbert’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 Represen- tations, 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 represen- tation 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 there- fore 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’shanski˘ı [13].

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

ISSN 0949–5932 / $2.50 c Heldermann Verlag

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 represen- tations 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 mod- ules, rings, or algebras. In this article we develop a coherent and comprehensive theory of these invariants. To illustrate how they arise naturally from the rep- resentation 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^{0}_{k} = SO_{k}(R). Then G_{k}
(resp. G^{0}_{k}) acts on V_{k} by right multiplication, and this induces an action of G_{k}
(resp. G^{0}_{k}) on A_{k}. Then the ring of G_{k} (resp. G^{0}_{k})-invariants is generated by
the constants and p^{0}_{k} = Pk

i=1X_{i}^{2}, where (X_{1}, . . . , X_{k}) = X ∈ V_{k}, and the G_{k}-
invariant differential operators are generated by 4k = p^{0}_{k}(D) = Pk

i=1∂_{i}^{2}, where

∂_{i}^{2} = _{∂X}^{∂}^{2}2

i . If Hk (resp. Hk^{d}) denote the subspace of harmonic polynomials (resp.

harmonic homogeneous of degree d), i.e., polynomials that are annihilated by 4k, then for k > 2 we have the “separation of variables” theorem

P^{(m)}(Vk) = X

i=0,... ,[m/2]

⊕(p^{0}_{k})^{(i)}H^{(m−2i)}k , (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^{0}_{k})^{(i)}H^{(m−2i)}k

is an irreducible Gk (resp. G^{0}_{k})-module of signature (m−2i,0, . . . ,0

| {z }

[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=S_{∞}

k=1G_{k} (resp. G^{0} =S_{∞}

k=1G^{0}_{k}); 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 P_{∞}

i=1X_{i}^{2} and X = (X_{1}, . . . , X_{k}, . . .)
denote the formal infinite row matrix, then Xg, g ∈G (i.e., g ∈Gk for some k),
equals ((X_{1}, . . . , X_{k})g, X_{k+1}, . . .), and it follows p^{0} is formally G-invariant. Set
4=P_{∞}

i=1∂_{i}^{2} and let 4 operate on A as follows:

If f ∈A then f ∈A_{k} for some k∈N, and 4f :=4kf.

Thus f ∈A is harmonic if 4f = 0, and H^{d} ={f ∈A^{d}| 4f = 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^{0}_{k} 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 hT^{n};n ∈Ni denote the
subalgebra of A_{∞} generated by the T^{n}’s, and J denote the subalgebra of G-
invariants in A_{∞}, then we can only show that hT^{n};n∈Ni 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^{j}_{i} :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^{k}_{i} =µ^{j}_{i} ◦µ^{k}_{j}.

Then the family {A_{i};µ^{j}_{i}} is called aninverse spectrum over the index set I
with connecting morphisms µ^{j}_{i}.

Form Q

i∈IA_{i} and let p_{i} denote its projection onto A_{i}. The subset
{a= (a_{i})∈Y

i∈I

A_{i} |a_{i} =p_{i}(a) = µ^{j}_{i} ◦p_{j}(a) = µ^{j}_{i}(a_{j}), whenever i≤j}

is called the inverse (or projective) limit of the inverse spectrum {A_{i};µ^{j}_{i}} 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_{∞} 6= ∅ 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^{j}_{i} 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
Q

i∈IA_{i} has a nontrivial product topology. Since each mapping µ^{j}_{i} is clearly
continuous it follows that the projection mapsp_{i}, and hence, the canonical maps µ_{i}
are continuous. It follows from Theorem 2.3, p. 428, of [5] that the sets {µ^{−1}_{i} (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 = (ai) of A_{∞}. Moreover, µ^{−}_{j}^{1}(aj) ⊂ µ^{−}_{i} ^{1}(ai)
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^{−1}_{i} (a_{i}) ⊂ µ^{−1}_{i} (U_{i}) ⊂ V. Since the set {a_{i}} is
open in A_{i}, µ^{−1}_{i} (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^{−1}_{j} (a_{j})) ={a_{j}}, or b_{j} =a_{j}. This implies that b_{i} =µ^{j}_{i}(b_{j}) = µ^{j}_{i}(a_{j}) = a_{i}, and
thus b∈µ^{−1}_{i} (a_{i}). This shows that µ^{−1}_{j} (a_{j})⊂µ^{−1}_{i} (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^{−1}_{i} (a_{i})| all i∈I, and all a= (a_{i})∈A_{∞}}, unless otherwise specified.

