Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 2, 411-421.

The Center of the Dipper Donkin Quantized Matrix Algebra

Hans Plesner Jakobsen Hechun Zhang^{1}^{2}

Mathematics Institute, Universitetsparken 5 DK–2100 Copenhagen Ø, Denmark e-mail: jakobsen@math.ku.dk zhang@math.ku.dk

Abstract. We compute the center in case q is a primitive root of unity.

1. Introduction

Let 𝔽 be a field of characteristic zero. There are several ways to quantize the matrix algebra M_{n}(𝔽). Among them, the most common is probably the algebra introduced by Faddeev, Reshetikhin, and Takhtajan in [4]. Denote that algebra by M_{n}(q). In [3] Dipper and Donkin defined another quantized matrix algebra which has many features which are different from M_{n}(q) e.g. the quantized determinant is not central. In [9] a two parameter quantized matrix algebra was defined for which the above two quantized matrix algebras are special cases. Even more general multi-parameter quantized algebras have been introduced by Artin, Schelter, and Tate [1], Sudbery [8], and Reshetikhin [7], independently.

To understand the structure of these quantized matrix algebras it is natural to determine their centers. In [6] the center of M_{n}(q) was determined by finding the degree of the algebra over its center and by investigating certain quantized minors. In this paper we compute the center of the quantized matrix algebra constructed by Dipper and Donkin [3] by similar methods. Throughout, the quantum parameter is denoted by q ∈ 𝔽^{∗}. In most applications q will be a primitive mth root of unity for some positive integer m.

1Permanent address: Dept. of applied Math, Tsinghua University, Beijing, 100084, P.R. China

2The second author is partially supported by NSF of China 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1997 Heldermann Verlag

H. P. Jakobsen, H. Zhang: The center of the Dipper Donkin quantized matrix algebra 412

The so-called Dipper Donkin quantized matrix algebra Mat_{n}(q) is an associative algebra over 𝔽 generated by elements Z_{i,j}, i, j = 1, 2, ⋯, n subject to the following relations:

Z_{i,j}Z_{s,t} = qZ_{s,t}Z_{i,j} if i > s and j ≤ t,  (1)
Z_{i,j}Z_{s,t} = Z_{s,t}Z_{i,j} + (q − 1)Z_{s,j}Z_{i,t}, if i > s and j > t,
Z_{i,j}Z_{i,k} = Z_{i,k}Z_{i,j} for all i, j, k.

In fact, the algebra Mat_{n}(q) is a bialgebra and its structure maps are defined as follows:

Δ(Z_{i,j}) = ∑_{k} Z_{i,k} ⊗ Z_{k,j} for i, j, k = 1, 2, ⋯, n,  (2)

ϵ(Z_{i,j}) = δ_{i,j} for i, j = 1, 2, ⋯, n.  (3)

We thank the referee for an interesting remark.

2. The degree of a prime algebra

The main tool used to compute the degree of Mat_{n}(q) is the theory developed in [2] by De Concini and Procesi. So let us first recall their set-up.

Let A be an algebra and let σ be an automorphism of A. A twisted derivation of A relative to σ is a linear map D : A ⟶ A such that:

D(ab) = D(a)b + σ(a)D(b), a, b ∈ A.  (4)

An element a induces an inner twisted derivation ad_{σ}a relative to σ by the formula:

ad_{σ}a(b) = ab − σ(b)a.  (5)

Given an automorphism σ and a twisted derivation D relative to σ we can define the twisted polynomial algebra A_{σ,D}[x] in the indeterminate x to be the 𝔽-module A ⊗ 𝔽[x] thought of as formal polynomials with multiplication defined by

xa = σ(a)x + D(a).  (6)

Let A be a prime algebra (i.e. aAb = 0 implies a = 0 or b = 0) over 𝔽 and let Z be the center of A. Then Z is a domain and A is a torsion free module over Z. Assume that A is a finite dimensional module over Z. Then A embeds in a finite dimensional central simple algebra Q(A) = Q(Z) ⊗_{Z} A where Q(Z) is the fraction field of Z. If \overline{Q(Z)} denotes the algebraic closure of Q(Z) we have that \overline{Q(Z)} ⊗_{Z} A is the full algebra M_{d}(\overline{Q(Z)}) of d × d matrices over \overline{Q(Z)}. Then d is called the degree of A.

