Quantized coinvariants at transcendental q

K R Goodearl and T H Lenagan^{∗}

Abstract

A general method is developed for deriving Quantum First and Second Fundamental Theorems of Coinvariant Theory from classical analogs in Invariant Theory, in the case that the quantization parameter q is transcendental over a base field. Several examples are given illustrating the utility of the method; these recover earlier results of various researchers including Domokos, Fioresi, Hacon, Rigal, Strickland, and the present authors.

2000 Mathematics Subject Classification: 16W35, 16W30, 20G42, 17B37, 81R50.

Keywords: coinvariants, First Fundamental Theorem, Second Fundamental Theorem, quantum group, quantized coordinate ring.

Introduction and background

In the classic terminology of Hermann Weyl [18], a full solution to any invariant theory problem should incorporate a First Fundamental Theorem, giving a set of generators (finite, where possible) for the ring of invariants, and a Second Fundamental Theorem, giving generators for the ideal of relations among the generators of the ring of invariants. Many of the classical settings of Invariant Theory have quantized analogs, and one seeks corresponding analogs of the classical First and Second Fundamental Theorems. However, the setting must be dualized before potential quantized analogs can be framed, since there are no quantum analogs of the original objects, only quantum analogs of their coordinate rings. Hence, one rephrases the classical results in terms of rings of \underline{co}invariants (see below), and then seeks quantized versions of these. A first stumbling block is that in general these coactions are not algebra homomorphisms, and so at the outset it is not even obvious that the coinvariants form a subalgebra. However, this can often be established; cf [9], Proposition 1.1; [4], Proposition 1.3.

Typically, a classical invariant theoretical setting is quantized uniformly with respect to a parameter q, that is, there is a family of rings of coinvariants to be determined, parametrized by nonzero scalars in a base field, such that the case q = 1 is the classical one. (Many authors, however, restrict attention to special values of q, such as those transcendental over the rational number field, and do not address the general case.) As in the classical setting, one usually has identified natural candidates to generate the ring of coinvariants. Effectively, then, one has a parametrized inclusion of algebras (candidate subalgebras inside the algebras of coinvariants), which is an equality at q = 1, and one seeks equality for other values of q. In the best of all worlds, the equality at q = 1 could be “lifted” by some general process to equality at arbitrary q. Lifting to transcendental q has been done succesfully in some cases, by ad hoc methods – see, for example, [6], [17]. Also, an early version of [9] for the transcendental case was obtained in this way. Quantum Second Fundamental Theorems can be approached in a similar manner. We develop here a general method for lifting equalities of inclusions from q = 1 to transcendental q, which applies to many analyses of quantized coinvariants.

∗This research was partially supported by NSF research grant DMS-9622876 and by NATO Collaborative Research Grant CRG.960250.

In order to apply classical results as indicated above, we must be able to transform invariants to coinvariants. For morphic actions of algebraic groups, the setting of interest to us, invariants and coinvariants are related as follows. Suppose that γ : G × V → V is a morphic action of an algebraic group G on a variety V. This action induces an action of G on O(V), where (x.f)(v) = f(x^{−1}.v) for x ∈ G, f ∈ O(V), and v ∈ V. The invariants for this action are, of course, those functions in O(V) which are constant on G-orbits. The comorphism of γ is an algebra homomorphism γ^{∗} : O(V) → O(G) ⊗ O(V), with respect to which O(V) becomes a left O(G)-comodule. Now a function f ∈ O(V) is a coinvariant in this comodule when γ^{∗}(f) = 1 ⊗ f. Since 1 ⊗ f corresponds to the function (x, v) ↦ f(v) on G × V, we see that γ^{∗}(f) = 1 ⊗ f if and only if f(x.v) = f(v) for all x ∈ G and v ∈ V, that is, if and only if f is an invariant function. To summarize:

O(V)^{co O(G)} = O(V)^{G}.

Quantized coordinate rings have been constructed for all complex semisimple algebraic groups G. These quantized coordinate rings are Hopf algebras, which we denote O_{q}(G), since we are concentrating on the single parameter versions. In those cases where a morphic action of G on a variety V has been quantized, we have a quantized coordinate ring O_{q}(V) which supports an O_{q}(G)-coaction. The coaction is often not an algebra homomorphism, but nonetheless – as mentioned above – the set of O_{q}(G)-coinvariants in O_{q}(V) is typically a subalgebra. The goal of Quantum First and Second Fundamental Theorems for this setting is to give generators and relations for the algebra O_{q}(V)^{co O_{q}(G)} of O_{q}(G)-coinvariants in O_{q}(V). We discuss several standard settings in later sections of the paper, and outline how our general method applies. We recover known Quantum First and Second Fundamental Theorems at transcendental q in these settings, with some simplifications to the original proofs, and in some cases extending the range of the theorems.

