Quantized trace rings

M. Domokos ^{∗} and T. H. Lenagan ^{†}

Abstract

The general linear group acts on m-tuples of N ×N matrices by simultaneous conjugation. Quantum deformations of the corresponding rings of invariants and the so-called trace rings are investigated.

2000 Mathematics Subject Classification. 16W35, 16R30, 20G42

1 Introduction

Write M_{N}^{m} for the space M_{N}(ℂ)⊕⋯⊕M_{N}(ℂ) of m-tuples of N ×N complex matrices. Denote by U(k) = (u(k)_{j}^{i})_{i,j=1}^{N} the function mapping an m-tuple to its kth component (k = 1, …, m). The N^{2}m-variable commutative polynomial algebra A = A(N, m) generated by the u(k)_{j}^{i} is the coordinate ring of M_{N}^{m}. Let us recall various subalgebras of A and M_{N}(A). The ℂ-subalgebra of M_{N}(A) generated by U(1), …, U(m) and the identity matrix, I, is known as the algebra G(N, m) of m generic N × N matrices. The complex special linear group SL_{N} acts on M_{N}^{m} by simultaneous conjugation. This induces an action on A by linear substitution of the variables: g ∈ SL_{N} maps u(k)_{j}^{i} to the (i, j)-entry of g^{−1}U(k)g. The ring R(N, m) of matrix invariants consists of

∗Supported through a European Community Marie Curie Fellowship. Partially supported by OTKA No. T034530, T046378, and a Bolyai Research Fellowship.

†Both authors partially supported by Leverhulme Research Interchange Grant F/00158/X.

R(N, m) = {f ∈ A(N, m) | ∀g ∈ SL_{N} : g · f = f}.

Let T (N, m) denote the algebra of SL_{N}-equivariant polynomial maps from M_{N}^{m} to M_{N}(ℂ). Identify polynomial maps M_{N}^{m} → M_{N}(ℂ) with elements of M_{N}(A) in the usual way. Then T (N, m) is the R(N, m)-subalgebra of M_{N}(A) generated by I, U(1), …, U(m). It is called the trace ring (or sometimes the algebra of matrix concomitants). The reason for the name is that R(N, m) is generated by elements of the form Tr(U(k_{1}) ⋯ U(k_{d})), where k_{1}, …, k_{d} ∈ {1, …, m}, d ≤ N^{2}, and Tr denotes the usual trace function (see [26], [24], [21]).

The algebra G(N, m) of generic matrices appeared in the theory of algebras satisfying polynomial identities, as a relatively free object. On the other hand, R(N, m) and T (N, m) are natural objects of invariant theory. By definition, G(N, m) ⊂ T (N, m); although this is a proper inclusion, G(N, m) turns out to be closely related to T (N, m). As a remarkable consequence, certain questions in non-commutative algebra can be approached by methods of classical invariant theory, see [21] and [9] for details.

Our aim in the present paper is to find analogues of the algebras R(N, m), T (N, m), G(N, m) in the context of quantum groups. Write F_{q}(SL_{N}) for the quantized function algebra of SL_{N}, defined for example in [25]. Here q ∈ ℂ^{×} is a non-zero parameter. F_{q}(SL_{N}) is a Hopf algebra, which is a non-commutative deformation of the coordinate ring F(SL_{N}) of SL_{N}. (For unexplained terminology and facts relating to Hopf algebras, coactions, and quantum groups, we refer to the books [14], [19].) Works of Majid [19] and Kulish and Sasaki [15] provide us with an appropriate noncommutative deformation A_{q}(N, m) of A(N, m), together with a right coaction β(m) : A_{q}(N, m) → A_{q}(N, m)⊗F_{q}(SL_{N}), which is a deformation of the comorphism A(N, m) → A(N, m)⊗F(SL_{N}) of the simultaneous conjugation action of SL_{N} on M_{N}^{m}. To simplify notation, when m = 1 we write A_{q}(N) instead of A_{q}(N, 1). We take for the algebra A_{q}(N) the braided matrices of [18] (called the reflection equation algebra in [16]), associated with the R-matrix belonging to the quantum SL_{N}. It is a right F_{q}(SL_{N})-comodule algebra, and for some problems relating the conjugation action of SL_{N} on M_{N}(ℂ), it appears to be a better quantum deformation of F(M_{N}), than the coordinate ring F_{q}(M_{N}) of quantum matrices, see for example [12], [6]. When m = 2, we take for A_{q}(N, 2) the braided tensor product (cf. [19]) A_{q}(N)\underline{⊗}A_{q}(N) of two copies of A_{q}(N). As a vector space, it is just the ordinary tensor product A_{q}(N) ⊗_{ℂ} A_{q}(N), endowed with the tensor product F_{q}(SL_{N})-comodule structure. Using that F_{q}(SL_{N}) is coquasitriangular, one can define a multiplication on this tensor product space that makes the coaction multiplicative, see Corollary 9.2.14 in [19], and the sign \underline{⊗} indicates the vector space tensor product endowed with this multiplication. With an iterated use of braided tensor products we arrive at A_{q}(N, m) for arbitrary m. The same algebra was defined in [15] in terms of generators and relations.

Now we place the algebra R_{q}(N, m) of β(m)-coinvariants into the centre of investigation, as well as a subalgebra T_{q}(N, m) of M_{N}(A_{q}(N, m)). Under certain assumptions on q we show that these algebras can be considered as quantum versions of the ring of matrix invariants and the trace ring, by investigating their Hilbert series, their generators, and ring theoretical properties. In the special case N = 2 we present some explicit calculations, leading to the somewhat surprising conclusion that T_{q}(2, 2) ≅ T (2, 2), and G_{q}(2, 2) ≅ G(2, 2).

2 Some quadratic algebras

To define F_{q}(SL_{N}) one starts with the R-matrix R = (R_{jl}^{ik})_{i,j,k,l=1}^{N}. This is an N^{2} × N^{2} matrix, whose rows and columns are indexed by pairs, and the entry in the crossing of the (i, k) row and the (j, l) column is

R_{jl}^{ik} = \begin{cases}q, & if i = j = k = l; \\ 1, & if i = j, k = l, i ≠ k; \\ q − q^{−1}, & if i > j, i = l, j = k; \\ 0, & otherwise. \end{cases}

It is usual to think of R as the matrix of a linear endomorphism of V ⊗ V with respect to the basis e_{1} ⊗ e_{1}, e_{1} ⊗ e_{2}, …, e_{1} ⊗ e_{N}, e_{2} ⊗ e_{1}, …, e_{N} ⊗ e_{N}. So identifying R and the endomorphism we have R(e_{j} ⊗ e_{l}) = R_{jl}^{ik}e_{i} ⊗ e_{k}, where the convention of summing over repeated indices is used (and will be used throughout the paper).