Let A be a prime algebra over 𝔽 generated by x_{1}, x_{2}, ⋯, x_{n} and let Z_{0} be a central subalgebra of A. For each i = 1, 2, ⋯, n, denote by A^{i} the subalgebra of A generated by x_{1}, x_{2}, ⋯, x_{i} and let Z_{0}^{i} = Z_{0} ∩ A^{i}. We assume that the following conditions hold for each i = 1, 2, ⋯, n :

1. x_{i}x_{j} = b_{i,j}x_{j}x_{i} + p_{i,j} if i > j, where b_{i,j} ∈ 𝔽^{∗} and p_{i,j} ∈ A^{i−1}.

2. A^{i} is a finite module over Z_{0}^{i}.

3. The formulas σ_{i}(x_{j}) = b_{i,j}x_{j} for j < i define an automorphism of A^{i−1} which is the identity on Z_{0}^{i−1}.

4. The formulas D_{i}(x_{j}) = p_{i,j} for j < i define a twisted derivation relative to σ_{i}.

It is easy to see that A then is an iterated twisted Ore extension. Let \overline{A} be an associative algebra generated by y_{1}, y_{2}, ⋯, y_{n} with defining relations y_{i}y_{j} = b_{i,j}y_{j}y_{i} for j < i. We call this algebra the associated quasipolynomial algebra of A. In [2] the following was proved

Theorem 2.1. Under the above assumptions the degree of A is equal to the degree of the associated quasipolynomial algebra \overline{A}.

Given an n × n skew-symmetric matrix H = (h_{i,j})_{i,j=1}^{n} over ℤ we construct the twisted polynomial algebra 𝔽_{H}[x_{1}, x_{2}, ⋯, x_{n}] as follows: It is the algebra generated by elements x_{1}, x_{2}, ⋯, x_{n} with the following defining relations:

x_{i}x_{j} = q^{h_{i,j}}x_{j}x_{i} for i, j = 1, 2, ⋯, n.  (7)

It can be viewed as an iterated twisted polynomial algebra with respect to any ordering of the indeterminates x_{i}. Given a = (a_{1}, a_{2}, ⋯, a_{n}) ∈ ℤ^{n} we write x^{a} = x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}}. We say that H is the defining matrix of the algebra.

Let q be a primitive mth root of unity. We consider the matrix H as a matrix of the homomorphism H : ℤ^{n} ⟶ (ℤ/mℤ)^{n} and we denote by K the kernel of H and by h the cardinality of the image of H. The following was proved in [2]:

Theorem 2.2. (a) The monomials x^{a} with a ∈ K ∩ ℤ_{+}^{n} form a basis of the center of 𝔽_{H}[x_{1}, x_{2}, ⋯, x_{n}].

(b) deg 𝔽_{H}[x_{1}, x_{2}, ⋯, x_{n}] = \sqrt{h}.

It is well-known that a skew-symmetric matrix over ℤ such as our matrix H can be brought into a block diagonal form by an element W ∈ SL(ℤ). Specifically, there is a W ∈ SL(ℤ)

and a sequence of 2 × 2 matrices S(m_{i}) = ( \begin{matrix}0 & −m_{i} \\ m_{i} & 0 \end{matrix} ) , i = 1, …, N, with m_{i} ∈ ℤ for each

i = 1, …, N, such that

W · H · W^{t} = \{ \begin{matrix}Diag (S(m_{1}), …, S(m_{N}), 0) with N = \frac{n^{2}−1}{2}, if n is odd \\ Diag (S_{1}(m_{1}), …, S(m_{N})) with N = \frac{n^{2}}{2} , if n is even \end{matrix} .  (8)

Definition 2.3. Any matrix of the form of the right-hand-side in (8) will be called a canonical form of H and will occasionally be denoted by J_{H}.

Thus, a canonical form of H reduces the algebra to the tensor product of twisted Laurent polynomial algebras in two variables with commutation relation xy = q^{r}yx. By Theorem 2.2 it follows in particular that the degree of a twisted Laurent polynomial algebra in two variables is equal to m/(m, r) where (m, r) is the greatest common divisor of m and r.

3. The degree of Mat_{n}(q)

It is well-known that Mat_{n}(q) is an iterated Ore extension (cf.[1]). By Theorem 2.1, we have

deg Mat_{n}(q) = deg \overline{Mat_{n}(q)}  (9)

where \overline{Mat_{n}(q)} is the associated quasipolynomial algebra of Mat_{n}(q).