Throughout, k will denote a field, which may be of arbitrary characteristic and need not be algebraically closed.

1 Reduction modulo q − 1

Throughout this section, we work with a field extension k^{∘} ⊂ k and a scalar q ∈ k \ k^{∘} which is transcendental over k^{∘}. Thus, the k^{∘}-subalgebra R = k^{∘}[q, q^{−1}] ⊂ k is a Laurent polynomial ring. Let us denote reduction modulo q − 1 by overbars, that is, given any Rmodule homomorphism ϕ : A → B, we write \overline{ϕ} : \overline{A} → \overline{B} for the induced map A/(q−1)A → B/(q − 1)B.

Proposition 1.1 Let A ⟶^{ϕ} B ⟶^{ψ} C be a complex of R-modules, such that C is torsionfree. Suppose that there are R-module decompositions

A = ⨁_{j∈J} A_{j},  B = ⨁_{j∈J} B_{j},  C = ⨁_{j∈J} C_{j}

such that B_{j} is finitely generated, ϕ(A_{j}) ⊆ B_{j}, and ψ(B_{j}) ⊆ C_{j} for all j ∈ J.

If the reduced complex \overline{A} ⟶^{\overline{ϕ}} \overline{B} ⟶^{\overline{ψ}} \overline{C} is exact, then so is

\xymatrix{k ⊗_{R} A \ar[r]^{id ⊗ϕ} & k ⊗_{R} B \ar[r]^{id ⊗ψ} & k ⊗_{R} C.}

Proof. The hypotheses and the conclusions all reduce to the direct sum components of the given decompositions, so it is enough to work in one component. Hence, there is no loss of generality in assuming that B is finitely generated.

Let S denote the localization of R at the maximal ideal (q − 1)R. Set

Ã = S ⊗_{R} A,  B̃ = S ⊗_{R} B,  C̃ = S ⊗_{R} C,

and let Ã \underrightarrow{ϕ̃} B̃ \underrightarrow{ψ̃} C̃ denote the induced complex of S-modules. Since Ã/(q − 1)Ã is naturally isomorphic to A/(q−1)A = \overline{A}, and similarly for B and C, there is a commutative diagram

\xymatrix{Ã \ar[r]^{ϕ̃} \ar[d]_{α} & B̃ \ar[r]^{ψ̃} \ar[d]_{β} & C̃ \ar[d]^{γ} \\ \overline{A} \ar[r]^{\overline{ϕ}} & \overline{B} \ar[r]^{\overline{ψ}} & \overline{C}}

where α, β, γ are epimorphisms with kernels (q − 1)Ã etc.

The bottom row of the diagram is exact by hypothesis, and we claim that the top row is exact. Consider an element x ∈ ker ψ̃. Chasing x around the diagram, we see that x − ϕ̃(y) = (q − 1)z for some y ∈ Ã and z ∈ B̃. Note that (q − 1)z ∈ ker ψ̃. Since C̃ is torsionfree, it follows that z ∈ ker ψ̃. Thus ker ψ̃ ⊆ ϕ̃(Ã) + (q − 1) ker ψ̃, whence

(ker ψ̃)/ϕ̃(Ã) = (q − 1)[(ker ψ̃)/ϕ̃(Ã)].

Since ker ψ̃ is a finitely generated S-module, it follows from Nakayama’s Lemma that (ker ψ̃)/ϕ̃(Ã) = 0, establishing the claim.

Since k is a flat S-module, the sequence \xymatrix{k ⊗_{S} Ã \ar[r]^{id ⊗ϕ} & k ⊗_{S} B̃ \ar[r]^{id ⊗ψ} & k ⊗_{S} C̃} is exact. This is isomorphic to the sequence

\xymatrix{k ⊗_{R} A \ar[r]^{id ⊗ϕ} & k ⊗_{R} B \ar[r]^{id ⊗ψ} & k ⊗_{R} C}

and therefore the latter is exact. □

Proposition 1.1 is useful in obtaining quantized versions of both First and Second Fundamental Theorems. For ease of application, we write out general versions of both situations with appropriate notation, as follows.

Theorem 1.2 Let H be an R-bialgebra and B a left H-comodule with structure map λ : B → H ⊗_{R} B, and let ϕ : A → B be an R-module homomorphism whose image is contained in B^{co H}. Assume that B and H are torsionfree R-modules, and suppose that there exist Rmodule decompositions A = ⨁_{j∈J} A_{j} and B = ⨁_{j∈J} B_{j} such that B_{j} is finitely generated, ϕ(A_{j}) ⊆ B_{j}, and λ(B_{j}) ⊆ H ⊗_{R} B_{j} for all j ∈ J.