The quantized coordinate ring F_{q}(M_{N}) is the ℂ-algebra generated by the entries of T = (t_{j}^{i})_{i,j=1}^{N}, subject to the relations RT_{1}T_{2} = T_{2}T_{1}R, with the matrix notation of [25]; that is, T_{1}, T_{2} are the matrix Kronecker products T_{1} = T ⊗ I, T_{2} = I ⊗ T, so (T_{1})_{jl}^{ik} = t_{j}^{i}δ_{l}^{k}, (T_{2})_{jl}^{ik} = δ_{j}^{i}t_{l}^{k}, where δ_{i}^{i} = 1, and δ_{j}^{i} = 0 for i ≠ j. The algebra F_{q}(M_{N}) is a bialgebra with comultiplication Δ(t_{j}^{i}) = t_{k}^{i} ⊗ t_{j}^{k} and counit ε(t_{j}^{i}) = δ_{j}^{i}. The quantum determinant det_{q} = ∑_{π∈S_{N}} (−q)^{length(π)}t_{π(1)}^{1} ⋯ t_{π(N)}^{N} is a central group-like element, and F_{q}(SL_{N}) is defined as the quotient of F_{q}(M_{N}) modulo the ideal ⟨det_{q} −1⟩. We keep the notation t_{j}^{i} ∈ F_{q}(SL_{N}) for the images of the generators under the natural surjection. Then F_{q}(SL_{N}) is a Hopf algebra with antipode S satisfying S(T)T = I = T S(T).

Denote by τ the flip endomorphism τ(v ⊗ w) = w ⊗ v of V ⊗ V. Define R̂ = τ ∘R and R_{21} = τ ∘R∘τ. In coordinates, we have R̂_{jl}^{ik} = R_{jl}^{ki} and (R_{21})_{jl}^{ik} = R_{lj}^{ki}. The reflection equation algebra A_{q}(N) is the ℂ-algebra generated by the entries of X = (x_{j}^{i})_{i,j=1}^{N} subject to the quadratic relations

X_{2}R̂X_{2}R̂ = R̂X_{2}R̂X_{2}.  (1)

The relations can be written also in the equivalent form

X_{2}R_{21}X_{1}R = R_{21}X_{1}RX_{2}.

By Proposition 10.3.5 in [19], A_{q}(N) is a right F_{q}(SL_{N})-comodule algebra in a natural way. Namely, the map X ↦ S(T)XT extends to an algebra homomorphism β : A_{q}(N) → A_{q}(N) ⊗ F_{q}(SL_{N}), and β is a right coaction of the Hopf algebra F_{q}(SL_{N}) on A_{q}(N). Here we treat the entries of S(T)XT (and more generally, products of the form txu with t, u ∈ F_{q}(SL_{N}), x ∈ A_{q}(N)) as elements of A_{q}(N) ⊗ F_{q}(SL_{N}) by assuming that the generators of F_{q}(SL_{N}) commute with the generators of A_{q}(N), and suppressing the ⊗ sign from the notation; so β(x_{j}^{i}) = x_{l}^{k} ⊗ S(t_{k}^{i})t_{j}^{l}. Taking into account that S(T) = T^{−1}, this notational device will be very convenient later.

Next take two copies of A_{q}(N). The matrix generators of them will be denoted by X = (x_{j}^{i}) and Y = (y_{j}^{i}), respectively. Define A_{q}(N, 2) as the algebra generated by the 2N^{2} elements x_{j}^{i}, y_{j}^{i}, subject to the additional relations

R^{−1}Y_{1}RX_{2} = X_{2}R^{−1}Y_{1}R.  (2)

By Theorem 10.3.1 of [19] this algebra can be identified with the braided tensor product A_{q}(N)\underline{⊗}A_{q}(N), by setting x_{j}^{i} ↦ x_{j}^{i}\underline{⊗}1, y_{j}^{i} ↦ 1\underline{⊗}x_{j}^{i}. The relations (2) are equivalent to

∀i, j, k, l : y_{j}^{i}x_{l}^{k} = (R^{−1})_{rc}^{an}R_{bl}^{sc}R_{am}^{id}(R̃)_{jd}^{bk}x_{n}^{m}y_{s}^{r},  (3)

where R̃^{t_{2}} = (R^{t_{2}})^{−1}, and for an N^{2} ×N^{2} matrix A, the matrix A^{t_{2}} is defined by (A^{t_{2}})_{jl}^{ik} = A_{jk}^{il}.

More generally, take m copies of A_{q}(N) with matrix generators

X(1) = (x(1)_{j}^{i}) , …, X(m) = (x(m)_{j}^{i}) ,

respectively, and define A_{q}(N, m) to be the algebra generated by {x(k)_{j}^{i} | k = 1, …, m; i, j = 1, …, N}, where in addition to the reflection equation (1) imposed for each X(k) = X, we impose the relations

∀r < s : R^{−1}X(s)_{1}RX(r)_{2} = X(r)_{2}R^{−1}X(s)_{1}R.  (4)

In other words, for each r < s the subalgebra of A_{q}(N, m) generated by the entries of X(r) and X(s) is isomorphic to A_{q}(N, 2) via an isomorphism x(r)_{j}^{i} ↦ x_{j}^{i}, x(s)_{j}^{i} ↦ y_{j}^{i}. This algebra was defined in [15]. It is clearly the same as the braided tensor product A_{q}(N)\underline{⊗} ⋯ \underline{⊗}A_{q}(N) (m copies) in the sense of [19]. Hence we may define the coaction β(m) : A_{q}(N, m) → A_{q}(N, m) ⊗ F_{q}(SL_{N}) by taking the mth tensor power of β. The multiplication in the braided tensor product is designed to make β(m) an algebra homomorphism. Therefore A_{q}(N, m) becomes an F_{q}(SL_{N})-comodule algebra. More explicitly, the coaction is given on the generators by β(m) : X(r) ↦ S(T)X(r)T for r = 1, …, m.

3 Polynormal sequences in the reflection equation algebra

It is standard that the ℤ-graded algebra F_{q}(M_{N}) has the same Hilbert series as F(M_{N}). On the other hand, a vector space isomorphism between the degree d homogeneous components of F_{q}(M_{N}) and A_{q}(N) is established in (7.37) of [19]. Consequently, the graded algebra A_{q}(N) has the same Hilbert series as the N^{2}-variable commutative polynomial algebra. We refine this statement by having a closer look at the defining relations of A_{q}(N). The matrix equality (1) is a shorthand for the following:

∀i, j, k, l : R̂_{st}^{ib}R̂_{jl}^{sa}x_{b}^{k}x_{a}^{t} = R̂_{sa}^{ik}R̂_{jb}^{st}x_{t}^{a}x_{l}^{b}  (5)

Taking into account the exact form of the R-matrix, one gets the following list of relations.

x_{j}^{i}x_{l}^{i} = qx_{l}^{i}x_{j}^{i}  for j < l, i ≠ j, i ≠ l;
q^{−1}x_{j}^{i}x_{i}^{i} − qx_{i}^{i}x_{j}^{i} = (q − q^{−1}) ∑_{s>i} x_{s}^{i}x_{j}^{s}  for i > j;
x_{l}^{i}x_{i}^{i} − x_{i}^{i}x_{l}^{i} = (1 − q^{−2}) ∑_{s>i} x_{s}^{i}x_{l}^{s}  for i < l;
x_{j}^{k}x_{j}^{i} = qx_{j}^{i}x_{j}^{k}  for i < k, j ≠ i, j ≠ k;
x_{i}^{i}x_{i}^{k} − x_{i}^{k}x_{i}^{i} = (1 − q^{−2}) ∑_{s>i} x_{s}^{k}x_{i}^{s}  for i < k;