Let M = ∑_{i≥j} E_{i,j} be the n × n matrix where E_{i,j} is the matrix unit in M_{n}(𝔽) with (i, j)-entry equal to 1 and zero elsewhere.

We define an ordering on the set of generators Z_{i,j}, i, j = 1, 2, ⋯, n, of Mat_{n}(q) by lexicographic ordering on the set of indices. Let J(n) be the defining matrix of \overline{Mat_{n}(q)} where the order of the generators is induced by that of Mat_{n}(q). It is easy to see that

J(n) = (\begin{matrix}0 & −M & −M & ⋯ & ⋯ & −M & −M & −M \\ M^{t} & 0 & −M & ⋯ & ⋯ & −M & −M & −M \\ M^{t} & M^{t} & 0 & ⋯ & ⋯ & −M & −M & −M \\ ⋯ & ⋯ & ⋯ \\ M^{t} & M^{t} & M^{t} & ⋯ & ⋯ & M^{t} & 0 & −M \\ M^{t} & M^{t} & M^{t} & ⋯ & ⋯ & M^{t} & M^{t} & 0 \end{matrix}) .  (10)

Theorem 3.1. The canonical form of J(n) is

(a) Diag(S_{1}, S_{2}, ⋯, S_{\frac{n^{2}}{2}} ) if n is even where S_{i} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , i = 1, 2, ⋯, \frac{n^{2}}{2} .

(b) Diag(S_{1}, S_{2}, ⋯, S_{\frac{n^{2}−1}{2}} , 0) if n is odd where S_{i} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , i = 1, 2, ⋯, \frac{n^{2}−1}{2}.

Proof. Let H_{1} = Diag(−M^{−1}, I, ⋯, I) where I is the n × n identity matrix. Then H_{1} ∈ SL(n^{2}, ℤ) and

J_{1} = H_{1}J(n)H_{1}^{t} = (\begin{matrix}0 & I & I & ⋯ & ⋯ & I & I & I \\ −I & 0 & −M & ⋯ & ⋯ & −M & −M & −M \\ −I & M^{t} & 0 & ⋯ & ⋯ & −M & −M & −M \\ ⋯ & ⋯ & ⋯ \\ −I & M^{t} & M^{t} & ⋯ & ⋯ & M^{t} & 0 & −M \\ −I & M^{t} & M^{t} & ⋯ & ⋯ & M^{t} & M^{t} & 0 \end{matrix}) .  (11)

Let H_{2} = (\begin{matrix}I & 0 & 0 & ⋯ & ⋯ & 0 & 0 \\ M & I & 0 & ⋯ & ⋯ & 0 & 0 \\ 0 & 0 & I & ⋯ & ⋯ & 0 & 0 \\ ⋯ & ⋯ & ⋯ \\ 0 & 0 & 0 & ⋯ & ⋯ & 0 & I \end{matrix}). Then

J_{2} = H_{2}J_{1}H_{2}^{t} = (\begin{matrix}0 & I & I & ⋯ & ⋯ & I & I & I \\ −I & M − M^{t} & 0 & ⋯ & ⋯ & 0 & 0 & 0 \\ −I & 0 & 0 & ⋯ & ⋯ & −M & −M & −M \\ ⋯ & ⋯ & ⋯ \\ −I & 0 & M^{t} & ⋯ & ⋯ & M^{t} & 0 & −M \\ −I & 0 & M^{t} & ⋯ & ⋯ & M^{t} & M^{t} & 0 \end{matrix}) .  (12)

Let H_{3} = (\begin{matrix}I & 0 & 0 & ⋯ & ⋯ & 0 & 0 \\ 0 & I & 0 & ⋯ & ⋯ & 0 & 0 \\ 0 & −I & I & ⋯ & ⋯ & 0 & 0 \\ ⋯ & ⋯ & ⋯ \\ 0 & −I & 0 & ⋯ & ⋯ & 0 & I \end{matrix}) . Then

J_{3} = H_{3}J_{2}H_{3}^{t} = (\begin{matrix}0 & I & 0 & ⋯ & ⋯ & 0 & 0 & 0 \\ −I & M − M^{t} & M^{t} − M & ⋯ & ⋯ & M^{t} − M & M^{t} − M & M^{t} − M \\ 0 & M^{t} − M & 0 & ⋯ & ⋯ & −M & −M & −M \\ ⋯ & ⋯ & ⋯ \\ 0 & M^{t} − M & M^{t} & ⋯ & ⋯ & M^{t} & 0 & −M \\ 0 & M^{t} − M & M^{t} & ⋯ & ⋯ & M^{t} & M^{t} & 0 \end{matrix}) .(13)