Now \overline{B} is a left comodule over the k^{∘}-bialgebra \overline{H}, and k ⊗_{R} B is a left comodule over the k-bialgebra k ⊗_{R} H. If \overline{B}^{co \overline{H}} equals the image of the induced map \overline{ϕ} : \overline{A} → \overline{B}, then [k ⊗_{R} B]^{co k⊗_{R}H} equals the image of the induced map id ⊗ϕ : k ⊗_{R} A → k ⊗_{R} B.

Proof. Since R is a principal ideal domain, H ⊗_{R} B is a torsionfree R-module. We may identify \overline{H ⊗_{R} B} with \overline{H} ⊗_{k^{∘}} \overline{B} and k ⊗_{R} (H ⊗_{R} B) with (k ⊗_{R} H) ⊗_{k} (k ⊗_{R} B). Now apply Proposition 1.1 to the complex A ⟶^{ϕ} B ⟶^{λ−γ} H ⊗_{R} B, where γ : B → H ⊗_{R} B is the map b ↦ 1 ⊗ b. □

Theorem 1.3 Let ϕ : A → B be a homomorphism of R-algebras, and I an ideal of A contained in ker ϕ. Assume that B is a torsionfree R-module, and suppose that there exist R-module decompositions

I = ⨁_{j∈J} I_{j},  A = ⨁_{j∈J} A_{j},  B = ⨁_{j∈J} B_{j}

such that A_{j} is finitely generated, I_{j} ⊆ A_{j}, and ϕ(A_{j}) ⊆ B_{j} for all j ∈ J.

If ker \overline{ϕ} equals the image of \overline{I}, then the kernel of the k-algebra homomorphism id ⊗ϕ : k ⊗_{R} A → k ⊗_{R} B equals the image of k ⊗_{R} I.

Proof. Apply Proposition 1.1 to the complex I ⟶^{η} A ⟶^{ϕ} B, where η is the inclusion map. □

Proposition 1.1 can easily be adapted to yield other exactness conclusions. In particular, suppose that, in addition to the given hypotheses, the cokernels of ϕ and ψ are torsionfree R-modules. Then one can show that for all nonzero scalars λ ∈ k^{∘}, the induced complex

A/(q − λ)A ⟶ B/(q − λ)B ⟶ C/(q − λ)C

is exact. Unfortunately, in the applications to quantized coinvariants, it appears to be a difficult task to verify the above additional hypotheses.

2 The interior action of GL_{t} on M_{m,t} × M_{t,n}

Let m, n, t be positive integers with t ≤ min{m, n}. The group GL_{t}(k) acts on the variety V := M_{m,t}(k) × M_{n,t}(k) via the rule

g · (A, B) = (Ag^{−1}, gB),

and consequently on the coordinate ring O(V) ≅ O(M_{m,t}(k)) ⊗ O(M_{t,n}(k)). The invariant theory of this action is closely related to the matrix multiplication map

μ : V ⟶ M_{m,n}(k)

and its comorphism μ^{∗}. The classical Fundamental Theorems for this situation ([2], Theorems 3.1, 3.4) are

(1) The ring of invariants O(V)^{GL_{t}(k)} equals the image of μ^{∗}.

(2) The kernel of μ^{∗} is the ideal of O(M_{m,n}(k)) generated by all (t + 1) × (t + 1) minors.

As discussed in the introduction, the comorphism of the action map γ : GL_{t}(k)×V → V is an algebra homomorphism γ^{∗} : O(V) → O(GL_{t}(k)) ⊗ O(V) which makes O(V) into a left comodule over O(GL_{t}(k)), and the coinvariants for this coaction equal the invariants for the action above: O(V)^{GL_{t}(k)} = O(V)^{co O(GL_{t}(k))}.

We may consider quantum versions of this situation, relative to a parameter q ∈ k^{×}, using the standard quantized coordinate rings O_{q}(M_{m,t}(k)), O_{q}(M_{t,n}(k)), and O_{q}(GL_{t}(k)). We write X_{ij} for the standard generators in each of these algebras; see, e.g., [8], (1.1), for the standard relations.

To quantize the classical coaction above, one combines quantized versions of the actions of GL_{t}(k) on M_{m,t}(k) and M_{n,t}(k) by right and left multiplication, respectively. The latter quantizations are algebra morphisms

