Full Text
Reports of the Academy of Sciences of the USSR
1967. Volume 174, No. 1
UDC 591.44
MATHEMATICS
Yu. A. DROZD
ON REPRESENTATIONS OF CUBIC \(Z\)-RINGS
(Presented by Academician A. I. Mal'cev on 30 VI 1966)
1. In the present note we use the terminology and notation of the paper \((^1)\). By a cubic \(Z\) \((Z_p)\)-ring we shall mean an order in a semisimple algebra of dimension three over the field \(Q\) \((Q_p)\) of rational \((p\)-adic) numbers. For such rings, B. N. Delone and D. K. Faddeev \((^{2,3})\) established a connection with classes of cubic binary forms. Namely, if
\[
(a,b,c,d)=ax^3-bx^2y+cxy^2-dy^3
\]
is an arbitrary integral form with nonzero discriminant, then to it there corresponds a ring \(\Lambda\) with basis \([1,\omega_1,\omega_2]\) and multiplication table:
\[
\begin{gathered}
\omega_1\omega_2=\omega_2\omega_1=ad,\\
\omega_1^2=b\omega_1+a\omega_2-ac,\\
\omega_2^2=d\omega_1+c\omega_2-bd.
\end{gathered}
\tag{1}
\]
Conversely, in every cubic ring one can choose a basis with multiplication table (1). Isomorphic rings correspond to unimodularly equivalent forms.
If \(M\) is the maximal overring of \(\Lambda\), then the factor group \(M/\Lambda\) is a direct sum of two cyclic groups of orders \(n_1(\Lambda)\) and \(n_2(\Lambda)\), where \(n_1(\Lambda)\) divides \(n_2(\Lambda)\).
Theorem. A cubic ring \(\Lambda\) has a finite number of nonisomorphic indecomposable representations if and only if \(n_1(\Lambda)\) is square-free.
The proof is divided into several parts. All subsequent considerations are local.
2. Proposition 1. If \(M=[1,\omega_1,\omega_2]\) is a maximal cubic \(Z_p\)-ring, and \(\Gamma=[1,p^2\omega_1,p^2\omega_2]\), then \(\Gamma\) has indecomposable representations of arbitrarily large dimension.
Every maximal cubic \(Z_p\)-ring is isomorphic to a ring with form \((1,b,c,d)\). The only exception is (for \(p=2\)) the maximal order in the algebra \(Q_2\oplus Q_2\oplus Q_2\), having form \((0,1,1,0)\).
We shall assume that the basis \([1,\omega_1,\omega_2]\) corresponds to the form \((1,b,c,d)\). Let \(A\) be a \(\Gamma\)-module with basis \([1,\omega_1,p\omega_2]\), and let \(\Omega_1\) \((\Omega_2)\) be the matrix of the operator \(p^2\omega_1\) \((p^2\omega_2)\) in the module \(A\),
\[
\alpha_1=
\begin{pmatrix}
0&pc&-p^2d\\
p&0&0\\
0&0&0
\end{pmatrix};
\qquad
\alpha_2=
\begin{pmatrix}
0&-pd&p^2bd\\
0&0&0\\
1&0&0
\end{pmatrix};
\]
\[
\beta_1=
\begin{pmatrix}
-pc&0&0\\
pb&pc&p^2d\\
1&0&0
\end{pmatrix};
\qquad
\beta_2=
\begin{pmatrix}
pd&0&0\\
0&-pd&p^2d\\
0&0&0
\end{pmatrix};
\]
\(E\) is the identity matrix of size \(k \times k\), \(J\) is a degenerate Jordan block of the same size
\[ J= \begin{pmatrix} 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&\cdots&1\\ 0&0&0&\cdots&0 \end{pmatrix}; \]
\(\otimes\) is the sign of tensor multiplication of matrices. Then the correspondence
\[ p^2\omega_i \to \begin{pmatrix} \Omega_i\otimes E & \alpha_i\otimes E+\beta_i\otimes J\\ 0 & \Omega_i\otimes E \end{pmatrix} \qquad (i=1,2) \]
is a representation of the ring \(\Gamma\), whose decomposition contains an indecomposable piece of dimension at least \(2k\). An analogous result holds also for \(p=2\) for the ring \(M\) with form \((0,1,1,0)\).
