L. A. SIMONYAN

ON TWO RADICALS OF LIE ALGEBRAS

(Presented by Academician A. I. Mal'tsev on 17 II 1964)

Throughout, by an “algebra” we mean a Lie algebra of infinite (not necessarily finite) dimension over a field of characteristic 0. A subalgebra of an algebra \(\Omega\) is called subinvariant if it is a member of some ascending normal series in \(\Omega\). An element of the algebra \(\Omega\) is called subinvariant if the one-dimensional subalgebra generated by it is subinvariant in \(\Omega\). An algebra each element of which is subinvariant is called a \(B\)-algebra. A subalgebra of an algebra \(\Omega\) that is a member of some finite normal series in \(\Omega\) is called attainable. An element is attainable if the one-dimensional subalgebra generated by it is attainable. An algebra each element of which is attainable is called an \(SB\)-algebra. The purpose of the present note is to define the \(B\)-radical of an algebra \(\Omega\), the subalgebra in \(\Omega\) generated by all its \(B\)-ideals (a \(B\)-ideal is an ideal that is a \(B\)-algebra), and the \(SB\)-radical. In connection with this it is proved that the sum of all \(B\)-ideals of an algebra is again a \(B\)-ideal, and the sum of all \(SB\)-ideals of an algebra is again an \(SB\)-ideal. The latter is a transfer to the case of Lie algebras of the following theorem proved in a paper of R. Baer (¹).

The product of normal divisors that are nilgroups is again a nilgroup.

Let \(\Omega\) be an algebra and \(D\) a derivation of the algebra \(\Omega\). We introduce the notation: \(xD^0=x\), \(xD\) is the image of the element \(x\) under this derivation, and \(xD^n=(xD^{n-1})D\). \(D\) is called a nilderivation if for every \(x\in\Omega\) there exists an \(n\) such that \(xD^n=0\).

Lemma 1. If \(D\) is a nilderivation of the algebra \(\Omega\), then \(\exp D=\)

\[ =\sum_{i=0}^{\infty}\frac{D^i}{i!} \]

is an automorphism of the algebra \(\Omega\).

An element \(y\) of the algebra \(\Omega\) is called a nil element if \(\operatorname{ad} y\) is a nilderivation. An algebra all of whose elements are nil elements is called a nilalgebra.

Lemma 2. A nil element \(x\) belongs to the normalizer of a subalgebra \(\mathfrak M\) of the algebra \(\Omega\) if and only if \(\mathfrak M\exp(\operatorname{ad} x)=\mathfrak M\).

By \(LN(\Omega)\) we shall denote the locally nilpotent radical of the algebra \(\Omega\) (²).

Lemma 3. If \(\mathfrak M\) is an ideal of a nilalgebra \(\Omega\), then \(LN(\mathfrak M)\subseteq LN(\Omega)\).

Lemma 4. Let the algebra \(\Omega\) possess a local system of subalgebras \(\{\Omega_\alpha\}\), and in each subalgebra \(\Omega_\alpha\), moreover, let an ideal \(\mathfrak M_\alpha\) be chosen, with \(\Omega_\alpha\subset \Omega_\beta\) always implying \(\mathfrak M_\alpha\subseteq \mathfrak M_\beta\). Denote by \(\mathfrak M\) the set-theoretic sum of all \(\mathfrak M_\alpha\). Then \(\mathfrak M\) is an ideal in \(\Omega\) with local system \(\{\mathfrak M_\alpha\}\).

With the aid of Lemmas 3 and 4 one proves:

Theorem 1. Every \(B\)-algebra is locally nilpotent.

Lemma 5. If \(\mathfrak M\) is a subinvariant subalgebra of a locally finite algebra \(\Omega\), and an element \(x\) from the normalizer of the subalgebra \(\mathfrak M\) is subinvariant in \(\Omega\), then \(\{\mathfrak M,x\}\) is subinvariant in \(\Omega\).

Let

\[ \mathfrak M=\mathfrak M_0\subset \mathfrak M_1\subset\cdots\subset \mathfrak M_\alpha\subset \mathfrak M_{\alpha+1}\subset\cdots\subset \mathfrak M_\gamma=\Omega \]

an increasing normal series in \(\Omega\). We show that the subalgebras

\[ \mathfrak N_\alpha=\bigcap_{n=-\infty}^{\infty}\mathfrak M_\alpha \exp(n\operatorname{ad}x) \]

form in \(\Omega\) an increasing normal series, possibly with repetitions, whose terms are invariant with respect to \(x\). From Lemma 2 it follows that \(\mathfrak N_0=\mathfrak M\), \(\mathfrak N_\gamma=\Omega\), and \([\mathfrak N_\alpha,x]\leqslant \mathfrak N_\alpha\). It is clear that \(\mathfrak N_\alpha\subseteq \mathfrak N_{\alpha+1}\). Further, since the subalgebra \(\mathfrak N_\alpha\exp(n\operatorname{ad}x)\) is invariant in the subalgebra \(\mathfrak N_{\alpha+1}\exp(n\operatorname{ad}x)\), it follows that \(\mathfrak N_{\alpha+1}\) belongs to the normalizer of this subalgebra, and therefore \(\mathfrak N_\alpha\) is an ideal in \(\mathfrak N_{\alpha+1}\). It remains to show that if \(\beta<\gamma\) is a limit number, then

\[ \mathfrak N_\beta=\bigcup_{\alpha<\beta}\mathfrak N_\alpha. \]