q^{−1}x_{j}^{j}x_{j}^{i} − qx_{j}^{i}x_{j}^{j} = (q − q^{−1}) ∑_{s>j} x_{s}^{i}x_{j}^{s}  for i < j;
x_{j}^{i}x_{l}^{k} = x_{l}^{k}x_{j}^{i}  for i < k, j < l, i ≠ l, j ≠ k;
x_{l}^{j}x_{j}^{i} − qx_{j}^{i}x_{l}^{j} = (q − q^{−1}) ∑_{s>j} x_{s}^{i}x_{l}^{s}  for i < j < l;
x_{j}^{i}x_{i}^{k} − qx_{i}^{k}x_{j}^{i} = (q − q^{−1}) ∑_{s>i} x_{s}^{k}x_{j}^{s}  for j < i < k;
x_{j}^{i}x_{l}^{k} − x_{l}^{k}x_{j}^{i} = (q − q^{−1})x_{j}^{k}x_{l}^{i}  for i > k, j < l, i ≠ l, j ≠ k, i ≠ j;
x_{j}^{i}x_{l}^{k} − x_{l}^{k}x_{j}^{i} = (q − q^{−1})x_{l}^{i}x_{j}^{k}  for i > k, j < l, i ≠ l, j ≠ k, k ≠ l;
x_{l}^{j}x_{j}^{i} − qx_{j}^{i}x_{l}^{j} = (q − q^{−1})(−x_{j}^{j}x_{l}^{i} + ∑_{s>j} x_{s}^{i}x_{l}^{s})  for j < l, j < i, i ≠ l;
x_{j}^{i}x_{i}^{k} − qx_{i}^{k}x_{j}^{i} = (q − q^{−1})(x_{j}^{k}x_{i}^{i} + ∑_{s>i} x_{s}^{k}x_{j}^{s})  for k < i, j < i, k ≠ j;
x_{j}^{i}x_{i}^{j} − x_{i}^{j}x_{j}^{i} = (1 − q^{−2})(x_{j}^{j}x_{i}^{i} − ∑_{s>j} x_{s}^{i}x_{i}^{s} + ∑_{t>i} x_{t}^{j}x_{j}^{t})  for j < i.

Proposition 3.1 (i) The algebra A_{q}(N) is noetherian.

(ii) Let x_{1}, …, x_{N^{2}} be an arbitrary permutation of x_{1}^{1}, …, x_{N}^{N}. Then the monomials of the form x_{1}^{a_{1}} ⋯ x_{N^{2}}^{a_{N^{2}}} constitute a ℂ-vector space basis of A_{q}(N).

(iii) A_{q}(N) is a domain.

Proof. (i) Denote by I_{j}^{i} the ideal of A_{q}(N) generated by the x_{t}^{s} with s ≥ i, t ≥ j, (s, t) ≠ (i, j). The relation indexed by i, j, k, l in (5) says that for some q-power q_{jl}^{ik}, we have x_{j}^{i}x_{l}^{k} − q_{jl}^{ik}x_{l}^{k}x_{j}^{i} ∈ I_{j}^{i} ∩ I_{l}^{k}. In particular, since I_{N}^{N} = 0, the element x_{N}^{N} is normal in A_{q}(N); that is, x_{N}^{N}A_{q}(N) = A_{q}(N)x_{N}^{N}. Next, observe that I_{N}^{N−1} = ⟨x_{N}^{N}⟩ = I_{N−1}^{N}. It follows that both the images of x_{N−1}^{N} and x_{N}^{N−1} are normal in the quotient A_{q}(N)⟨x_{N}^{N}⟩. Continuing this way it is easy to see that the generators can be arranged into a so-called polynormal sequence. To be more precise, let x_{1}, …, x_{N^{2}} be a permutation of x_{1}^{1}, …, x_{N}^{N}. Define the function p by x_{p(i,j)} = x_{j}^{i}. Assume that

for all i, j, k, l with i ≤ k, j ≤ l we have that p(k, l) ≤ p(i, j).  (6)

Then x_{1}, …, x_{N^{2}} is a polynormal sequence: for all i, the image of x_{i} is normal in A_{q}(N)/⟨x_{1}, …, x_{i−1}⟩. Condition (6) holds for example for the sequence x_{N}^{N}, x_{N}^{N−1}, …, x_{N}^{1}, x_{N−1}^{N}, x_{N−1}^{N−1}, …, x_{1}^{1}. So the graded algebra A_{q}(N) is generated by a polynormal sequence, hence is noetherian (both left and right) by Lemma 8.2 of [1]. (Alternatively, one may apply Proposition I.8.17 from [2] for a generating sequence satisfying (6) to conclude that A_{q}(N) is noetherian.)

(ii) Write W for the free semigroup generated by the variables x_{j}^{i}. Define the function h on W by

h(x_{j}^{i} ⋯ x_{l}^{k}) = ij + ⋯ + kl.

Note that if w′ is obtained by permuting the factors of the monomial w, then h(w′) = h(w). Introduce a partial order ≺ on W: for v, w ∈ W, we set v ≺ w if h(v) > h(w). Clearly, ≺ is compatible with multiplication in the sense that v ≺ w implies avb ≺ awb for all a, b ∈ W ∪ {1}. In A_{q}(N), the relation indexed by i, j, k, l in (5) says that x_{j}^{i}x_{l}^{k} − q_{jl}^{ik}x_{l}^{k}x_{j}^{i} is a linear combination of monomials w with w ≺ x_{j}^{i}x_{l}^{k} and w ≺ x_{l}^{k}x_{j}^{i} (where q_{jl}^{ik} is a q-power, hence nonzero). Restrict attention to the degree d homogeneous component A_{q}(N)^{(d)} of A_{q}(N). By an iterated use of the quadratic relations, we can rewrite w = x_{j_{1}}^{i_{1}} ⋯ x_{j_{d}}^{i_{d}} as a non-zero scalar multiple of some w′ = (x_{1})^{a_{1}} ⋯ (x_{N^{2}})^{a_{N^{2}}} (obtained by permuting the factors of w), modulo the span of monomials v with v ≺ w, v ≺ w′. This obviously implies that the monomials of the form (x_{1})^{a_{1}} ⋯ (x_{N^{2}})^{a_{N^{2}}} span A_{q}(N)^{(d)}. Their number coincides with the dimension of A_{q}(N)^{(d)} (which we know already), so they form a basis.

(iii) Given a homogeneous element f of A_{q}(N), express it in terms of the basis described in (ii). Among the monomials of f that are maximal with respect to the partial ordering ≺ choose the one which is largest with respect to the lexicographic ordering of the monomials of the form (x_{1})^{i_{1}} ⋯ (x_{N^{2}})^{i_{N^{2}}}, and call it the leading term of f. If f, g are non-zero homogeneous elements of A_{q}(N) with leading term (x_{1})^{i_{1}} ⋯ (x_{N^{2}})^{i_{N^{2}}} and (x_{1})^{j_{1}} ⋯ (x_{N^{2}})^{j_{N^{2}}}, then by the rewriting process sketched in the proof of (ii), the leading term of fg is (x_{1}^{i_{1}+j_{1}}) ⋯ (x_{N^{2}})^{i_{N^{2}}+j_{N^{2}}}, so fg is non-zero. Thus A_{q}(N) does not contain zero-divisors. □

4 Quantum trace rings

After these preparations we are in position to define R_{q}(N, m) as the subalgebra of β(m)-coinvariants in A_{q}(N, m); that is,

R_{q}(N, m) = {f ∈ A_{q}(N, m) | β(m)f = f ⊗ 1}.

