FROBENIUS FULL MATRIX ALGEBRAS AND GORENSTEIN TILED ORDERS

Hisaaki Fujita and Yosuke Sakai

Let D be a discrete valuation ring with a unique maximal ideal πD, and let Λ be a Dorder. It is standard to reduce homological properties of Λ to those of the factor algebras Λ/πΛ and such factor algebras are deserving of further study. (See [2].)

Let n be an integer with n ≥ 2. In [1], we introduced an n × n 𝔸-full matrix algebra over a field K, whose multiplication is determined by a structure system 𝔸, that is, an n-tuple of n × n matrices with certain properties. 𝔸-full matrix algebras are associative, basic, connected K-algebras. A prototype of 𝔸-full matrix algebras is the class of factor algebras Λ/πΛ of tiled D-orders Λ. Studying representation matrices of certain modules over 𝔸-full matrix algebras, Frobenius 𝔸-full matrix algebras are characterized by the shape of their structure systems 𝔸. For a Gorenstein tiled D-order Λ, the factor algebra Λ/πΛ is a Frobenius 𝔸-full matrix algebra. In this paper we study the converse of this fact. Our main result is the following.

Theorem 1. (1) For 2 ≤ n ≤ 7, all Frobenius n × n 𝔸-full matrix algebras are isomorphic to Λ/πΛ for some Gorenstein tiled D-orders Λ. Moreover a list of them (up to isomorphism) is obtained.

(2) For each n ≥ 8, there is a Frobenius n × n 𝔸-full matrix algebra having no corresponding Gorenstein tiled D-orders.

1. 𝔸-Full matrix algebras

We begin by recalling 𝔸-full matrix algebras. Let K be a field and n ≥ 2 an integer. Let 𝔸 = (A_{1}, …, A_{n}) be an n-tuple of n × n matrices A_{k} = (a_{ij}^{(k)}) ∈ 𝕄_{n}(K) (1 ≤ k ≤ n) satisfying the following three conditions.

(A1) a_{ij}^{(k)}a_{il}^{(j)} = a_{il}^{(k)}a_{kl}^{(j)} for all i, j, k, l ∈ {1, …, n},
(A2) a_{kj}^{(k)} = a_{ik}^{(k)} = 1 for all i, j, k ∈ {1, …, n}, and
(A3) a_{ii}^{(k)} = 0 for all i, k ∈ {1, …, n} such that i ≠ k.

Let A = ⨁_{1≤i,j≤n} Ku_{ij} be a K-vector space with basis {u_{ij} | 1 ≤ i, j ≤ n}. Then we define multiplication of A by using 𝔸, that is,

u_{ik}u_{lj} := \{ \begin{matrix}a_{ij}^{(k)}u_{ij} & if k = l \\ 0 & otherwise. \end{matrix}

Then u_{11}, …, u_{nn} are orthogonal primitive idempotents such that u_{11} + ⋯ + u_{nn} = 1_{A} an identity of A and u_{ii}Au_{jj} ≅ K. Hence A is an associative, basic, connected K-algebra.

The detailed version of this paper will be submitted for publication elsewhere.

We call A an n × n 𝔸-full matrix algebra with a structure system 𝔸. We note that gl.dimA = ∞, because every entry of the Cartan matrix of A is 1.

In what follows, we assume that a_{ij}^{(k)} = 0 or 1 for all 1 ≤ i, k, j ≤ n.

2. Tiled orders

Let D be a discrete valuation ring with a unique maximal ideal πD and n ≥ 2 an integer. Let {λ_{ij} | 1 ≤ i, j ≤ n} be a set of integers satisfying

λ_{ij} ≥ 0, λ_{ii} = 0, λ_{ik} + λ_{kj} ≥ λ_{ij}, and λ_{ij} + λ_{ji} > 0 (if i ≠ j)

for all 1 ≤ i, j, k ≤ n. Then Λ = (π^{λ_{ij}}D) is a subring of 𝕄_{n}(D), which we call an n × n tiled D-order.