ρ_{q}^{∗} : O_{q}(M_{m,t}(k)) ⟶ O_{q}(M_{m,t}(k)) ⊗ O_{q}(GL_{t}(k))
λ_{q}^{∗} : O_{q}(M_{t,n}(k)) ⟶ O_{q}(GL_{t}(k)) ⊗ O_{q}(M_{t,n}(k))

which are both determined by X_{ij} ↦ ∑_{l=1}^{t} X_{il} ⊗ X_{lj} for all i, j. The right comodule structure on O_{q}(M_{m,t}(k)) determines a left comodule structure in the standard way, via the map τ ∘ (id ⊗S) ∘ ρ_{q}^{∗}, where τ is the flip and S is the antipode in O_{q}(GL_{t}(k)). We set O_{q}(V) = O_{q}(M_{m,t}(k)) ⊗ O_{q}(M_{t,n}(k)), which is a left comodule over O_{q}(GL_{t}(k)) in the standard way. The resulting structure map γ_{q}^{∗} : O_{q}(V) → O_{q}(GL_{t}(k)) ⊗ O_{q}(V) is determined by the rule

γ_{q}^{∗}(a ⊗ b) = ∑_{(a),(b)} S(a_{1})b_{−1} ⊗ a_{0} ⊗ b_{0}

for a ∈ O_{q}(M_{m,t}(k)) and b ∈ O_{q}(M_{t,n}(k)), where ρ_{q}^{∗}(a) = ∑_{(a)} a_{0} ⊗ a_{1} and λ_{q}^{∗}(b) = ∑_{(b)} b_{−1} ⊗ b_{0}. It is, unfortunately, more involved to work with γ_{q}^{∗} than with its classical predecessor γ^{∗} because γ_{q}^{∗} is not an algebra homomorphism.

The analog of μ^{∗} is the algebra homomorphism μ_{q}^{∗} : O_{q}(M_{m,n}(k)) → O_{q}(V) such that X_{ij} ↦ ∑_{l=1}^{t} X_{il} ⊗ X_{lj} for all i, j. We can now state the Fundamental Theorems for the quantized coinvariants in the current situation:

(1_{q}) The set of coinvariants O_{q}(V)^{co O_{q}(GL_{t}(k))} equals the image of μ_{q}^{∗} (and so is a subalgebra of O_{q}(V)).

(2_{q}) The kernel of μ_{q}^{∗} is the ideal of O_{q}(M_{m,n}(k)) generated by all (t+1)×(t+1) quantum minors.

These theorems were proved for arbitrary q in [9], Theorem 4.5, and [8], Proposition 2.4, respectively.

To illustrate the use of Theorems 1.2 and 1.3, we specialize to the case that q is transcendental over a subfield k^{∘} ⊂ k. Set R = k^{∘}[q, q^{−1}] as in the previous section. Since the construction of quantum matrix algebras requires only a commutative base ring and an invertible element in that ring, we can form O_{q}(M_{m,t}(R)) and O_{q}(M_{t,n}(R)). These R-algebras are iterated skew polynomial extensions of R, and thus are free R-modules, as is the algebra O_{q}(V_{R}) := O_{q}(M_{m,t}(R)) ⊗_{R} O_{q}(M_{t,n}(R)). Similarly, O_{q}(M_{t}(R)) is a free R-module and an integral domain, and so the localization O_{q}(GL_{t}(R)), obtained by inverting the quantum determinant, is a torsionfree R-module, as well as a Hopf Ralgebra. Since the restrictions of ρ_{q}^{∗} and λ_{q}^{∗} make O_{q}(M_{m,t}(R)) and O_{q}(M_{t,n}(R)) into right and left comodules over O_{q}(GL_{t}(R)), respectively, the restriction γ_{R}^{∗} of γ_{q}^{∗} makes O_{q}(V_{R}) into a left O_{q}(GL_{t}(R))-comodule. Finally, μ_{q}^{∗} restricts to an R-algebra homomorphism μ_{R}^{∗} : O_{q}(M_{m,n}(R)) → O_{q}(V_{R}). It is easily checked that the image of μ_{R}^{∗} is contained in O_{q}(V_{R})^{co O_{q}(GL_{t}(R))}, and that the kernel of μ_{R}^{∗} contains the ideal I generated by all (t + 1) × (t + 1) quantum minors (cf [9], Proposition 2.3 and [8], (2.1)).

All of the quantum matrix algebras O_{q}(M_{•,•}(R)) are positively graded R-algebras, with the generators X_{ij} having degree 1, and O_{q}(V_{R}) inherits a positive grading from its two factors. In each of these algebras, the homogeneous components are finitely generated free R-modules. It is easily checked that

μ_{R}^{∗}(O_{q}(M_{m,n}(R))_{j}) ⊆ O_{q}(V_{R})_{2j} and γ_{R}^{∗}(O_{q}(V_{R})_{j}) ⊆ O_{q}(GL_{t}(R)) ⊗_{R} O_{q}(V_{R})_{j}

for all j ≥ 0. Moreover, the ideal I of O_{q}(M_{m,n}(R)) is homogeneous with respect to this grading. Note that when we come to apply Theorem 1.3, we should replace the grading on O_{q}(V_{R}) by the vector space decomposition ⨁_{j=0}^{∞}(O_{q}(V_{R})_{2j} ⊕ O_{q}(V_{R})_{2j+1}), for instance.