Let H_{4} = (\begin{matrix}I & 0 & 0 & ⋯ & ⋯ & 0 & 0 \\ M^{t} & I & 0 & ⋯ & ⋯ & 0 & 0 \\ M^{t} − M & 0 & I & ⋯ & ⋯ & 0 & 0 \\ ⋯ & ⋯ & ⋯ \\ M^{t} − M & 0 & 0 & ⋯ & ⋯ & 0 & I \end{matrix}). Then

H_{4}J_{3}H_{4}^{t} = (\begin{matrix}0 & I & 0 & ⋯ & ⋯ & 0 & 0 & 0 \\ −I & 0 & 0 & ⋯ & ⋯ & 0 & 0 & 0 \\ 0 & 0 & 0 & ⋯ & ⋯ & −M & −M & −M \\ ⋯ & ⋯ & ⋯ \\ 0 & 0 & M^{t} & ⋯ & ⋯ & M^{t} & 0 & −M \\ 0 & 0 & M^{t} & ⋯ & ⋯ & M^{t} & M^{t} & 0 \end{matrix}) .  (14)

If n is even we repeat the above procedure. Then

J(n) ⟶ Diag(M_{1}, M_{2}, ⋯, M\frac{n}{2} )  (15)

where M_{i} = (\begin{matrix}0 & I \\ −I & 0 \end{matrix}) , i = 1, 2, ⋯, \frac{n}{2}. This proves our assertion in case n is even.

If n is odd we also repeat the above procedure. Then

J(n) ⟶ Diag(L_{1}, L_{2}, ⋯, L_{\frac{n−1}{2}} )  (16)

where

L_{i} = (\begin{matrix}0 & I \\ −I & 0 \end{matrix}) ,  (17)

for i = 1, 2, ⋯, (n − 3)/2, and

L_{\frac{n−1}{2}} = (\begin{matrix}0 & −M & −M \\ M^{t} & 0 & −M \\ M^{t} & M^{t} & 0 \end{matrix}) .  (18)

Similar to the above we can easily find a transformation

L_{\frac{n−1}{2}} ⟶ (\begin{matrix}0 & I & 0 \\ −I & 0 & 0 \\ 0 & 0 & M − M^{t} \end{matrix})  (19)

and we can find a transformation

M − M^{t} ⟶ Diag(N_{1}, N_{2}, ⋯, N_{\frac{n−1}{2}} , 0)  (20)

where

N_{j} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , j = 1, 2, ⋯, \frac{n − 1}{2}.  (21)

This proves our assertion in case n is odd. □

4. The center of Mat_{n}(q)

For any B = (b_{i,j})_{i,j=1}^{n} ∈ M_{n}(ℤ_{+}) we define

Z^{B} = ΠZ_{i,j}^{b_{i,j}},  (22)

where the factors are arranged according to lexicographic ordering. We denote the generators of \overline{Mat_{n}(q)} by X_{i,j}, i, j = 1, 2, ⋯, n, and define the symbol X^{B} analogously in terms of the same ordering.

Let C be the center of Mat_{n}(q) and \overline{C} the center of \overline{Mat_{n}(q)}. For any P ∈ C the leading term of P must be of the form cZ^{B}, c ∈ 𝔽, for some B = (a_{i,j})_{i,j=1}^{n} ∈ M_{n}(ℤ_{+}). For any Z_{k,l} the leading term of P Z_{k,l} is cq^{r_{k,l}}Z^{B+E_{k,l}} where r_{k,l} = ∑_{i>k} ∑_{j≤l} a_{i,j}. The leading term of Z_{k,l}P is cq^{l_{k,l}}Z^{B+E_{k,l}} where l_{k,l} = ∑_{i<k} ∑_{j≥l} a_{i,j}. Since P Z_{k,l} = Z_{k,l}P we get q^{l_{k,l}} = q^{r_{k,l}} and this implies that cX^{B} is a central element of the twisted polynomial algebra \overline{Mat_{n}(q)}. Hence we can define a map

Λ : C ⟶ \overline{C}  (23)

by

P ↦ cX^{B}  (24)

if the leading term of P is cZ^{B}, c ∈ 𝔽.