Example 2. Let Λ = (π^{λ_{ij}}D) be an n × n tiled D-order. Put A := Λ/πΛ, K := D/πD and u_{ij} := π^{λ_{ij}}e_{ij} + πΛ ∈ A, where e_{ij}’s are the matrix units in 𝕄_{n}(D). Define A_{k} = (a_{ij}^{(k)}) ∈ 𝕄_{n}(K) (1 ≤ k ≤ n) by

a_{ij}^{(k)} := \{ \begin{matrix}1 & if λ_{ik} + λ_{kj} = λ_{ij} \\ 0 & otherwise, \end{matrix}

and set 𝔸 := (A_{1}, …, A_{n}). Then note that

u_{ik}u_{lj} = \{ \begin{matrix}a_{ij}^{(k)}u_{ij} & if k = l \\ 0 & otherwise. \end{matrix}

Hence A is an 𝔸-full matrix algebra.

3. Representation matrices

Let A = ⨁_{1≤i,j≤n} u_{ij}K be an n × n 𝔸-full matrix algebra, where 𝔸 = (A_{1}, …, A_{n}) and A_{k} = (a_{ij}^{(k)}) ∈ 𝕄_{n}(K) (1 ≤ k ≤ n). Let M be a right A-module with dimension type \underline{dim}M = (1, …, 1). Then M has a K-basis \{v_{i} | 1 ≤ i ≤ n\} such that v_{i}u_{ii} = v_{i} for all 1 ≤ i ≤ n. Hence there exists a matrix S = (s_{ij}) ∈ 𝕄_{n}(K) such that v_{i}u_{ij} = s_{ij}v_{j} for all 1 ≤ i, j ≤ n. We call S a representation matrix of M.

Proposition 3. For each indecomposable projective right A-module u_{ii}A, \underline{dim}u_{ii}A = (1, …, 1) and it has a representation matrix (a_{ij}^{(k)})_{k,j}, that is, an n × n matrix whose (k, j)-entry is a_{ij}^{(k)}. Moreover u_{ii}A is isomorphic to an injective Hom_{K}(Au_{ll}, K) if and only if a_{il}^{(k)} = 1 for all 1 ≤ k ≤ n.

Example 4. Let A be an 𝔸-full matrix algebra where

𝔸 = ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 \end{matrix} 0 1 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 0 \begin{matrix}0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \end{matrix} ) .

Then representation matrices of u_{11}A, …, u_{44}A are given by

( \begin{matrix}1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix} ) .

Hence u_{11}A, u_{22}A and u_{44}A are injective but not u_{33}A.

4. Frobenius 𝔸-full matrix algebras

By means of structure systems, we can characterize Frobenius 𝔸-full matrix algebras. Let A = ⨁_{1≤i,j≤n} u_{ij}K be an n × n 𝔸-full matrix algebra, where 𝔸 = (A_{1}, …, A_{n}) and A_{k} = (a_{ij}^{(k)}) ∈ 𝕄_{n}(K) (1 ≤ k ≤ n).

Proposition 5. The following are equivalent for an 𝔸-full matrix algebra A. (1) A is a Frobenius algebra.

(2) There exists a permutation σ of the set {1, …, n} such that σ(i) ≠ i for all 1 ≤ i ≤ n and that a_{iσ(i)}^{(k)} = 1 for all 1 ≤ i, k ≤ n.

In this case σ is a Nakayama permutation of A, that is, soc(u_{ii}A) ≅ top(u_{σ(i)σ(i)}A) for all 1 ≤ i ≤ n. Moreover, for all 1 ≤ i, k, j ≤ n, a_{ij}^{(k)} = a_{kσ(i)}^{(j)} holds.

Using Proposition 5, we can find structure systems 𝔸 which define Frobenius 𝔸-full matrix algebras. Let σ be a permutation of the set {1, …, n} such that σ(i) ≠ i for all 1 ≤ i ≤ n. Let T be the set of triples (i, k, j) of integers 1 ≤ i, k, j ≤ n. Then we have a bijection

φ : T → T, (i, k, j) ↦ (k, j, σ(i)).

Decompose T into φ-orbits {T_{α}}_{α}, and put I := ∪{T_{α}| (i, k, σ(i)) ∈ T_{α}}, Z := ∪{T_{α}| (i, k, i) ∈ T_{α}, i ≠ k}, and X := ∪{T_{α}| T_{α} ⊄ I∪Z}. Then we have T = I∪Z∪X (disjoint).