The classical First and Second Fundamental Theorems (1) and (2) say that \overline{μ_{R}^{∗}} maps \overline{O_{q}(M_{m,n}(R))} onto the coinvariants of \overline{O_{q}(V_{R})}, and that the kernel of \overline{μ_{R}^{∗}} equals the image of \overline{I}. Therefore Theorems 1.2 and 1.3 yield the quantized Fundamental Theorems (1_{q}) and (2_{q}) with no further work in the transcendental case.

3 The right action of SL_{r} on M_{n,r}

Fix positive integers r < n. In this section, we consider the right action of SL_{r}(k) on M_{n,r}(k) by multiplication: A.g = Ag for A ∈ M_{n,r}(k) and g ∈ SL_{r}(k). The First Fundamental Theorem for this case (cf [7], Proposition 2, p 138) says that

(1) The ring of invariants O(M_{n,r}(k))^{SL_{r}(k)} equals the subalgebra of O(M_{n,r}(k)) generated by all r × r minors.

To state the Second Fundamental Theorem for this case, let k[X] be the polynomial ring in a set of variables X_{I}, where I runs over all r-element subsets of {1, …, n}. In view of the theorem above, there is a natural homomorphism of ϕ : k[X] → O(M_{n,r}(k))^{SL_{r}(k)}.

(2) The kernel of ϕ is generated by the Plücker relations ([7], Proposition 2, p 138).

To quantize this situation, we make O_{q}(M_{n,r}(k)) into a right comodule over O_{q}(SL_{r}(k)) via the algebra homomorphism ρ : O_{q}(M_{n,r}(k)) → O_{q}(M_{n,r}(k)) ⊗ O_{q}(SL_{r}(k)) such that ρ(X_{ij}) = ∑_{l=1}^{r} X_{il} ⊗ X_{lr} for all i, j. The First Fundamental Theorem for the quantized coinvariants is the statement

(1_{q}) The set of coinvariants O_{q}(M_{n,r}(k))^{co O_{q}(SL_{r}(k))} equals the subalgebra of O_{q}(M_{n,r}(k)) generated by all r × r quantum minors.

This follows from work of Fioresi and Hacon under the assumptions that k is algebraically closed of characteristic 0 and q is transcendental over some subfield of k ([6], Theorem 3.12) – in fact, they prove (in our notation) that (1_{q}) holds with k replaced by a Laurent polynomial ring k^{∘}[q, q^{−1}]. The result is obtained for any nonzero q in an arbitrary field k in [11], Theorem 5.2. The Fioresi-Hacon version of the Second Fundamental Theorem yields a presentation of O_{q}(M_{n,r}(k))^{co O_{q}(SL_{r}(k))} in terms of generators λ_{i_{1}···i_{r}} where i_{1} ⋯ i_{r} runs over all unordered sequences of r distinct integers from {1, …, n}. Each λ_{i_{1}···i_{r}} is mapped to (−q)^{ℓ(π)}D_{{i_{1},...,i_{r}}} where ℓ(π) denotes the length of the permutation π sending l ↦ i_{l} for l = 1, …, r, and D_{{i_{1},...,i_{r}}} denotes the r × r quantum minor with row index set {i_{1}, …, i_{r}}.

(2_{q}) The algebra O_{q}(M_{n,r}(k))^{co O_{q}(SL_{r}(k))} is isomorphic to the quotient of the free algebra k⟨λ_{i_{1}···i_{r}}⟩ modulo the ideal generated by the relations (c) and (y) given in [6], Theorem 3.14.

The hypotheses on k and q are as above.

Now let k^{∘}, k, q, R be as in Section 2, and define everything over R. In particular, O_{q}(M_{n,r}(R)) is a right comodule algebra over O_{q}(SL_{r}(R)). As already mentioned, Fioresi and Hacon have proved the R-algebra versions of (1_{q}) and (2_{q}), assuming that k is algebraically closed of characteristic zero. The easier parts of their work – proving inclusions rather than equalities in these R-algebra forms – lead directly to the k-algebra forms of these theorems, as follows.