Let G_{q}(N, m) denote the unitary subalgebra of M_{N}(A_{q}(N, m)) (the algebra of N × N matrices over A_{q}(N, m)) generated by X(1), …, X(m). And finally, consider

T_{q}(N, m) = {F ∈ M_{N}(A_{q}(N, m)) | β(m)F = S(T)F T },

where β(m)F stands for the matrix obtained by applying β(m) to each entry of F, and S(T)XT is identified with an element of M_{N}(A_{q}(N, m))⊗F_{q}(SL_{N}) in the way indicated in Section 2.

Proposition 4.1 T_{q}(N, m) is a subalgebra of M_{N}(A_{q}(N, m)).

Proof. This follows from the formula S(T) = T^{−1}, together with the multiplicativity of the coaction β(m) : A_{q}(N, m) → A_{q}(N, m) ⊗ F_{q}(SL_{N}). □

The generic matrices X(k) belong to T_{q}(N, m) by definition of β(m), hence all products X(k_{1}) ⋯ X(k_{d}) belong to T_{q}(N, m) by Proposition 4.1. Also, if f ∈ R_{q}(N, m), then the scalar matrix fI is contained in T_{q}(N, m).

One can produce β(m)-coinvariants using the so-called quantum trace operation. For an arbitrary N × N matrix F with entries from a ℂ-algebra, define Tr_{q}(F) = Tr(QF), where Q is the diagonal matrix with diagonal entries q^{N−1}, q^{N−3}, q^{N−5}, …, q^{−N+1}. In particular, Tr_{q}(I) = [N]_{q}, the Nth q-integer. Now Tr_{q}(X) is a β-coinvariant in A_{q}(N); more generally, if F ∈ T_{q}(N, m), then Tr_{q}(F) is contained in R_{q}(N, m) (see the proof of Theorem 4.2). In particular, the elements Tr_{q}(X(k_{1}) ⋯ X(k_{d})) are all contained in R_{q}(N, m).

Note that there is a natural ℤ^{m}-grading on the algebra A_{q}(N, m): the multidegree of the entries of X(k) is the kth standard basis vector of ℤ^{m}. This induces a ℤ^{m}-grading on M_{N}(A_{q}(N, m)) as well. Since the multihomogeneous components of A_{q}(N, m) are subcomodules with respect to β(m), we have that R_{q}(N, m) is a ℤ^{m}-graded subalgebra of A_{q}(N, m), and similarly, T_{q}(N, m) is a ℤ^{m}-graded subalgebra of M_{N}(A_{q}(N, m)).

Theorem 4.2 Assume that q ∈ ℂ^{×} is not a root of unity. Then the ℤ^{m}- graded algebra R_{q}(N, m) has the same Hilbert series as its classical counterpart R(N, m), and T_{q}(N, m) has the same Hilbert series as T (N, m).

Proof. Consider the multihomogeneous component A_{q}(N, m)^{(d_{1},...,d_{m})}, it is a subcomodule of A_{q}(N, m) with respect to β. Since q is assumed to be not a root of unity, F_{q}(SL_{N}) is cosemisimple (as in the case that q = 1), and this finite dimensional comodule decomposes as the direct sum of simple subcomodules. By definition, the coefficient of t_{1}^{d_{1}} ⋯ t_{m}^{d_{m}} in the Hilbert series H(R_{q}(N, m); t_{1}, …, t_{m}) is the multiplicity of the trivial comodule as a direct summand. This multiplicity can be computed by “restricting the coaction to the diagonal subgroup” (see for example [4] for a more detailed explanation). To be more precise, denote by F(K) the coordinate Hopf algebra of the diagonal subgroup of SL_{N}(ℂ), so F(K) = ℂ[z_{1}, …, z_{N} | z_{1} ⋯ z_{N} = 1]. There is a surjective homomorphism π : F_{q}(SL_{N}) → F(K), π(t_{i}^{i}) = z_{i}, and π(t_{j}^{i}) = 0 for i ≠ j. Consider the coaction (id ⊗ π)β(m) : A_{q}(N, m) → A_{q}(N, m) ⊗ F(K) of F(K). It follows from Proposition 3.1 (ii) and the interpretation of A_{q}(N, m) as a braided tensor product of copies of A_{q}(N) that the F(K)-comodule A_{q}(N, m)^{(d_{1},...,d_{m})} is isomorphic to its counterpart in the classical case q = 1. That is, formally the same computation gives the coefficients of the Hilbert series of R_{q}(N, m) as that of R(N, m).

The proof of the statement about the Hilbert series of T_{q}(N, m) is similar, one just needs an appropriate interpretation of the elements of T_{q}(N, m) in terms of corepresentation theory. Denote by φ the restriction of the coaction β to the N^{2}-dimensional space X = Span_{ℂ}{x_{j}^{i}}_{i,j=1}^{N}. As in the classical case q = 1, this corepresentation decomposes as φ ≅ triv ⊕ φ_{0}, where triv is the trivial corepresentation, and φ_{0} is an irreducible (N^{2} − 1)-dimensional corepresentation. It is well known (and easy to show; we shall comment on it later in this Section) that triv is realized on the subspace of X spanned by Tr_{q}(X). Thus the decomposition of X as a sum of simple subcomodules is X = ℂTr_{q}(X) ⊕ X_{0}, where X_{0} is spanned by the entries of X − \frac{Tr_{q}(X)}{[N]_{q}}I, with [N]_{q} standing for the q-integer q^{N−1} + q^{N−3} + ⋯ + q^{−N+1}.

By definition, a matrix F = (f_{j}^{i}) belongs to T_{q}(N, m) if and only if its entries span a β(m)-subcomodule of A_{q}(N, m), such that X → Span_{ℂ}{f_{j}^{i}}, x_{j}^{i} ↦ f_{j}^{i} is a homomorphism of F_{q}(SL_{N})-comodules. It follows in particular that if F ∈ T_{q}(N, m), then Tr_{q}(F) ∈ R_{q}(N, m). Therefore G = F − \frac{Tr_{q}(F)}{[N_{q}]}I ∈ T_{q}(N, m) with Tr_{q}(G) = 0, and the entries of G span a subcomodule of A_{q}(N, m) which is a homomorphic image of the simple comodule X_{0}. Conversely, given an element f ∈ R_{q}(N, m), the scalar matrix fI obviously belongs to T_{q}(N, m). Now take a subcomodule Y of A_{q}(N, m) which is isomorphic to X_{0}. Fix a comodule surjection X → Y, x_{j}^{i} ↦ f_{j}^{i}. (Note that by Schur’s Lemma, this homomorphism is determined up to a scalar multiple.) Then the matrix F = (f_{j}^{i}) belongs to T_{q}(N, m) by construction, and has the property Tr_{q}(F) = 0.