Clearly, Λ(P) = 0 implies P = 0.

Theorem 4.1. (a) If n is even and q is a primitive m-th root of unity then the center of \overline{Mat_{n}(q)} is generated by

{X_{i,j}^{m} | i, j = 1, 2, ⋯, n}.  (25)

(b) If n is odd and q is a primitive m-th root of unity then the center of \overline{Mat_{n}(q)} is generated by

{X_{i,j}^{m}, X^{A} | i, j = 1, 2, ⋯, n}  (26)

where A = (a_{i,j})_{i,j=1}^{n} ∈ M_{n}(ℤ_{+}) is given by a_{i,j} = 1 if i + j is even and a_{i,j} = m − 1 if i + j is odd.

Proof. Our assertions follow immediately from Theorem 3.1 and Theorem 2.2. □

Lemma 4.2. Let n ∈ ℤ_{+} and assume that i > s and j > t. Then

Z_{i,j}Z_{s,t}^{n} = Z_{s,t}^{n}Z_{i,j} + (1 − q^{−n})Z_{i,t}Z_{s,j}Z_{s,t}^{n−1},  (27)

Z_{s,t}Z_{i,j}^{n} = Z_{i,j}^{n}Z_{s,t} + (1 − q^{n})Z_{s,j}Z_{i,t}Z_{i,j}^{n−1}.  (28)

Proof. The assertions follow from (1) by using induction on n. □

Corollary 4.3. If q is an m-th root of unity then Z_{i,j}^{m} is a central element for all i, j = 1, 2, ⋯, n.

The quantum determinant of Mat_{n}(q) is defined by the following formula:

det_{q} = Σ_{σ∈S_{n}}(−1)^{l(σ)}Z_{1,σ(1)}Z_{2,σ(2)} ⋯ Z_{n,σ(n)}  (29)

where S_{n} is the symmetric group on n letters and l() is the standard length function on S_{n}. In [3], Dipper and Donkin proved that

Theorem 4.4.

det_{q} = Σ_{σ∈S_{n}}(−q)^{−l(σ)}Z_{σ(1),1}Z_{σ(2),2} ⋯ Z_{σ(n),n}.  (30)

Furthermore, the following result was obtained.

Theorem 4.5.

Z_{i,j}det_{q} = q^{i−j}det_{q}Z_{i,j} for 1 ≤ i and j ≤ n.  (31)

Let r ≤ n be a positive integer. Given any two subsets I = {i_{1}, i_{2}, ⋯, i_{r}} and J = {j_{1}, j_{2}, ⋯, j_{r}} of {1, 2, ⋯, n}, each having cardinality r, clearly the subalgebra of Mat_{n}(q) generated by all Z_{i_{a},j_{b}}, a, b = 1, 2, ⋯, r, is isomorphic to Mat_{r}(q). Hence we can talk about its determinant. Such a determinant is called a subdeterminant (or minor) of det_{q} and will be denoted by det_{q}(I, J). If I = {1, 2, ⋯, n} \ {i}, J = {1, 2, ⋯, n} \ {j}, det_{q}(I, J) will be denoted by A(i, j).

We have [3, page 203] Lemma 4.6. Let i, k ≤ n be fixed integers. Then

δ_{i,k}det_{q} = ∑_{j=1}^{n}(−1)^{k−j}Z_{i,j}A(k, j).  (32)

δ_{i,k}det_{q} = ∑_{j=1}^{n}(−1)^{k−j}Z_{j,i}A(j, k).  (33)

The above formulas are called the quantum Laplace expansions.

Definition 4.7. An element x ∈ Mat_{n}(q) is called covariant if for any Z_{i,j} there exists an integer n_{ij} such that

xZ_{i,j} = q^{n_{i,j}}Z_{i,j}x.  (34)

Clearly, Z_{1,n} and Z_{n,1} are covariant.

Let I = {t + 1, t + 2, ⋯, n}, J = {1, 2, ⋯, n − t}, and let ϕ_{t} = det_{q}(I, J). Let I^{∗} = {1, 2, ⋯, t}, J^{∗} = {n − t + 1, n − t + 2, ⋯, n} and let ϕ_{t}^{∗} = det_{q}(I^{∗}, J^{∗}). Then we have

Theorem 4.8. ϕ_{t} and ϕ_{t}^{∗} are covariant for all t.

