MATHEMATICS

B. Yu. WEISFEILER

CLASSIFICATION OF SEMISIMPLE LIE ALGEBRAS OVER A \(p\)-ADIC FIELD

(Presented by Academician S. L. Sobolev, 11 IV 1964)

The problem of classifying simple Lie algebras over a field of characteristic 0 has recently been considered by many authors \((^{1-3})\) and others. A. Weil obtained results that make it possible to give a classification of classical Lie groups not containing components of type \(D_4\), over an arbitrary field \(k\) of characteristic 0. Analogues of Weil’s theorems were proved for groups of types \(G_2\) and \(F_4\). In the present note a classification is given of all simple Lie algebras over a \(p\)-adic field \(k\). The results are obtained by general methods, making use mainly of root techniques.

By a Lie algebra we shall mean a Lie algebra over the universal domain. It is defined over a field \(k\) if the commutation operation is defined over the field \(k\); in this case the set \(\mathfrak g_k\) of points rational over \(k\) of the algebra \(\mathfrak g\) is a Lie algebra over \(k\) in the classical sense. An algebra defined over a field \(k\) is said to be \(k\)-simple if it contains no ideals defined over \(k\). An algebra is called absolutely simple if it is \(L\)-simple for every extension \(L\) of its field of definition \(k\).

Let \(\mathfrak g\) be a semisimple Lie algebra defined over a field \(k\). It decomposes over \(k\) into a direct sum of \(k\)-simple algebras. Therefore, for our purposes it suffices to consider the case where \(\mathfrak g\) is a \(k\)-simple Lie algebra. Then there exists a finite separable extension \(K\) of the field \(k\) and an absolutely simple algebra \(\mathfrak g'\), defined over \(K\), such that \(\mathfrak g\) is obtained from \(\mathfrak g'\) by restricting the base field from \(K\) to \(k\): \(\mathfrak g=R_{K/k}\mathfrak g'\) \((^{4})\). Thus the problem of classifying semisimple Lie algebras defined over the field \(k\) is reduced to the problem of classifying absolutely simple Lie algebras defined over \(k\) and its finite extensions. In the classification, Satake’s results \((^{3})\) were used extensively.

Let \(k\) be a \(p\)-adic field (a finite extension of the field of \(p\)-adic numbers), and let \(\mathfrak g\) be an absolutely simple Lie algebra defined over \(k\).

Theorem 1. There exists a maximal subalgebra in \(\mathfrak g\), consisting of nilpotent elements and defined over an unramified extension \(\mathfrak K\) of the field \(k\).

Using Theorem 1 and some results of \((^{3})\), one can find a normal splitting field \(K\) of the algebra \(\mathfrak g\), containing the field \(\mathfrak K\), such that: a) if \(\mathfrak g\) is an algebra of type \(B_n, C_n, E_7, E_8, G_2\), or \(F_4\), then \(K=\mathfrak K\); b) if \(\mathfrak g\) is an algebra of type \(A_n, D_n\) \((n\ne4)\), or \(E_6\), then the Galois group \(\Gamma(K/k)\) is abelian and the group \(\Gamma(\mathfrak K/k)\) has index 1 or 2 in \(\Gamma(K/k)\); c) if \(\mathfrak g\) is of type \(D_4\), then the group \(\Gamma(K/k)\) is either trivial, or isomorphic to one of the cyclic groups \(Z_2, Z_3\), or isomorphic to the symmetric group on three letters \(S_3\).

The formulated results make it possible to give a classification of absolutely simple Lie algebras defined over the field \(k\). By analogy with the real case, we shall call Lie algebras defined over \(k\) \(k\)-forms.

* After the paper had already been written, the author learned that T. A. Springer had proved a more general theorem \((^{5})\).

Before turning to the description of \(k\)-forms, let us introduce some notation and definitions.

If \(\Sigma\) is the root system of a semisimple Lie algebra, and \(\Gamma\) is a subgroup of the group \(\operatorname{Aut}\Sigma\), then a \(\Gamma\)-order is such a linear order that, if \(\alpha\in\Sigma\), \(\alpha>0\), and \(\sum_{\sigma\in\Gamma}\sigma\alpha\ne0\), then \(\sigma\alpha>0\) for all \(\sigma\in\Gamma\). The fundamental system of roots with respect to a \(\Gamma\)-order is called a \(\Gamma\)-fundamental system of roots \((^3)\).