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

Lemma 2.3. Let S ⊂A_{∞}. Then x∈S¯ 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^{n}_{i} =x_{i}
whenever n≥N_{i}.

Proof. By definition the sequence {x^{n}} converges to x if: “for every neighbor-
hood 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^{−1}_{i} (x_{i}), ∀i∈ I. This means that x^{n} →x if
and only if

“∀i∈I ∃N_{i} ∀n≥N_{i} :x^{n}_{i} =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^{n}_{i} =a_{i}”,

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

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

“∀i∈I ∃N_{i} ∀n≥N_{i} (a·b)_{i} =a_{i} ·b_{i} =a^{n}_{i} ·b^{n}_{i} = (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 Si ⊂Ai and assume that µ^{j}_{i}(Sj)⊂ Si whenever i ≤ j.
Then {S_{i};µ^{j}_{i}|Sj} 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_{∞}∩Q

i∈IS_{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_{∞}= ¯S.

Proof. Let s ∈ S; then s_{i} = µ_{i}(s) ∈ S_{i}, ∀i ∈ I, and µ^{j}_{i}(s_{j}) = µ^{j}_{i} ◦µ^{j}(s) =
µ_{i}(s) = s_{i}. Thus µ^{j}_{i}(S_{j}) ⊂ S_{i} and S ⊂ S_{∞}. Let us show that S_{∞} is closed in
A_{∞}. Let s^{0} ∈ S¯_{∞}; 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^{n}_{i} =s^{0}_{i} whenever n≥N_{i}. This implies that s^{0}_{i} ∈S_{i} for every i∈I,
and hence, s^{0} ∈ S_{∞}. Thus S_{∞} is closed, and it follows that ¯S ⊂ 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^{j}_{i} =µ^{j}_{i}(s^{j}_{j}) =µ^{j}_{i}(s_{j}) =s_{i}. It follows
that, for every i∈ I, there exists Ni =i such that s^{j}_{i} =si whenever j ≥Ni =i.

Theorem 2.4 implies that s^{j} →s, and thus s ∈S¯. Therefore, S_{∞}⊂S¯, and hence
S¯=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^{j}_{i} |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, Si are subrings of Ri such that µ^{j}_{i}(Sj)⊂Si 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 Ri. Moreover, S_{∞} is also a subring of R_{∞} such that S_{∞}= ¯S.

Theorem 2.8. Let {R_{i};µ^{j}_{i} |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^{j}_{i}(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_{∞}= ¯I.

Let R be a unital commutative ring and let S 6= ∅ be any subset of R. Let hSi denote the subring generated by S ; i.e., the smallest subring containing S. Similarly if S 6=∅ 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 P

finiter_{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 hSii =µi(hSi),
(S)_{i} =µ_{i}((S)), all i∈I. Then the following hold:

(i) lim

← hSii 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^{j}_{i}} 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^{j}_{i}(b_{j}) =µ^{j}_{i}(r_{j}a_{j}) =µ^{j}_{i}(r_{j})µ^{j}_{i}(a_{j}), or (2.1)
riai =µ^{j}_{i}(rj)ai.

If I = {0} then obviously I is closed in R_{∞}. If I 6= {0} then we may assume
without loss of generality that a_{i} 6= 0 for sufficiently large i. For such an i, Eq.

(2.1) implies that µ^{j}_{i}(rj) = ri since Ii is an integral domain. Set r = (ri); then
since µ^{j}_{i}(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^{j}_{i}(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^{j}_{i}(I_{j}) ⊂ I_{i} it follows
that a_{i} =µ^{j}_{i}(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 Xij, 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 theα_{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}| = P

i,jα_{ij}. Then every polynomial p_{k} ∈ A_{k} can be
written in exactly one way in the form

p_{k} = X

|(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 pk; the c(α)_{k}X^{(α)}^{k} are called theterms of pk. For l≥k
every polynomial p_{l} =P

c_{(α)}_{l}X^{(α)}^{l} of A_{l} can be written uniquely in the form
p_{l}=X

(α^{0})l

c_{(α}0)lX^{(α}^{0}^{)}^{l}+ X

(α^{00})l

c_{(α}00)lX^{(α}^{00}^{)}^{l} (2.3)

where in each (α^{0})_{l} all the integers α^{0}_{ij} are zero whenever j > k, and in each (α^{00})_{l}
there must be an integer α^{00}_{ij} >0 whenever k < j≤l. Identify each (α^{0})_{l} with an
element (α)k and define the map µ^{l}_{k} :Al →Ak by

p_{k} =µ^{l}_{k}(p_{l}) = X

(α^{0})_{l}

c_{(α}0)_{l}X^{(α}^{0}^{)}^{l} =X

(α)_{k}

c_{(α)}_{k}X^{(α)}^{k}. (2.4)