Putting this all together, we reach the conclusion that the dimension of T_{q}(N, m)^{(d_{1},...,d_{m})} is the sum of the dimension of R_{q}(N, m)^{(d_{1},...,d_{m})} and the multiplicity of the irreducible corepresentation φ_{0} as a direct summand of β(m) on A_{q}(N, m)^{(d_{1},...,d_{m})}. This latter multiplicity can be computed by restriction to the corresponding F(K)-coaction, which by Proposition 3.1 (ii) is isomorphic to its classical counterpart. Finally, note that the dimension of T (N, m)^{(d_{1},...,d_{m})} in the classical case clearly can be expressed in the same way in terms of multiplicities of two types of irreducible summands in A(N, m). □ Remark 4.3 Let us extract from the above proof explicitly that the ℂvector space T_{q}(N, m) can be identified with the space of F_{q}(SL_{N})-comodule homomorphisms Hom_{F_{q}(SL_{N})}(X , A_{q}(N, m)). There is a natural vector space isomorphism Hom_{ℂ}(X , A_{q}(N, m)) ≅ A_{q}(N, m) ⊗ X^{∗}. The dual space X^{∗} of X is an F_{q}(SL_{N})-comodule via the contragredient corepresentation of φ (cf. section 11.1.3, page 398 in [14]), and consider the tensor product comodule structure on A_{q}(N, m) ⊗ X^{∗}. It is straightforward to check that under the isomorphism Hom_{ℂ}(X , A_{q}(N, m)) ≅ A_{q}(N, m) ⊗ X^{∗} the space of coinvariants (A_{q}(N, m) ⊗ X^{∗})^{F_{q}(SL_{N})} is mapped onto Hom_{F_{q}(SL_{N})}(X , A_{q}(N, m)). In particular, we have T_{q}(N, m) ≅ (A_{q}(N, m) ⊗ X^{∗})^{F_{q}(SL_{N})} as vector spaces.

Under a stronger restriction on q we are able to present explicit generators of R_{q}(N, m) and T_{q}(N, m).

Theorem 4.4 Assume that q is transcendental over the rationals. Then R_{q}(N, m) is the ℂ-subalgebra of A_{q}(N, m) generated by the elements of the form Tr_{q}(X(i_{1}) ⋯ X(i_{d})), where i_{1}, …, i_{d} ∈ {1, …, m}, and d ≤ N^{2}. Furthermore, T_{q}(N, m) is the left (respectively, right) R_{q}(N, m)-submodule of M_{N}(A_{q}(N, m)) generated by the elements X(i_{1}) ⋯ X(i_{d}) with d ≤ N^{2} − 1.

Proof. These statements are known to hold when q = 1. Take a set of multihomogeneous monomials whose traces form a basis of R(N, m) when q = 1. Take the q-traces of formally the same monomials (but interpret them as matrices with entries from A_{q}(N, m)). We obtain some multihomogeneous elements in R_{q}(N, m). Since q is assumed to be transcendental, we may conclude by a standard argument that these elements in R_{q}(N, m) are linearly independent (because of the linear independency of the corresponding elements for q = 1). Therefore they constitute a basis of R_{q}(N, m) by Theorem 4.2. The proof of the statement about T_{q}(N, m) is the same. □

When q is not a root of unity, the multilinear component A_{q}(N, m)^{(1,...,1)} can be naturally identified with a homomorphic image of the Hecke algebra of the symmetric group S_{m}. This follows from some natural linear algebra isomorphisms (mentioned already in Remark 4.3). Indeed, consider the fundamental corepresentation ω : V → V ⊗ F_{q}(SL_{N}), e_{j} ↦ e_{i} ⊗ t_{j}^{i}, where V is an N-dimensional vector space with basis e_{1}, …, e_{N}. Write ε^{1}, …, ε^{N} for the corresponding dual basis in V^{∗}. Then using the notation of the proof of Theorem 4.2, X → V^{∗} ⊗ V, x_{j}^{i} ↦ ε^{i} ⊗ e_{j} intertwines φ and ω^{∗} ⊗ ω. Therefore as a comodule, the multilinear component A_{q}(N, m)^{(1,...,1)} can be identified with V^{∗} ⊗ V ⊗ ⋯ ⊗ V^{∗} ⊗ V. With a repeated use of the comodule isomorphism V ⊗ V^{∗} → V^{∗} ⊗ V, e_{i} ⊗ ε^{j} ↦ R̃_{il}^{kj}ε^{l} ⊗ e_{k}, and the other natural isomorphisms mentioned in Remark 4.3 (note also that (A ⊗ B)^{∗} ≅ B^{∗} ⊗ A^{∗} for any finite dimensional F_{q}(SL_{N})-comodules A, B) we obtain an isomorphism A_{q}(N, m)^{(1,...,1)} ≅ End_{ℂ}(V ⊗ ⋯ ⊗ V), which restricts to R_{q}(N, m)^{(1,...,1)} ≅ End_{F_{q}(SL_{N})}(V^{⊗m}). The latter is known to be generated as an algebra by R̂_{i,i+1} (operating as R̂ on the ith and (i + 1)th tensor component, and operating as the identity on the other components), i = 1, …, m − 1. That way we get an explicit vector space homomorphism from the Hecke algebra of the symmetric group S_{m} onto R_{q}(N, m)^{(1,...,1)}. The kernel of this homomorphism is a known two-sided ideal of the Hecke algebra. For example, in the special case m = 1, the element id ∈ End_{F_{q}(SL_{N})}(V) is mapped to q^{−2}Tr_{q}(X). However, it does not seem to be easy to exploit this connection for explicit calculations, because of the excessive use of the R-matrix in some of the above identifications. For example, in the case N = m = 2, we have

id_{V ⊗V} ↦ e_{i} ⊗ e_{j} ⊗ ε^{j} ⊗ ε^{i} ↦ R̃_{kt}^{si}R̃_{in}^{ml}R̃_{jl}^{kj}x_{m}^{n}y_{t}^{s},

which is q^{−2}Tr_{q}(XY) + (q^{−5} − q^{−3})Tr_{q}(X)Tr_{q}(Y) ∈ A_{q}(2, 2).

Another interesting fact about A_{q}(N, m) is the centrality of certain β(m)- coinvariants. The β-coinvariants are central in A_{q}(N) (see e.g. [19]), and Tr_{q}(X(k)^{n}) is central in A_{q}(N, m), as it was observed in [15]. Using the approach of [19] it is easy to show that if f ∈ R_{q}(N, m) belongs to the subalgebra generated by the entries of X(k) for some k, then f is central in A_{q}(N, m). It is sufficient to show that f commutes with the entries of X(l) for l ≠ k (since we know already that f commutes with the entries of X(k)). So one can reduce the question to the case m = 2. Let x, y be elements of A_{q}(N, 2), x depending only on the entries of X, and y depending only on the entries of Y. With Sweedler’s notation, β(x) = ∑ x_{0} ⊗ x_{1}, and β(y) = ∑ y_{0} ⊗ y_{1}. By definition of the multiplication in the braided tensor product A_{q}(N)\underline{⊗}A_{q}(N), we have

yx = ∑ r(y_{1}, x_{1})x_{0}y_{0},  (7)

where r is the universal r-form in the coquasitriangular Hopf algebra F_{q}(SL_{N}) (see e.g. Proposition 35, 10.3.2 in [14]). If β(x) = x ⊗ 1, then we get yx = ∑ r(y_{1}, 1)xy_{0} = x ∑ ε(y_{1})y_{0} = xy. Similarly, β(y) = y ⊗ 1 implies yx = xy.