Proposition 6. (1) Suppose that 𝔸 is a Frobenius structure system with Nakayama permutation σ. Then there exists a φ-invariant subset Y of X such that

a_{ij}^{(k)} = \{ \begin{matrix}1 & if (i, k, j) ∈ I ∪ Y \\ 0 & otherwise. \end{matrix}

(2) Let Y be a φ-invariant subset of X, and define 𝔸(Y) = (a_{ij}^{(k)}) by

a_{ij}^{(k)} := \{ \begin{matrix}1 & if (i, k, j) ∈ I ∪ Y \\ 0 & otherwise. \end{matrix}

Then 𝔸(Y) is a Frobenius structure system whenever (A1) holds for 𝔸(Y).

(3) For the empty subset ∅ of X, 𝔸(∅) is a Frobenius structure system.

5. φ-orbits for a cyclic permutation

In this section, we clarify the φ-orbits of T for a cyclic permutation σ = (1 2 ⋯ n). First we count the number of φ-orbits of T.

Proposition 7. (1) For a φ-orbit T_{α} of T, the number |T_{α}| of elements in T_{α} is 3n or n.

(2) T has a φ-orbit T_{α} with |T_{α}| = n if and only if n is not divisible by 3. In this case, T has a unique φ-orbit having n elements, which is contained in X.

(3) I has n − 1 φ-orbits.

(4) Z has n − 2 φ-orbits.

(5) If n is divisible by 3, then X has (n − 1)^{2}/3 φ-orbits.

(6) If n is not divisible by 3, then X has (n − 2)(n − 4)/3 + 1 φ-orbits.

Next we clarify the members of each φ-orbit of T.

Proposition 8. Let T_{α} be a φ-orbit of T and put T_{α}^{(r)} := {(i, k, j) ∈ T_{α} | k = r} for all 1 ≤ r ≤ n.

(1) Suppose that |T_{α}| = 3n. Then |T_{α}^{(r)}| = 3 for each r = 1, …, n. If (i, 1, j) ∈ T_{α}, then

T_{α}^{(1)} = {(i, 1, j), (σ^{−j+1}(1), 1, σ^{−j+2}(i)), (σ^{−i}(j), 1, σ^{−i+1}(1))}.

(2) Suppose that |T_{α}| = n. Then |T_{α}^{(r)}| = 1 for each r = 1, …, n. If n = 3t + 1 then T_{α}^{(1)} = {(t + 1, 1, 2t + 2)}. If n = 3t + 2 then T_{α}^{(1)} = {(2t + 2, 1, t + 2)}.

6. Minimal Frobenius structure systems

Let A be a Frobenius 𝔸-full matrix algebra with Nakayama permutation σ. Then it follows from Proposition 6 (1) that 𝔸 is determined by a φ_{σ}-invariant subset Y of X. We call 𝔸 a minimal Forbenius structure system if Y is minimal among φ_{σ}-invariant subset of X which define Frobenius full matrix algebras. For a cyclic permutation, minimal Frobenius structure systems are determined by the following theorem.

Theorem 9. Let n be an integer with n ≥ 4, and let σ = (1 2 ⋯ n) be a cyclic permutation. Then the following statements hold.

(1) Let n be even. Then the φ-invariant subsets defining minimal Frobenius structure systems are just φ-orbits contained in X.

(2) Let n be odd and n = 2s + 1 for some s. Then the φ-invariant subsets defining minimal Frobenius structure systems are just φ-orbits X_{β} contained in X such that X_{β} does not contain any element of the form (s + 1, 1, k) for any k with k ≢ s^{2} + 1 (mod n).

The following example illustrates Theorem 9.

Example 10. Let n = 7. Then X has 6 φ-orbits X_{i} (1 ≤ i ≤ 6) such that

X_{1}^{(1)} = {(4, 1, 3), (6, 1, 3), (6, 1, 5)}
X_{2}^{(1)} = {(2, 1, 5), (3, 1, 7), (4, 1, 6)}
X_{3}^{(1)} = {(2, 1, 6), (4, 1, 7), (3, 1, 5)}
X_{4}^{(1)} = {(5, 1, 3), (5, 1, 4), (6, 1, 4)}
X_{5}^{(1)} = {(2, 1, 4), (2, 1, 7), (5, 1, 7)}
X_{6}^{(1)} = {(3, 1, 6)}

Since 7 = 2 · 3 + 1 and 3^{2} + 1 ≡ 3 (mod 7), there are minimal Frobenius structure systems corresponding to X_{1}, X_{4}, X_{5}, X_{6}, but not to X_{2}, X_{3}.