Using Eqs. (2.3) and (2.4) we can easily deduce that µ^{l}_{k} is an algebra homo-
morphism and we have µ^{m}_{k} = µ^{l}_{k} ◦µ^{m}_{l} 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^{l}_{k} are surjective. These connecting homomorphisms are called
truncation homomorphisms.

Let A_{∞} denote the inverse limit of the inverse spectrum {A_{k};µ^{l}_{k}}. 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 Th´eor`eme 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^{l}_{k}(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^{i}_{k} ∈ A_{k} by p^{i}_{k} = Pk

j=1X_{ij}^{2}. Consider the thread
f^{i} = (p^{i}_{k}) in A_{∞}. As remarked above for each k, p^{i}_{k} can be considered as an
element of A_{∞} so that if we set f^{i,k} =p^{i}_{k} 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^{l}_{k}(p^{i}_{l}) = p^{i}_{k} = f_{k}^{i} whenever l ≥ k. Theorem 2.4
implies that lim

k→∞f^{i,k} = f^{i}. In general, if S^{k} = Pk

i=1g^{i}, k ∈ N, is a convergent
sequence in A_{∞}, then we write its limit as P_{∞}

i=1g^{i}. Similarly, if P^{k} =Qk

i=1g^{i} is

a convergent sequence in A_{∞}, then we write its limit as Q_{∞}

i=1g^{i}. Thus with this
convention we have f^{i} =P∞

j=1X_{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} =Pn

i=1X_{ii}f^{i}; then {S^{n}|n ∈N} is a sequence in I such that
S_{k}^{n}=µ_{k}(S^{n}) =

n

X

i=1

µ_{k}(X_{ii})µ_{k}(f^{i}) =

( Pn

i=1X_{ii}f_{k}^{i}, k > n,
Pk

i=1X_{ii}f_{k}^{i}, k≤n. (2.5)
Set S = P_{∞}

i=1X_{ii}f^{i} and let us show that S ∈ A_{∞} and lim

n→∞S^{n} = S. First
consider {(Pk

i=1X_{ii}f_{k}^{i})_{k} | k ∈ N}. Then µ^{l}_{k}(Pl

i=1X_{ii}f_{l}^{i}) =Pk

i=1X_{ii}f_{k}^{i} whenever
l ≥ k. Thus ((Pk

i=1X_{ii}f_{k}^{i})_{k}) is a thread in A_{∞} and S = ((Pk

i=1X_{ii}f_{k}^{i})_{k}). Now
we have from Eq. (2.5) “∀k ∈N ∃Nk =k ∀n≥k :S_{k}^{n} =Sk”, which means that

n→∞limS^{n}=S ∈I¯.

Now let us show that S /∈ I. A general element in I is of the form g = Pm

i=1h^{i}f^{i} where h^{i} ∈ A_{∞}, 1 ≤ i ≤ m. Then g_{k} =µ_{k}(g) = Pm

i=1µ_{k}(h^{i})µ_{k}(f^{i}) =
Pm

i=1h^{i}_{k}f_{k}^{i} where the h^{i}_{k} belong to A_{k}, 1≤i≤m. Thus each h^{i}_{k} 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} = Pk

i=1X_{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 adirect (orinductive)system with index
set I and connecting morphisms λ_{ij}.

The image of y_{i} ∈Y_{i} under any connecting morphism is called asuccessor
of y_{i}. Let Y = S

i∈IY_{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 : S

i∈IY_{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^{∞} = S

k∈IG_{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 {Ai;µ^{j}_{i}} 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^{k}_{j} and λ_{jk} satisfy the following
condition

g ·(µ^{k}_{j}(ak)) = µ^{k}_{j}(λjk(g)·ak), (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

(g·a)_{k} :=g·a_{k},

(g·a)_{n} :=λ_{kn}(g)·a_{n}, if n≥k,
(g·a)j :=µ^{k}_{j}(g ·ak), if j ≤k,

∀g ∈G_{k}, ∀a= (a_{k})∈A_{∞}.