List of \(k\)-forms of absolutely simple Lie algebras (the list does not contain \(k\)-forms decomposable over \(k\))

If the algebra \(\mathfrak g_k\) contains no nilpotent elements, then the algebra \(\mathfrak g\) is called \(k\)-compact \((^3)\). According to Satake’s theorem \((^3)\), a \(k\)-form of a semisimple algebra \(\mathfrak g\) is completely determined by specifying its normal splitting field \(L\), a \(\Gamma(L/k)\)-fundamental system of roots \(\Delta\), the action of the group \(\Gamma(L/k)\) on \(\Delta\), and the \(k\)-compact \(k\)-form corresponding to the \(\Gamma(L/k)\)-subsystem

\[ \Delta_0=\{\alpha\in\Delta:\sum_{\sigma\in\Gamma}\sigma\alpha=0\}. \]

If \(\mathfrak A\) is an associative algebra defined over \(k\), \(\mathfrak B\) is its commutator algebra, and \(\mathfrak Z\) is the center of the algebra \(\mathfrak B\), then by \(\mathfrak g(\mathfrak A)\) we denote the Lie algebra \(\mathfrak B/\mathfrak Z\). If \(\mathfrak A_k\) is a division algebra, then \(\mathfrak g(\mathfrak A)\) is a \(k\)-compact algebra. From our results it follows that the converse assertion is true:

Theorem 2. Let \(\mathfrak g\) be a \(k\)-simple \(k\)-compact Lie algebra. There exists a finite extension \(L\) of the field \(k\) and a division algebra \(\mathfrak D\) over \(L\) such that

\[ \mathfrak g=R_{L/k}\mathfrak g(\mathfrak D). \]

It is known that the isomorphism classes of division algebras of order \(n\) over \(k\) can be numbered by integers \(m\):

\[ 0<m<n,\qquad (m,n)=1. \]

We shall denote the division algebra over \(k\) specified by the pair \((n;m)\) by \(\mathfrak D_{n,m}\). Put further \(k_{n,m}=\mathfrak g(\mathfrak D_{n,m})\); then we have \(k_{n,m}=k_{n,n-m}\).

For the description of \(k\)-forms we use \(\Gamma(K/k)\)-fundamental systems of roots (\(K\) is the splitting field constructed above) (see the scheme). Roots belonging to \(\Delta_0\) are denoted by black circles. The schemes of \(k\)-simple \(k\)-compact \(k\)-forms are singled out, and next to them it is indicated which \(k\)-compact \(k\)-form they correspond to; to one isolated black circle there always corresponds the algebra \(k_{2,1}\). If in the group \(\Gamma(K/k)\) there are elements preserving

system of simple roots, then the corresponding substitution is described by arrows.

If \(\mathfrak P\) (respectively \(\mathfrak p\)) is a prime ideal of the field \(K\) (respectively \(k\)), we put \(\mathfrak p=\mathfrak P^e\), \(N_{K/k}(\mathfrak P)=\mathfrak p^f\). For all \(k\)-forms the splitting field \(K\) will be specified by giving the numbers \(e\) and \(f\) and the group \(\Gamma(K/k)\). In this case, distinct fields with the same invariants \(e\) and \(f\) and group \(\Gamma(K/k)\) correspond to distinct \(k\)-forms, and to every field with the indicated characteristics there corresponds a \(k\)-form.

The methods applied to the study of Lie algebras over a \(p\)-adic field, together with known results on associative algebras, make it possible to obtain the following propositions (in which we call a Lie algebra defined over \(k\) \(k\)-quasinormal if it contains a maximal subalgebra defined over \(k\) and consisting of nilpotent elements).

Theorem 3. Let \(k\) be an algebraic number field; \(V\) the set of its inequivalent valuations; \(k_v\) the completion of \(k\) with respect to \(v\in V\); \(\mathfrak g\) a Lie algebra defined over \(k\). Then: 1) for almost all \(v\in V\) the algebra \(\mathfrak g\) is \(k\)-quasinormal; 2) if for all \(v\in V\) the algebra \(\mathfrak g\) is \(k_v\)-quasinormal (respectively split over \(k_v\)), then it is \(k\)-quasinormal (respectively split over \(k\)); 3) there exists a cyclic extension \(K\) of the field \(k\) such that the algebra \(\mathfrak g\) is \(K\)-quasinormal.

I take this opportunity to express my deep gratitude to E. B. Vinberg for the attention he has given to the present work.

Moscow
Electric-Lamp Factory

Received
1 IV 1964

CITED LITERATURE

\(^{1}\) A. Weil, J. Indian Math. Soc., 24, 289 (1960).
\(^{2}\) J. Tits, Coll. sur la théorie des groupes algébriques, Bruxelles, 1962, p. 137.
\(^{3}\) I. Satake, J. Math. Soc. Japan, 15, No. 2, 210 (1963).
\(^{4}\) A. Weil, Adeles and Algebraic Groups, Princeton, 1961.
\(^{5}\) T. A. Springer, Coll. sur la théorie des groupes algébriques, Bruxelles, 1962, p. 129.