We mention also that the entries of XY satisfy the defining relations of A_{q}(N); this is stated in [15]. Indeed,

(XY)_{2}R_{21}(XY)_{1}R = X_{2}Y_{2}τRτX_{1}Y_{1}R = X_{2}τ(Y_{1}RX_{2})τY_{1}R
= X_{2}τRX_{2}R^{−1}Y_{1}RτY_{1}R = (X_{2}R_{21}X_{1})τR^{−1}τ(Y_{2}R_{21}Y_{1}R)
= R_{21}X_{1}RX_{2}R^{−1}R_{21}^{−1}R_{21}Y_{1}RY_{2} = R_{21}X_{1}R(X_{2}R^{−1}Y_{1}R)Y_{2}
= R_{21}X_{1}RR^{−1}Y_{1}RX_{2}Y_{2} = R_{21}X_{1}Y_{1}RX_{2}Y_{2}
= R_{21}(XY)_{1}R(XY)_{2},

so XY satisfies the matrix equality (1). Furthermore, the map X ↦ X(1)X(2), Y ↦ X(3) extends to an algebra homomorphism A_{q}(N, 2) → A_{q}(N, 3). All one needs to check is that the matrices X(1)X(2) and X(3) satisfy the equality (2). This can be done as in the calculation above. Alternatively, think of A_{q}(N, 3) as A_{q}(N, 2)\underline{⊗}A_{q}(N); the multiplication in the braided tensor product is expressed in terms of the F_{q}(SL_{N})-comodule algebra structures of the factors (see (7)). Since the linear isomorphisms X ↦ X(1)X(2), Y ↦ X(3) intertwine the corresponding coactions, the cross relations (3) hold between the entries of X(1)X(2) and X(3).

5 The 2 × 2 case

Throughout this section we assume N = 2. Recall from Section 3 that A_{q}(2) is generated by x_{1}^{1}, x_{2}^{1}, x_{1}^{2}, x_{2}^{2}, subject to the relations

x_{2}^{2}x_{2}^{1} = q^{2}x_{2}^{1}x_{2}^{2};  x_{1}^{1}x_{2}^{1} = x_{2}^{1}x_{1}^{1} + (q^{−2} − 1)x_{2}^{1}x_{2}^{2};  (8)
x_{1}^{1}x_{2}^{2} = x_{2}^{2}x_{1}^{1};  x_{1}^{2}x_{2}^{1} = x_{2}^{1}x_{1}^{2} + (q^{−2} − 1)x_{2}^{2}(x_{2}^{2} − x_{1}^{1});
x_{1}^{2}x_{2}^{2} = q^{2}x_{2}^{2}x_{1}^{2};  x_{1}^{2}x_{1}^{1} = x_{1}^{1}x_{1}^{2} + (q^{−2} − 1)x_{2}^{2}x_{1}^{2}.

The cross relations in A_{q}(2, 2) between the entries of X and Y are the following:

y_{1}^{1}x_{1}^{1} = x_{1}^{1}y_{1}^{1} + (q^{−4} − q^{−2})x_{1}^{2}y_{2}^{1};  (9)
y_{2}^{1}x_{1}^{1} = x_{1}^{1}y_{2}^{1};
y_{1}^{1}x_{2}^{1} = x_{2}^{1}y_{1}^{1} + (1 − q^{−2})x_{1}^{1}y_{2}^{1} + (q^{−2} − 1)x_{2}^{2}y_{2}^{1};
y_{2}^{1}x_{2}^{1} = q^{2}x_{2}^{1}y_{2}^{1};
y_{1}^{2}x_{1}^{1} = x_{1}^{1}y_{1}^{2} + (q^{−2} − 1)x_{1}^{2}y_{2}^{2} + (1 − q^{−2})x_{1}^{2}y_{1}^{1};
y_{2}^{2}x_{1}^{1} = x_{1}^{1}y_{2}^{2} + (1 − q^{−2})x_{1}^{2}y_{2}^{1};
y_{1}^{2}x_{2}^{1} = q^{−2}x_{2}^{1}y_{1}^{2} + (q^{−2} − 1)x_{1}^{1}y_{1}^{1} + (1 − q^{−2})x_{1}^{1}y_{2}^{2} + (1 − q^{−2})x_{2}^{2}y_{1}^{1} +
+(q^{−2} − 1)x_{2}^{2}y_{2}^{2} + (q^{−4} − q^{−2} − 1 + q^{2})x_{1}^{2}y_{2}^{1};
y_{2}^{2}x_{2}^{1} = x_{2}^{1}y_{2}^{2} + (1 − q^{2})x_{1}^{1}y_{2}^{1} + (q^{2} − 1)x_{2}^{2}y_{2}^{1};
y_{1}^{1}x_{1}^{2} = x_{1}^{2}y_{1}^{1};
y_{2}^{1}x_{1}^{2} = q^{−2}x_{1}^{2}y_{2}^{1};
y_{1}^{1}x_{2}^{2} = x_{2}^{2}y_{1}^{1} + (1 − q^{−2})x_{1}^{2}y_{2}^{1};
y_{2}^{1}x_{2}^{2} = x_{2}^{2}y_{2}^{1};
y_{1}^{2}x_{1}^{2} = q^{2}x_{1}^{2}y_{1}^{2};
y_{2}^{2}x_{1}^{2} = x_{1}^{2}y_{2}^{2};
y_{1}^{2}x_{2}^{2} = x_{2}^{2}y_{1}^{2} + (q^{2} − 1)x_{1}^{2}y_{2}^{2} + (1 − q^{2})x_{1}^{2}y_{1}^{1};
y_{2}^{2}x_{2}^{2} = x_{2}^{2}y_{2}^{2} + (1 − q^{2})x_{1}^{2}y_{2}^{1}.

The algebra A_{q}(2, m) is generated by the entries of the 2 × 2 matrices X(1), …, X(m). For each k, the entries of X(k) satisfy (8) with X ↦ X(k), and for each k < l, the entries of X(k) and X(l) satisfy (9) with X ↦ X(k), Y ↦ X(l). The monomials

w(1) … w(m) with w(k) = (x(k)_{2}^{1})^{a_{k}}(x(k)_{2}^{2})^{b_{k}}(x(k)_{1}^{1})^{c_{k}}(x(k)_{1}^{2})^{d_{k}} (10)

form a basis of A_{q}(2, m). Above we have presented the relations of A_{q}(2, m) in such a form that makes obvious an algorithm to rewrite an arbitrary monomial in the generators as a linear combination of these basis elements. This allows us to perform calculations in A_{q}(2, m).

The quantum version of the Cayley-Hamilton identity for the matrix X of the generators of A_{q}(N) was found in [23] (see also [13]). In the special case N = 2 it takes the form

X^{2} − q^{−1}Tr_{q}(X)X + \frac{1}{[2]_{q}} (q^{−1}Tr_{q}(X)^{2} − Tr_{q}(X^{2}))I = 0.  (11)

We note that \frac{1}{q+q^{−1}} (q^{−1}Tr_{q}(X)^{2} − Tr_{q}(X^{2})) = q^{−2}x_{2}^{2}x_{1}^{1} − x_{2}^{1}x_{1}^{2}.