7. Gorenstein tiled orders

A D-order Λ is Gorenstein if Hom_{D}(Λ, D) is projective as a right (or left) Λ-module. It is known that for an n × n tiled D-order Λ = (π^{λ_{ij}}D), Λ is Gorenstein if and only if there exists a permutation σ such that λ_{ik} + λ_{kσ(i)} = λ_{iσ(i)} for all 1 ≤ i, k ≤ n. (See [4].) Since id_{Λ}Λ = id_{Λ/πΛ}Λ/πΛ + 1 and id_{Λ}Λ = lattice-inj.dim_{Λ}Λ + 1, Λ is Gorenstein if and only if Λ/πΛ is Frobenius. (We note that this fact follows from Proposition 5 in our context.) Hence some Frobenius 𝔸-full matrix algebras are isomorphic to the factor algebras Λ/πΛ for some Gorenstein tiled D-orders Λ. However the converse is not true.

Theorem 11. For every integer n ≥ 8, there exists a Frobenius n × n full matrix algebra which has no corresponding Gorenstein tiled orders.

Proof. Let σ be a cyclic permutation (1 2 ⋯ n). Let X_{β} be a φ-orbit containing (2, 1, 5). Then, since n ≥ 8, it follows from Theorem 9 that X_{β} defines a minimal Frobenius structure system 𝔸. Suppose that there exists a Gorenstein tiled D-order Λ = (π^{λ_{ij}}D) such that Λ/πΛ is a Frobenius 𝔸-full matrix algebra. We may assume that λ_{1j} = 0 for all 1 ≤ j ≤ n by [3, Lemma 1.1]. Then we have λ_{53} = 0, which implies that λ_{54} = 0, so that (3, 1, 5) ∈ X_{β}, a contradiction. □

In the ending part of the proof, we need a technical argument not included in this report, but the following example may be helpful to see the proof.

Example 12. When n = 8, the exponent matrix (λ_{ij}) of Gorenstein tiled D-orders Λ = (π^{λ_{ij}}D) is given by

( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a & 0 & a & b & c & d & c & b \\ b & 0 & 0 & b & r & s & s & r \\ c & 0 & x & 0 & c & s & t & s \\ d & 0 & y & y & 0 & d & s & s \\ c & 0 & z & w & z & 0 & c & r \\ b & 0 & y & w & w & y & 0 & b \\ a & 0 & x & y & z & y & x & 0 \end{matrix} )

where x := a − b, y := a − c, z := a − d, w := a + b − c − d, r := b + c − a, s := c + d − a, and t := 2c + d − b − a.

8. The case of 2 ≤ n ≤ 7

For each n (2 ≤ n ≤ 7), we can verify that every Frobenius n × n 𝔸-full matrix algebra A has a Gorenstein tiled D-order Λ such that Λ/πΛ ≅ A. The following table shows that how many isomorphism classes of Frobenius 𝔸-full matrix algebras are for each n = 2, …, 7.

n no. of iso. classes
2  1
3  1
4  3
5  4
6  21
7  17

References

[1] H. Fujita, Full matrix algebras with structure systems, Colloq. Math. 98(2) (2003), 249–258.

[2] K. R. Goodearl and B. Huisgen-Zimmermann, Repetitive resolutions over classical orders and finite dimensional algebras, Algebras and Modules II (Geiranger, 1996), CMS Conf. Proc. 24, Amer. Math. Soc., Providence, RI, (1998), 205–225.

[3] V. A. Jategaonkar, Global dimensions of tiled orders over a discrete valuation ring, Trans. Amer. Math. Soc. 196 (1974), 313–330.

[4] K. W. Roggenkamp, V. V. Kirichenko, M. A. Khibina and V. N. Zhuravlev, Gorenstein tiled orders, Comm. in Algebra 29(9) (2001), 4231–4247.

Institute of Mathematics

University of Tsukuba

Tsukuba, Ibaraki 305-8571 JAPAN

E-mail: fujita@math.tsukuba.ac.jp, ysksakai@math.tsukuba.ac.jp