Let R⟨λ_{•}⟩ denote the free R-algebra on generators λ_{i_{1}···i_{r}}, and let ϕ_{q} : R⟨λ_{•}⟩ → O_{q}(M_{n,r}(R)) be the R-algebra homomorphism sending each λ_{i_{1}···i_{r}} to (−q)^{ℓ(π)}D_{{i_{1},...,i_{r}}}. The image of ϕ_{q} is the R-subalgebra generated by the D_{{i_{1},...,i_{r}}}, which is contained in O_{q}(M_{n,r}(R))^{co O_{q}(SL_{r}(R))} by [6], Lemma 3.5. Let I_{q} be the ideal of R⟨λ_{•}⟩ generated by the relations (c) and (y) of [6], Theorem 3.14; that I_{q} ⊆ ker ϕ_{q} was checked by Fioresi ([5], Proposition 2.21 and Theorem 3.6). Now O_{q}(M_{n,r}(R)) and R⟨λ_{•}⟩ are positively graded R-algebras, with the X_{ij} having degree 1 and the λ_{•} having degree r. The homogeneous components with respect to these gradings are finitely generated free R-modules, I_{q} is a homogeneous ideal of R⟨λ_{•}⟩, and the map ϕ_{q} is homogeneous of degree 0.

The classical Fundamental Theorems (1) and (2) say that \overline{ϕ_{q}} maps \overline{R⟨λ_{•}⟩} onto the coinvariants of \overline{O_{q}(M_{n,r}(R))}, and the kernel of \overline{ϕ_{q}} equals \overline{I_{q}}. (The latter statement requires a change of relations using the results of [7], Chapter 8, as observed in [6], p 435.) Therefore Theorems 1.2 and 1.3 yield the quantized Fundamental Theorems (1_{q}) and (2_{q}). Note that the hypotheses of algebraic closure and characteristic zero on k are not needed, although q is still assumed to be transcendental over a subfield of k.

4 The right action of Sp_{2n} on M_{m,2n}

Fix positive integers m and n, and consider the right action of Sp_{2n}(k) on M_{m,2n}(k) by multiplication. Here we take Sp_{2n}(k) to preserve the standard alternating bilinear form on k^{2n}, which we denote ⟨−, −⟩, and we view M_{m,2n}(k) as the variety of m vectors of length 2n. Thus, each row x_{i} = (X_{i1}, …, X_{i,2n}) of generators in O(M_{m,2n}(k)) corresponds to the i-th of m generic 2n-vectors. We shall need the functions describing the values of ⟨−, −⟩ on two of these generic vectors:

z_{ij} := ⟨x_{i}, x_{j}⟩ = ∑_{l=1}^{n}(X_{i,2l−1}X_{j,2l} − X_{i,2l}X_{j,2l−1})

for i, j = 1, …, m.

Next, consider the variety Alt_{m}(k) of alternating m × m matrices over k; its coordinate ring is the algebra

O(Alt_{m}(k)) = O(M_{m}(k))/⟨X_{ii}, X_{ij} + X_{ji} | i, j = 1, …, m⟩.

Let us write Y_{ij} for the coset of X_{ij} in O(Alt_{m}(k)). There is a morphism ν : M_{m,2n}(k) → Alt_{m}(k) given by the rule ν(A) = ABA^{tr}, where B is the matrix of the symplectic form ⟨−, −⟩, and the comorphism ν^{∗} : O(Alt_{m}(k)) → O(M_{m}(k)) sends Y_{ij} ↦ z_{ij} for all i, j. For any even number h of distinct indices i_{1}, …, i_{h} from {1, …, m}, let [i_{1}, …, i_{h}] ∈ O(Alt_{m}(k)) be the Pfaffian of the submatrix of (Y_{ij}) obtained by taking the rows and columns with indices i_{1}, …, i_{h}.

The First and Second Fundamental Theorems for the present situation ([2], Theorems 6.6, 6.7) state that

(1) The ring of invariants O(M_{m,2n}(k))^{Sp_{2n}(k)} equals the subalgebra of O(M_{m,2n}(k)) generated by the z_{ij} for i < j, that is, the image of ν^{∗}.

(2) If m ≤ 2n + 1, the kernel of ν^{∗} is zero, while if m ≥ 2n + 2, the kernel of ν^{∗} is the ideal of O(Alt_{m}(k)) generated by all the Pfaffians [i_{1}, …, i_{2n+2}].

A quantized version of this situation was studied by Strickland [17], who quantized the U(𝔰𝔭_{2n}(k))-module action on O(M_{2n,m}(k)) rather than its dual, the O(Sp_{2n}(k)) coaction. We transpose the matrices and indices from her paper in order to match the notation for the classical situation used above. Since the approaches to quantized enveloping algebras for 𝔰𝔭_{2n}(k) and quantized coordinate rings for Sp_{n}(k) become more complicated at roots of unity, let us restrict our discussion to a quantum parameter q ∈ k^{×} which is not a root of unity.