Classically one can polarize the 2 × 2 Cayley-Hamilton identity to obtain an equivalent bilinear identity depending on two matrix variables. This bilinear identity has an analogues in the quantum setting:

Proposition 5.1 The following equality holds in T_{q}(2, 2):

XY + q^{2}Y X − qTr_{q}(X)Y − qTr_{q}(Y)X
+(Tr_{q}(X)Tr_{q}(Y) − q^{−1}Tr_{q}(XY))I = 0.  (12)

Proof. Direct computation, using the above mentioned basis and rewriting algorithm in A_{q}(2, 2). □

We take a digression and for illustrative purposes we list a couple of other equalities in A_{q}(2, 2) and in A_{q}(2, 3). We have

q^{2}Tr_{q}(Y X) − q^{−2}Tr_{q}(XY) = (q − q^{−1})Tr_{q}(X)Tr_{q}(Y) in A_{q}(2, 2) (13)

(this can be checked by direct computation again). To simplify notation, write X, Y, Z (instead of X(1), X(2), X(3)) for the matrices of the generators of A_{q}(2, 3). Multiplying (12) by Z from the right and taking q-trace one obtains

Tr_{q}(X)Tr_{q}(Y)Tr_{q}(Z) − q^{−1}Tr_{q}(XY)Tr_{q}(Z) − qTr_{q}(X)Tr_{q}(Y Z)
−qTr_{q}(Y)Tr_{q}(XZ) + q^{2}Tr_{q}(Y XZ) + Tr_{q}(XY Z) = 0 in A_{q}(2, 3), (14)

whereas making the substitution X ↦ Y, Y ↦ Z in (12), multiplying by X from the left, and taking q-trace one gets

Tr_{q}(X)Tr_{q}(Y)Tr_{q}(Z) − qTr_{q}(XY)Tr_{q}(Z) − q^{−1}Tr_{q}(X)Tr_{q}(Y Z)
−qTr_{q}(XZ)Tr_{q}(Y) + q^{2}Tr_{q}(XZY) + Tr_{q}(XY Z) = 0 in A_{q}(2, 3). (15)

From either of the above two equalities one recovers the so-called fundamental trace identity of 2 × 2 matrices as the special case q = 1. However, we have no canonical choice to single out one of them as “the fundamental q-trace identity”, because as is indicated by (13), the number of the different trilinear products of q-traces is larger than 6, whereas dim_{ℂ}(A_{q}(2, 3)^{(1,1,1)}) = 5, as in the classical case. Furthermore, make the substitutions X ↦ XY, Y ↦ Z (respectively X ↦ X, Y ↦ Y Z) in the identity (13); by the discussion at the end of Section 4, we obtain the equalities

q^{2}Tr_{q}(ZXY) − q^{−2}Tr_{q}(XY Z) = (q − q^{−1})Tr_{q}(XY)Tr_{q}(Z),  (16)
q^{2}Tr_{q}(Y ZX) − q^{−2}Tr_{q}(XY Z) = (q − q^{−1})Tr_{q}(X)Tr_{q}(Y Z).  (17)

Finally, multiply (12) by Z from the left and take its q-trace; from this, (15), (14), (16), and (17) one gets that modulo the subalgebra generated by qtraces of monomials of degree ≤ 2, we have Tr_{q}(XY Z) ≡ −q^{2}Tr_{q}(Y XZ) ≡ −q^{2}Tr_{q}(XZY) ≡ q^{4}Tr_{q}(Y ZX) ≡ q^{4}Tr_{q}(ZXY) ≡ −q^{6}Tr_{q}(ZY X).

Next we investigate T_{q}(2, 2). In order to simplify notation, set

A = q^{−1}Tr_{q}(X)I; B = q^{−1}Tr_{q}(Y)I;
C = (q^{−2}x_{2}^{2}x_{1}^{1} − x_{2}^{1}x_{1}^{2})I; D = (q^{−2}y_{2}^{2}y_{1}^{1} − y_{2}^{1}y_{1}^{2})I;

E = AB − q^{−3}Tr_{q}(XY)I.

Theorem 5.2 Suppose that q is transcendental over the rationals. Then we have the following.

(i) R_{q}(2, 2) is a 5-variable commutative polynomial algebra generated by Tr_{q}(X), Tr_{q}(X^{2}), Tr_{q}(Y), Tr_{q}(Y^{2}), Tr_{q}(XY).

(ii) T_{q}(2, 2) is a free left R_{q}(2, 2)-module generated by I, X, Y, XY.

(iii) As a ℂ-algebra, T_{q}(2, 2) is generated by X, Y, A, B, C, D, E, and a complete list of relations among these generators is the following:

A, B, C, D are central;
X^{2} = AX − C; Y^{2} = BY − D;
Y X = −q^{−2}XY + AY + BX − E;
XE = q^{2}EX + (1 − q^{2})ABX + (q^{−2} − q^{2})CY +
+(1 − q^{−2})AXY + (q^{2} − 1)CB;
Y E = q^{−2}EY + (1 − q^{−2})ABY + (1 − q^{−4})DX +
+(q^{−4} − q^{−2})BXY + (q^{−2} − 1)AD.

Proof. (i) As we noted in Section 4, Tr_{q}(X), Tr_{q}(X^{2}), Tr_{q}(Y), Tr_{q}(Y^{2}) are central in A_{q}(2, 2), hence the given five elements pairwise commute. They are algebraically independent because of the algebraic independence of the corresponding elements in the case q = 1 (here we use the assumption on q, in the same way as in the proof of Theorem 4.4). We know from Theorem 4.2 that R_{q}(2, 2) has the same Hilbert series as a polynomial algebra generated by two degree 1 elements and three degree 2 elements. This implies that R_{q}(2, 2) coincides with its subalgebra ℂ[Tr_{q}(X), Tr_{q}(X^{2}), Tr_{q}(Y), Tr_{q}(Y^{2}), Tr_{q}(XY)].

(ii) T_{q}(2, 2) contains I, X, Y, XY, hence the left R_{q}(2, 2)-module generated by them. This is a free module, because the corresponding module is free when q = 1 (again the transcendentality of q is used). Consequently, it has the same Hilbert series as T (2, 2) (computed in [11]). Hence by Theorem 4.2, the left R_{q}(2, 2)-module generated by I, X, Y, XY coincides with T_{q}(2, 2).

(iii) It follows from (ii) that X, Y, A, B, C, D, E generate T_{q}(2, 2) as an algebra. By the remarks in Section 4 about the centrality of certain β(m)- coinvariants, we have that A, B, C, D are central in T_{q}(2, 2). The relations X^{2} = AX − C, Y^{2} = BY − D are the Cayley-Hamilton identities of [23], see (11). The next relation in the list is the bilinear Cayley-Hamilton identity of Proposition 5.1. The remaining two relations are verified by direct calculation, using the basis and rewriting algorithm in A_{q}(2, 2) explained at the beginning of this Section. Thus the relations we listed all hold. It is straightforward that modulo these relations an arbitrary product of the elements X, Y, A, B, C, D, E can be rewritten as an element of the left ℂ[A, B, C, D, E]- module generated by I, X, Y, XY. Hence by (ii), the given list of relations is complete. □

Remark 5.3 A rather lengthy calculation using the relations in (iii) yields that XY E = EXY. Knowing that A, B are central, this is equivalent to the assertion that Tr_{q}(XY) commutes with XY. However, this latter statement immediately follows (for arbitrary N) from the existence of the algebra homomorphism A_{q}(N) → A_{q}(N, 2), X ↦ XY (discussed in Section 4), and the centrality of Tr_{q}(X) in A_{q}(N).

Now consider the ordinary trace ring T (2, 2), and write x, y for the matrix generators, so x, y are 2 × 2 generic matrices with pairwise commuting algebraically independent entries. The algebra T (2, 2) probably appeared first in [27]; see [10] for a recent survey. We set

a = Tr(x)I; b = Tr(y)I; c = det(x)I; d = det(y)I;
e = ab − Tr(xy)I.