- In what follows we shall use the same notation for a representation and its module. If \(A_1,\ldots,A_s\) are representation modules, then the module \(B\) will be called of type \((A_1,\ldots,A_s)\) if \(B\) can be brought to the form:
\[ B= \begin{pmatrix} A_1 & & *\\ & A_2 & \\ & & \ddots & \\ 0 & & & A_s \end{pmatrix}. \]
For rings \(\Lambda\) not contained in \(\Gamma\) (upper rings), one can prove the finiteness of the number of nonisomorphic indecomposable representations and describe these representations. Obviously, it suffices to restrict oneself to self-nonconjugate rings \((\Lambda\not\approx\Lambda^*)\), since if \(\Lambda\approx\Lambda^*\), then every exact indecomposable module is invertible and therefore coincides with \(\Lambda\) (1). The result depends on the type of the algebra \(\hat{\Lambda}=\mathfrak A\).
a) \(\mathfrak A\) is an unramified field. In the maximal order \(M\) one can choose a basis \([1,\omega_1,\omega_2]\) with form \((\varepsilon,0,c,1)\), where the polynomial \(\varepsilon t^3+ct-1\) has no roots mod \(p\), and \(\varepsilon\) is not divisible by \(p\). In \(\mathfrak A\) there is one self-nonconjugate upper order \(\Lambda=[1,p\omega_1,p\omega_2]\).
Proposition 2. \(\Lambda\) has 4 nonisomorphic indecomposable representations: \(\Lambda,\Lambda^*,M\), and one representation of type \((\Lambda,\Lambda^*)\).
b) \(\mathfrak A\) is a ramified field. In the maximal order \(M\) one can choose a basis \([1,\omega_1,\omega_2]\) with form \((\varepsilon,0,c,p)\) (\(c\) divisible by \(p\)). The exceptions (for \(p=3\)) are rings with forms \((\varepsilon,0,3,1)\) \((\varepsilon\equiv -1 \pmod 3,\ \varepsilon\not\equiv 5 \pmod 9)\). In \(\mathfrak A\) there are two self-nonconjugate upper orders: \(\Lambda=[1,p\omega_1,p\omega_2]\), \(\Sigma=[1,p^2\omega_1,p\omega_2]\).
Proposition 3. \(\Lambda\) has 6 indecomposable representations: \(\Lambda,\Lambda^*\), \(K=[1,p\omega_1,\omega_2]\), \(M\), one representation of type \((\Lambda,\Lambda^*)\), and one of type \((\Lambda,M)\).
The ring \(\Sigma\) has 18 indecomposable representations: those listed above, and also \(\Sigma,\Sigma^*\), one representation of type \((\Sigma,K)\), two of type \((\Sigma,M)\), one of type \((\Sigma,\Sigma^*)\), one of type \((\Sigma,\Lambda^*)\), one of type \((\Sigma\oplus\Sigma,M)\), one of type \((\Sigma,M\oplus\Lambda^*)\), one of type \((\Sigma\oplus\Lambda,\Lambda^*)\), one of type \((\Sigma\otimes\Sigma,M\oplus\Lambda^*)\), and one of type \((\Sigma\otimes\Lambda,M\oplus\Lambda^*)\).
An analogous result holds (for \(p=3\)) also for subrings of the maximal ring with form \((\varepsilon,0,3,1)\).
c) \(\mathfrak A=Q_p\oplus Q_p\oplus Q_p\). Let \(e_1,e_2,e_3\) be a system of minimal idempotents in \(\mathfrak A\). Then every upper self-nonconjugate ring is isomorphic to one of the following:
\[ \Lambda_k=[1,pe_2,p^k e_3]\qquad (k>0), \]
\[ L_{k,\theta}=[1,pe_2+\theta p^{k-1}e_3,p^k e_3]\qquad (k>1,\ \theta\text{ is a unit in }Z_p). \]
Irreducible—
decomposable \(\Lambda\)-modules are \(A_i\) \((i=1,2,3)\), where \(A_i\) is a free \(Z_p\)-module of rank 1 with generators \(a_i\), \(a_i e_j=\delta_{ij}a_i\).