Recall that the quantized coordinate rings of the different classical groups coact on different quantized coordinate rings of affine spaces. In particular, O_{q}(Sp_{2n}(k)) coacts on an algebra that we denote O_{q}(𝔰𝔭 k^{2n}), the quantized coordinate ring of symplectic 2n-space ([16], Definition 14; see [15], §1.1, for a simpler set of relations). Thus, O(M_{m,2n}(k)) needs to be quantized by a suitable algebra with m · 2n generators, each row of which generates a copy of O_{q}(𝔰𝔭 k^{2n}). Let us call this algebra O_{q}((𝔰𝔭 k^{2n})^{m}) and take it to be the k-algebra with generators X_{ij} = x_{j,i} (for i = 1, …, m and j = 1, …, 2n) satisfying the relations given by Strickland in [17], Equations (2.1), (2.2), (2.3). (The x_{j,i} are Strickland’s generators for the algebra she denotes B; we transpose the indices for the reasons indicated above.) Similarly, let us write O_{q}(Alt_{m}(k)) for the k-algebra with generators Y_{ij} = a_{j,i} for 1 ≤ j < i ≤ m satisfying the relations given in [17], Equations (1.1). As in [17], Theorem 2.5(2), there is a k-algebra homomorphism ϕ : O_{q}(Alt_{m}(k)) → O_{q}((𝔰𝔭 k^{2n})^{m}) such that

ϕ(Y_{ts}) = ∑_{l=1}^{n} q^{l−1−n}X_{sl}X_{t,2n−l+1} − ∑_{l=1}^{n} q^{n−l+1}X_{s,2n−l+1}X_{tl}

for s < t.

Set B_{q} = O_{q}((𝔰𝔭 k^{2n})^{m}) for the time being. Strickland defines an action of U_{q}(𝔰𝔭_{2n}(k)) on B_{q} ([17], p 87). It is clear from the definition of this action that B_{q} is a locally finite dimensional left U_{q}(𝔰𝔭_{2n}(k))-module. Hence, B_{q} becomes a right comodule over the Hopf dual U_{q}(𝔰𝔭_{2n}(k))^{∘} in the standard way ([14], Lemma 1.6.4(2)), and the U_{q}(𝔰𝔭_{2n}(k))-invariants coincide with the U_{q}(𝔰𝔭_{2n}(k))^{∘}-coinvariants ([14], Lemma 1.7.2(2)). The standard quantized coordinate ring of Sp_{2n}(k), which we denote O_{q}(Sp_{2n}(k)), is a sub-Hopf-algebra of U_{q}(𝔰𝔭_{2n}(k))^{∘}, and one can check that the coaction B_{q} → B_{q} ⊗ U_{q}(𝔰𝔭_{2n}(k))^{∘} actually maps B_{q} into B_{q} ⊗ O_{q}(Sp_{2n}(k)). Consequently, B_{q} is a right comodule over O_{q}(Sp_{2n}(k)), and its O_{q}(Sp_{2n}(k))-coinvariants coincide with its U_{q}(𝔰𝔭_{2n}(k))^{∘}-coinvariants.

Statements of First and Second Fundamental Theorems for quantized coinvariants in the symplectic situation above can be given as follows, where the q-Pfaffians [i_{1}, …, i_{h}] are defined as in [17], p 82:

(1_{q}) The set of coinvariants O_{q}((𝔰𝔭 k^{2n})^{m})^{co O_{q}(Sp_{2n}(k))} equals the image of ϕ.

(2_{q}) If m ≤ 2n + 1, the kernel of ϕ is zero, while if m ≥ 2n + 2, the kernel of ϕ is the ideal of O_{q}(Alt_{m}(k)) generated by all the q-Pfaffians [i_{1}, …, i_{2n+2}].

Both statements have been proved by Strickland (modulo the changes of notation discussed above) under the assumptions that char(k) = 0 and q is transcendental over a subfield of k ([17], Theorem 2.5). Via Theorems 1.2 and 1.3, we obtain the transcendental cases of (1_{q}) and (2_{q}) from the classical results (1) and (2) in arbitrary characteristic. (We note that part of Strickland’s development also involves reduction modulo q − 1. See the proof of [17], Theorem 1.5.) As far as we are aware, it is an open question whether (1_{q}) and (2_{q}) hold when q is algebraic over the prime subfield of k.

5 The conjugation action of GL_{n} on M_{n}

Fix a positive integer n, and assume that k has characteristic zero. In this section, we consider the quantum analogue of the classical conjugation action of GL_{n}(k) on M_{n}(k). We shall need the trace functions tr_{i} for i = 1, …, n, where tr_{i} is the sum of the i × i principal minors. Note that tr_{1} is the usual trace function, and that tr_{n} is the determinant function. The First and Second Fundamental Theorems for this situation ([13], Satz 3.1) can be stated as follows:

(1) The ring of invariants O(M_{n}(k))^{GL_{n}(k)} equals the subalgebra of O(M_{n}(k)) generated by tr_{1}, …, tr_{n}.

(2) There are no relations among the tr_{i}, that is, tr_{1}, …, tr_{n} are algebraically independent over k.