Proof. Let M_{t}^{−} = {(i, j) ∈ ℕ^{2} | 1 ≤ i ≤ t and 1 ≤ j ≤ n − t}, M_{t}^{+} = {(i, j) ∈ ℕ^{2} | t + 1 ≤ i ≤ n and n − t + 1 ≤ j ≤ n}, M_{t}^{l} = {(i, j) ∈ ℕ^{2} | t + 1 ≤ i ≤ n and 1 ≤ j ≤ n − t}, and M_{t}^{r} = {(i, j) ∈ ℕ^{2} | 1 ≤ i ≤ t and n − t + 1 ≤ j ≤ n}. We use induction on t to prove that

Z_{i,j}ϕ_{t} = q^{−j}ϕ_{t}Z_{ij} if (i, j) ∈ M_{t}^{−},  (35)
Z_{i,j}ϕ_{t} = q^{n−i}ϕ_{t}Z_{i,j} if (i, j) ∈ M_{t}^{+},
Z_{i,j}ϕ_{t} = q^{i−t−j}ϕ_{t}Z_{i,j} if (i, j) ∈ M_{t}^{l}, and
Z_{i,j}ϕ_{t} = q^{−t}ϕ_{t}Z_{i,j} if (i, j) ∈ M_{t}^{r}.

If t = n − 1 then ϕ_{n−1} = Z_{n,1} which is clearly covariant.

For (i, j) ∈ M_{t}^{l} ∪ M_{t}^{r} the assertions follow from Theorem 4.5 and the defining relations of Mat_{n}(q), respectively.

Now assume that (i, j) ∈ M_{t}^{−}. If j = n − t the assertion follows immediately from the defining relations of Mat_{n}(q). For j < n − t we have

Z_{i,j}ϕ_{t} = ∑_{k=t+1}^{n}(−1)^{n−t+k}Z_{i,j}Z_{k,n−t}A_{t}(k, n − t)  (36)

= ∑_{k=t+1}^{n}(−1)^{n−t+k}[Z_{k,n−t}Z_{i,j}A_{t}(k, n − t) − (1 − q)Z_{i,n−t}Z_{k,j}A_{t}(k, n − t)  (37)

where A_{t}(k, n − t) is the subdeterminant of ϕ_{t}. By the inductive hypothesis and Lemma 4.6 we have

Z_{i,j}ϕ_{t} = q^{j}ϕ_{t}Z_{i,j}  (38)

for (i, j) ∈ M_{t}^{−}. By the same method it follows that

Z_{i,j}ϕ_{t} = q^{n−i}ϕ_{t}Z_{i,j}  (39)

for (i, j) ∈ M_{t}^{+}. Hence ϕ_{t} is covariant.

In a similar way it can be proved that ϕ_{t}^{∗} is covariant. This completes the proof. □

Theorem 4.9. Let q be a primitive m-th root of unity, let Ω_{n}^{+} = Π_{t=1}^{n−1}(ϕ_{t})^{a_{1,t+1}}, and let Ω_{n}^{−} = Π_{t=1}^{n−1}(ϕ_{t}^{∗})^{a_{t,n}}, with the a_{i,j} as in Theorem 4.1. Then Ω_{n} = Ω_{n}^{+}det_{q}Ω_{n}^{−} is a central element of Mat_{n}(q) provided that n is odd.

Proof. Obviously the leading term of Ω_{n} is Z^{A}. By Theorem 4.8 we know that Ω_{n} is covariant. It is easy to see that

Z_{i,j}Ω_{n} = Z^{A}Z_{i,j} + lower terms.  (40)

Hence Z_{i,j}Ω_{n} = Ω_{n}Z_{i,j} for all 1 ≤ i, j ≤ n. This completes the proof. □

Theorem 4.10. If n is even and q is a primitive m-th root of unity then the center of Mat_{n}(q) is generated by

{Z_{i,j}^{m} | i, j = 1, 2, ⋯, n}.  (41)

Proof. Let C′ be the subalgebra generated by the elements Z_{i,j}^{m}, i, j = 1, 2, ⋯, n. We use induction on the order of the leading term to prove that any P ∈ C also belongs to C′. By Theorem 4.1, Λ(P) is of the form cX^{mB} for some c ∈ 𝔽 and B ∈ M_{n}(ℤ_{+}). Hence there exists an element P′ ∈ C′ such that Λ(P) = Λ(P′). Hence the order of the leading term of P −P′ is smaller than that of leading term of P. So P − P′ ∈ C′. Therefore, P ∈ C′. This completes the proof. □

Lemma 4.11. Let I, J be two subsets of {1, 2, ⋯, n}, each having cardinality s. We use |I| (resp. |J|) to denote the sum of the elements in I (resp. in J). Then

det_{q}(I, J)det_{q} = q^{|I|−|J|}det_{q}det_{q}(I, J).  (42)

Proof. The assertion follows from Theorem 4.5. □

The following theorem gives a full solution to the problem raised in Remark 1.3.4 in [3].