Proposition 4. The ring \(L_{k,\theta}\) has \(4k+1\) indecomposable representations: \(A_1,A_2,A_3\), one representation \(U\) of type \((A_1,A_2)\), one \(V\) of type \((A_2,A_3)\), \(k-1\) of type \((A_1,A_3)\), \(k\) of type \((U,A_3)\), \(k-1\) of type \((A_1,V)\), \(k-2\) of type \((U,V)\). The ring \(\Lambda_k\) has \(4(k+1)\) indecomposable representations (one more representation of each of the types \((A_1,A_3)\), \((A_1,V)\), \((U,V)\) is added).
c) \(\mathfrak A=Q_p\oplus \mathfrak A_1\) \((\mathfrak A_1=Q_p(\sqrt{\alpha}),\ \alpha\in Z_p\) and is not divisible by \(p^2\)). Let \(e_1,e_2\) be orthogonal idempotents in \(\mathfrak A\), \(\dim_{Q_p}(e_1\mathfrak A)=1\). A maximal order in \(\mathfrak A_1\) has basis \([e_2,\omega]\), \(\omega^2=\alpha e_2\) (except for the case \(p=2,\ \alpha=-3\)). The upper nonconjugate orders in \(\mathfrak A\) have the form
\[ \Lambda_k=[1,p^k\omega,pe_1]\qquad (k>0), \]
\[
L_{k,\theta}=[1,p^k\omega+\theta pe_1,\ p^2e_1]\qquad (k>0,\ \theta\text{ a unit in }Z_p),\quad K=[1,p\omega,p^2e_1]
\]
(only when \(\alpha\) is divisible by \(p\)).
The rings \(\Lambda_k\) and \(L_{k,\theta}\) have one irreducible representation \(A\), belonging to the first component, and \(k+1\) representations belonging to the second component: \(U_0,U_1,\ldots,U_k\).
Proposition 5. If \(\alpha\) is not divisible by \(p\), then the ring \(\Lambda_k\) has \(4k+2\) indecomposable representations: \(A,U_i\) \((i=0,\ldots,k)\), one of type \((U_0,A)\), two each of types \((U_i,A)\) \((i=1,\ldots,k-1)\), one of type \((U_k,A)\), and one each of types \((U_i,A\oplus A)\) \((i=0,\ldots,k-1)\). The ring \(L_{k,\theta}\) has \(4k+3\) indecomposable representations (to the \(k\) listed above one more of type \((U_k,A)\) is added).
Proposition \(5'\). If \(\alpha\) is divisible by \(p\), then the ring \(\Lambda_k\) \((L_{k,\theta})\) has \(4k+3\) (respectively, \(4(k+1)\)) indecomposable representations (to the \(k\) listed above one more of type \((U_0,A)\) is added). The ring \(K\) has 12 indecomposable representations: \(A,U_0,U_1\), three of type \((U_0,A)\), two of type \((U_1,A)\), two of type \((U_0,A\oplus A)\), one of type \((U_0\oplus U_1,A)\), and one of type \((U_0\oplus U_1,A\oplus A)\).
An analogous result also holds for \(p=2,\ \mathfrak A_1=Q_p(\sqrt{-3})\). From Propositions 3–5, \(5'\) it follows that all upper cubic \(Z_p\)-rings have a finite number of nonisomorphic indecomposable representations. The globalization of this result gives the main theorem formulated in § 1.
The author expresses deep gratitude to D. K. Faddeev and A. V. Roiter, as well as to the participants of the seminar on representation theory at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR.
Kyiv State University
named after T. G. Shevchenko
Received
29 VI 1966
REFERENCES
- D. K. Faddeev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 80, 145 (1965).
- B. N. Delone, D. K. Faddeev, ibid., 11 (1940).
- D. K. Faddeev, ibid., 80, 183 (1965).