Theorem 5.4 Suppose that q is transcendental over the rationals. Then the map

X ↦ x, Y ↦ y, A ↦ a, B ↦ b, C ↦ c, D ↦ d,

E ↦ e + (1 − q^{−2})xy extends to an algebra isomorphism T_{q}(2, 2) → T (2, 2). In particular, we have G_{q}(2, 2) ≅ G(2, 2) are isomorphic algebras, hence the Hilbert series of G_{q}(2, 2) is

\frac{(1 − s)(1 − t)(1 − st) + st}{(1 − s)^{2}(1 − t)^{2}(1 − st)}.

Proof. The elements a, b, c, d, e are obviously central in T (2, 2), the Cayley-Hamilton Theorem asserts x^{2} = ax−c, y^{2} = by−d, and its bilinearization says yx = −xy + ay + bx − e. With these equalities at hand, direct computation verifies that the elements x, y, a, b, c, d, e + (1 − q^{−2})xy satisfy the relations of X, Y, A, B, C, D, E. Therefore there exists an algebra homomorphism ϕ : T_{q}(2, 2) → T (2, 2) mapping the generators to the prescribed images. These images clearly generate the whole T (2, 2), hence ϕ is surjective. On the other hand, ϕ is a grading preserving map between two spaces with the same Hilbert series. So ϕ must be an isomorphism. Finally, note that under this isomorphism, G_{q}(2, 2) is mapped onto G(2, 2). The Hilbert series of G(2, 2) was determined in [11]. □

Remark 5.5 Note that the above isomorphism T_{q}(2, 2) → T (2, 2) is nontrivial in the sense that the subalgebra R_{q}(2, 2)I is not mapped into R(2, 2)I: the element E is mapped outside R(2, 2)I.

Remark 5.6 The algebras G(N, m) are highly non-trivial objects from the combinatorial point of view. For example, they are not finitely presented algebras: the ideal of relations among the generators is not finitely generated (although it is finitely generated as a T-ideal); see the papers [7], [8], [22],

[17] for some computations in the 2 × 2 case. Therefore the isomorphism G_{q}(2, 2) ≅ G(2, 2) looks a bit surprising to us.

Not all β(m)-coinvariants are central in A_{q}(N, m), and the algebra R_{q}(N, m) is not commutative in general, as the following Proposition shows.

Proposition 5.7 Suppose that q^{2} ≠ 1.

(i) The elements Tr_{q}(XY) and Tr_{q}(XZ) do not commute in A_{q}(2, 3).

(ii) The element Tr_{q}(XY) is not central in A_{q}(2, 2).

Proof. (i) The expansion of the commutator [Tr_{q}(XY), Tr_{q}(XZ)] in the basis (10) is ∑_{i,j,k,l} f_{jl}^{ik}y_{j}^{i}z_{l}^{k}, where f_{jl}^{ik} is a homogeneous quadratic expression of the variables x_{t}^{s}. We claim that f_{11}^{22}, the coefficient of y_{1}^{2}z_{1}^{2}, is non-zero. An inspection of (8), (9) shows that y_{1}^{2} (respectively, z_{1}^{2}) appears on the right hand side of a relation if and only if y_{1}^{2} (respectively, z_{1}^{2}) appears on the left hand side of the relation. Therefore the coefficient of y_{1}^{2}z_{1}^{2} is the same as in [qx_{2}^{1}y_{1}^{2}, qx_{2}^{1}z_{1}^{2}], hence f_{11}^{22} = (1 − q^{2})x_{2}^{1}x_{2}^{1}.

(ii) This follows from the relations in Theorem 5.2 (iii), but it is easy to check directly, similarly to (i). Write [Tr_{q}(XY), x_{1}^{1}] as ∑ f_{j}^{i}y_{j}^{i}, where f_{j}^{i} is a linear combination of the entries of X. We claim that f_{1}^{2} ≠ 0. By the same observation on the relations we used in (i), we see that f_{1}^{2} is the same as the coefficient of y_{1}^{2} in [qx_{2}^{1}y_{1}^{2}, x_{1}^{1}], so f_{1}^{2} = q[x_{2}^{1}, x_{1}^{1}] ≠ 0. □

Remark 5.8 If we endow the coordinate ring of pairs of 2 × 2 matrices with the Poisson bracket that is the classical limit of the tensor product of two copies of the quantum coordinate ring of 2 × 2 matrices, then a calculation similar to the above proof yields that R(2, 2) is not a Poisson subalgebra. So roughly speaking, there is no subalgebra in the tensor product of two copies of the quantum coordinate ring of 2 × 2 matrices that would lie above R(2, 2). This indicates that we are forced to move to the reflection equation algebra and braided tensor products to find quantum analogues of the rings of matrix invariants.

We continue with the ring theoretic study of our algebras.

Theorem 5.9 (i) The algebra A_{q}(2, m) is noetherian.

(ii) Suppose that q is not a root of unity. Then R_{q}(2, m) is noetherian and is finitely generated as a ℂ-algebra.

(iii) Suppose that q is not a root of unity. Then T_{q}(2, m) is finitely generated as a left (respectively right) R_{q}(2, m)-module. In particular, it is noetherian and is finitely generated as a ℂ-algebra.

Proof. (i) One may build up A_{q}(2, m) by adjoining step-by-step the variables x(k)_{j}^{i} in the following order:

x(1)_{2}^{1}, x(1)_{2}^{2}, x(1)_{1}^{1}, x(1)_{1}^{2}, x(2)_{2}^{1}, x(2)_{2}^{2}, x(2)_{1}^{1}, x(2)_{1}^{2}, ….

For notational simplicity, rename the members of this ordered sequence as x_{1}, x_{2}, …, x_{4m}. Denote by A_{i} the subalgebra generated by x_{1}, …, x_{i}. Then it is easy to see from the defining relations of A_{q}(2, m) that for all i, we have that A_{i}x_{i+1} is contained in A_{i}+x_{i+1}A_{i}, and x_{i+1}A_{i} is contained in A_{i}+A_{i}x_{i+1}. This implies successively that the algebras A_{i} are noetherian (see 1.2.10 in [20]), and thus A_{q}(2, m) is noetherian.

(ii) The Haar functional of the cosemisimple Hopf algebra F_{q}(SL_{N}) can be used to construct a projection (Reynolds operator) A_{q}(N, m) → R_{q}(N, m) with certain good properties (see for example [5] for details). Then both statements can be derived from (i) by the well known argument of Hilbert.

(iii) In order to show that T_{q}(2, m) is a finitely generated left (right) module over R_{q}(2, m), we recall from Remark 4.3 the ℂ-vector space isomorphism T_{q}(2, m) ≅ (A_{q}(2, m) ⊗ X^{∗})^{F_{q}(SL_{2})}. This is clearly an isomorphism of left (right) R_{q}(2, m)-modules (note that A_{q}(2, m)⊗X^{∗} is naturally a left and right A_{q}(2, m)-module). Now A_{q}(2, m) ⊗ X^{∗} is a finitely generated module over A_{q}(2, m), hence is noetherian by (i). Hence (A_{q}(2, m) ⊗ X^{∗})^{F_{q}(SL_{2})} is a finitely generated module over A_{q}(2, m)^{F_{q}(SL_{2})} = R_{q}(2, m) by a well known argument using the Reynolds operator A_{q}(2, m) → R_{q}(2, m) again; see for example page 71 in [3] for this argument. □

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M. Domokos: Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary E-mail: domokos@renyi.hu

T. H. Lenagan: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland E-mail: tom@maths.ed.ac.uk