Theorem 4.12. Suppose that n is odd and q is a primitive m-th root of unity. Let C′ be the central subalgebra of Mat_{n}(q) generated by Z_{i,j}^{m}, for all i, j = 1, 2, ⋯, n. Then C is generated over C′ by Ω_{n}. The minimal polynomial of Ω_{n} is x^{m} − a for some a ∈ C′.

Proof. The first statement follows as in the proof of Theorem 4.10. Hence

C = C′ + C′Ω_{n} + ⋯ + C′Ω_{n}^{l} + · · · .  (43)

By Theorem 4.5, det_{q}^{m} ∈ C. Since deg det_{q} < deg Ω_{n} if n > 1 we see that det_{q}^{m} ∈ C′. Similarly, ϕ_{s}^{m}, ϕ_{t}^{∗}^{m} ∈ C′.

Let Ω̃_{n}^{−} = det_{q} Ω_{n}^{−}. We know that for any (i, j) there exists a b_{i,j} ∈ ℤ such that

Z_{i,j}Ω̃_{n}^{−} = q^{b_{i,j}} Ω̃_{n}^{−}Z_{i,j}.  (44)

We now compute the b_{i,j} explicitly. For this is enough to work in the associated quasipolynomial algebra \overline{Mat_{n}(q)} since Ω̃_{n}^{−} is covariant.

Let D = (d_{i,j})_{i,j=1}^{n} where

d_{i,j} = \{ \begin{matrix}0 if i > j, \\ 1 if i ≤ j and i + j is even, \\ m − 1 if i ≤ j and i + j is odd. \end{matrix}  (45)

Note that the leading term of Ω̃_{n}^{−} is Z^{D} so when we work in the associated quasipolynomial algebra \overline{Mat_{n}(q)} we have

X_{i,j}X^{D} = q^{b_{i,j}}X^{D}X_{i,j}.  (46)

It follows that

b_{i,j} = \{ \begin{matrix}\frac{i}{2} − \frac{j}{2} & mod m & if both i, j are even, \\ \frac{i}{2} − \frac{j−1}{2} & mod m & if i is even and j is odd, \\ \frac{i−1}{2} − \frac{j}{2} & mod m & if i is odd and j is even, \\ \frac{i−1}{2} − \frac{j−1}{2} & mod m & if both i and j are odd. \end{matrix}  (47)

It is easy to see that b_{i,j} = g_{i} − g_{j} where g_{i} = \frac{i}{2} if i is even g_{i} = \frac{i−1}{2} if i is odd. Then it follows directly that for all t = 1, …, n − 1, Ω̃_{n}^{−}ϕ_{t} = q^{x_{t}}ϕ_{t}Ω̃_{n}^{−} for some integer x_{t}. Therefore, Ω_{n}^{m} = q^{x}(Ω_{n}^{+})^{m}(det_{q}Ω_{n}^{−})^{m} for some x ∈ ℤ.

By Lemma 4.11 we have

ϕ_{s}ϕ_{t} = q^{x_{s,t}}ϕ_{t}ϕ_{s} and  (48)
ϕ_{s}^{∗}ϕ_{t}^{∗} = q^{y_{s,t}}ϕ_{t}^{∗}ϕ_{s}^{∗}

for some integers x_{s,t} and y_{s,t} and this implies that Ω_{n}^{m} = q^{w}Π_{t}ϕ_{t}^{m} det_{q}^{m} Π_{s}ϕ_{s}^{∗}^{m} for some integer w. Hence Ω_{n}^{m} ∈ C′. Since Ω_{n} is of degree m over Q(C′), the fraction field of C′, the minimal polynomial of Ω_{n} over C′ is x^{m} − a for some a ∈ C′. This completes the proof. □

References

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Received November, 11, 1996