The coinvariants of a quantum analogue of the conjugation action have been studied in [3]. The right conjugation coaction of O_{q}(GL_{n}(k)) on O_{q}(M_{n}(k)) is the right coaction β : O_{q}(M_{n}(k)) ⟶ O_{q}(M_{n}(k)) ⊗ O_{q}(GL_{n}(k)) given by β(u) := ∑_{(u)} u_{2} ⊗ S(u_{1})u_{3}, where we are using the Sweedler notation. In particular, β(X_{ij}) = ∑_{l,m} X_{lm} ⊗ S(X_{il})X_{mj} for all i, j. However, as with the interior coaction studied in Section 2, the map β is not an algebra homomorphism.

Recall that if I and J are subsets of {1, …, n} of the same size, then the quantum determinant of the quantum matrix subalgebra generated by the X_{ij} with i ∈ I and j ∈ J is denoted by [I|J] and called a quantum minor of the relevant quantum matrix algebra. For each i = 1, …, n, define the weighted sums of principal quantum minors by τ_{i} := ∑_{I} q^{−2w(I)}[I|I], where I runs through all i element subsets of {1, …, n} and w(I) is the sum of the entries in the index set I. These weighted sums of principal quantum minors provide quantized coinvariants for β by [3], Proposition 7.2. One can state First and Second Fundamental Theorems for quantized coinvariants in this situation as follows:

(1_{q}) The set of coinvariants O_{q}(M_{n}(k))^{co O_{q}(GL_{n}(k))} is equal to the subalgebra of O_{q}(M_{n}(k)) generated by τ_{1}, …, τ_{n}.

(2_{q}) The subalgebra k[τ_{1}, …, τ_{n}] is a commutative polynomial algebra of degree n.

These statements were proved for k = ℂ and q not a root of unity in [3], Theorem 7.3, by using the corepresentation theory of the cosemisimple Hopf algebra O_{q}(GL_{n}(ℂ)). We give a partial extension below.

Now let k^{∘} = ℚ. Suppose that q ∈ k^{×} is transcendental over ℚ, set R = ℚ[q, q^{−1}] and define everything over R. Then O_{q}(M_{n}(R)) is a right comodule over O_{q}(GL_{n}(R)) by using β as above. A straightforward calculation shows that the R-subalgebra generated by τ_{1}, …, τ_{n} is contained in O_{q}(M_{n}(R))^{co O_{q}(GL_{n}(R))}; see, for example, [3], Proposition 7.2. Also, note that when q = 1 the coinvariants τ_{1}, …, τ_{n} coincide with the classical traces tr_{1}, …, tr_{n}. Let F (R) denote the free R-algebra on generators γ_{1}, …, γ_{n}, and consider F (R) to be graded by setting deg(γ_{i}) = i. Let ϕ_{q} be the R-algebra homomorphism sending γ_{i} to τ_{i}. Then ϕ_{q} is homogeneous of degree 0 and the image of ϕ_{q} is contained in O_{q}(M_{n}(R))^{co O_{q}(GL_{n}(R))}. The homogeneous components of O_{q}(M_{n}(R)) and F (R) are finitely generated free R-modules with ϕ_{q}(F (R)_{j}) ⊆ O_{q}(M_{n}(R))_{j} and β(O_{q}(M_{n}(R))_{j}) ⊆ O_{q}(M_{n}(R))_{j} ⊗ O_{q}(GL_{n}(R)). The classical First Fundamental Theorem (1) says that \overline{ϕ_{q}} maps \overline{F (R)} onto the coinvariants of \overline{O_{q}(M_{n}(R))}. Theorem 1.2 now yields the quantized First Fundamental Theorem (1_{q}) for the case that char(k) = 0 and q is transcendental over ℚ.

In [1], Corollary 2.3, Cohen and Westreich show that the τ_{i} commute, by exploiting the coquasitriangular structure of O_{q}(GL_{n}(k)) (see, e.g., [10], Theorem 3.1 and Proposition 4.1; a more detailed proof for the case k = ℂ is given in [12], Theorem 10.9). Thus, if we set I to be the ideal of F (R) generated by the commutators γ_{i}γ_{j} − γ_{j}γ_{i}, we see that I ⊆ ker(ϕ_{q}). It is obvious that I is a homogeneous ideal. The classical theory shows that ker(\overline{ϕ_{q}}) = \overline{I}. Thus, Theorem 1.3 yields the Second Fundamental Theorem (2_{q}) in the case under discussion.

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K R Goodearl, Department of Mathematics, University of California, Santa Barbara, CA 93106, USA email: goodearl@math.ucsb.edu

T H Lenagan, School of Mathematics, James Clerk Maxwell Building, Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland email: tom@maths.ed.ac.uk