(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^{l}_{i}((g·a)l) whenever l≥i.

If l ≥i≥k then by Eq. (3.1) we have

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

=λ_{ki}(g)·(µ^{l}_{i}(a_{l})) = λ_{ki}(g)·a_{i} = (g·a)_{i}.
If l ≥k > i then by definition we have

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

=µ^{k}_{i}(g·µ^{l}_{k}(a_{l})) =µ^{k}_{i}(g·a_{k}) = (g·a)_{i}.
If k > l≥i then by definition we have

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

Now let us show that Eq. (3.2) defines an action of G^{∞} on A_{∞}. Let g1 ∈Gi

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

(g1g2)·a=g1·(g2·a), for all a= (ak)∈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}

=λkn(g1)·(λkn(g2)·an) =λkn(g1)·(g2·a)n = (g1·(g2·a))n. For j ≤k we have

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

=µ^{k}_{j}(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^{k}_{j}(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 ˜p denote the polynomial function obtained by substituting X_{ij}
by x_{ij} ∈ F. Since F is an infinite field the mapping p → p˜ 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^{l}_{k} be the truncation homomorphisms described in Example
2.12. Let A_{∞} denote the inverse limit of the inverse spectrum {A_{k};µ^{l}_{k}}. 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 Jk denote the subalgebra of Gk-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^{l}_{k}(Jl)⊂Jk 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) = ¯J. Thus, finally we have the chain of inclusions.

J ⊂J¯= lim

← µk(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 gener- alize 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 alge-
braically 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 bealgebraically 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 h{f^{α}}^{α}∈Λi=
A_{∞}, where h{f^{α}}α∈Λi 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 poly- nomials 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}^{α}

0}α∈Λ is algebraically independent
in A_{k}_{0} then {f^{α}}α∈Λ is also algebraically independent in A_{∞} and h{f^{α}}α∈Λi 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 h{f^{α}}α∈Λi is closed we suppose
that ϕ is the limit of a sequence{ϕ^{n}} inh{f^{α}}α∈Λiand verify that ϕ∈ h{f^{α}}α∈Λi.
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^{n}_{i} = ϕ_{i} whenever n ≥ N_{i}. In particular,
for i = k_{0} there exists N_{k}_{0} such that ϕ^{n}_{k}

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

where p_{n} is a polynomial depending on n. Since µ^{i}_{k}_{0} is an algebraic homomor-
phism, ϕk0 =ϕ^{n}_{k}_{0} =µ^{i}_{k}_{0}(ϕ^{n}_{i}) =pn({f_{k}^{α}_{0}}). The fact that {f_{k}^{α}_{0}}^{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 ∈ h{f^{α}}^{α}∈Λi, 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
h{f_{i}^{α}}α∈Λi=A_{i}. Then Theorem 3.6 implies that {f^{α}}α∈Λ is also an algebraic ba-
sis 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^{l}_{k}} denote the inverse spectrum with connecting homomorphisms
µ^{l}_{k} : A_{l} → A_{k}, l ≥ k, l, k ∈ N. The µ^{l}_{k} are truncation homomorphisms, which
in this case can be defined simply by setting µ^{l}_{k}(x_{j}) = x_{j} for 1 ≤ j ≤ k and
µ^{l}_{k}(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^{l}_{k}}. 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 Gk acts linearly. Let C[Vk] 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 p1, . . . , ps 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∈Λk 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 Λk ⊂ Λl. Set Λ = S

k∈IΛk. 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 Jk. If
f^{α} = lim

←f_{k}^{α} then the family {f^{α}}α∈Λ generates J. In particular, if {f_{k}^{α}}α∈Λk is
an algebraic basis for J_{k} then {f^{α}}α∈Λ is an inverse limit basis for J.

Proof. Let J^{0} =h{f^{α}};α∈Λi; then by assumption µ_{k}(J^{0}) =J_{k}, for all k ∈I.

By Theorem 2.6, ¯J^{0} =J_{∞}. By Theorem 3.4, J_{∞} =J. Therefore ¯J^{0} =J. Now if
in addition the sets {f_{k}^{α}}α∈Λk 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.