Let \(y\in\mathfrak N_\beta\). This means that for every \(n\) there is a \(y_n\) in \(\mathfrak M_\beta\) such that \(y=y_n\exp(n\operatorname{ad}x)\). The latter is equivalent to saying that all \(y\exp(-n\operatorname{ad}x)\) are contained in \(\mathfrak M_\beta\). The subalgebra generated by all \(y\exp(-n\operatorname{ad}x)\) will be contained in the finite-dimensional subalgebra generated by the elements \(x\) and \(y\). Therefore there exists an \(\alpha<\beta\) such that, for all \(n\), \(y\exp(-n\operatorname{ad}x)\) is contained in \(\mathfrak M_\alpha\). Consequently, \(y\in\mathfrak N_\alpha\) and

\[ \mathfrak N_\beta=\bigcup_{\alpha<\beta}\mathfrak N_\alpha. \]

We now show that \(\{\mathfrak M,x\}\) is subinvariant in \(\Omega\). Since \(x\) is subinvariant in \(\{\mathfrak N_{\alpha+1},x\}\) and \(\mathfrak N_\alpha\) is an ideal in \(\{\mathfrak N_{\alpha+1},x\}\), there exists in \(\{\mathfrak N_{\alpha+1},x\}\) an increasing normal series whose first term is \(\{\mathfrak N_\alpha,x\}\). It is easy to verify that the terms of all such series form in \(\Omega\) an increasing normal series, possibly with repetitions. Consequently, the subalgebra \(\{\mathfrak M,x\}\) is subinvariant in \(\Omega\).

Lemma 6. Every subalgebra with a finite number of generators of a \(B\)-algebra is subinvariant in it.

Let \(\Omega\) be a \(B\)-algebra and \(\mathfrak M\) its subalgebra with a finite number of generators. By Theorem 1, \(\mathfrak M\) is a finite-dimensional nilpotent algebra. Consequently, in \(\mathfrak M\) one can construct a finite normal series with one-dimensional factors. From Lemma 5, by induction on the length of this series, it follows that \(\mathfrak M\) is subinvariant in \(\Omega\).

Lemma 7. If an ideal \(\mathfrak M\) of an algebra \(\Omega\) is a \(B\)-algebra and if there exists a subinvariant element \(x\) in \(\Omega\) such that \(\{\mathfrak M,x\}=\Omega\), then \(\Omega\) is also a \(B\)-algebra.

Let \(y\in\mathfrak M\). Since \(\mathfrak M\) is locally nilpotent (Theorem 1), \(\Omega\) is also locally nilpotent \((^{2})\). The subalgebra \(\mathfrak N\), generated by the elements \(x\) and \(y\), is a finite-dimensional nilpotent subalgebra. Consequently, \(\mathfrak M\cap\mathfrak N\) is subinvariant in \(\Omega\) (Lemma 6). Since \(\mathfrak M\cap\mathfrak N\) is an ideal in \(\mathfrak N\) and \(\{\mathfrak M\cap\mathfrak N,x\}=\mathfrak N\), by Lemma 5, \(\mathfrak N\) is subinvariant in \(\Omega\). From the fact that \(x+y\) is subinvariant in \(\mathfrak N\), it now follows that \(x+y\) is subinvariant in \(\Omega\).

Theorem 2. The sum of all \(B\)-ideals of an algebra is again a \(B\)-ideal.

Let \(\Omega\) be an algebra, let \(\mathfrak M\) and \(\mathfrak N\) be its \(B\)-ideals, and let \(x\in\mathfrak M+\mathfrak N\). The element \(x\) can be represented in the form \(x=y+z\), where \(y\in\mathfrak M\) and \(z\in\mathfrak N\). Since \(y\) is subinvariant in \(\Omega\), the subalgebra \(\{\mathfrak N,y\}\) is subinvariant in \(\Omega\). By Lemma 7, \(\{\mathfrak N,y\}\) is a \(B\)-algebra. From the fact that \(\{\mathfrak N,y\}=\{\mathfrak N,x\}\), it now follows that \(x\) is subinvariant in \(\Omega\).

By induction it is proved that the sum of any finite set of \(B\)-ideals is again a \(B\)-ideal. If \(B(\Omega)\) is the sum of all \(B\)-ideals of the algebra \(\Omega\) and \(x\in B(\Omega)\), then \(x\) is contained in the sum of some finite number of \(B\)-ideals. Consequently, \(x\) is subinvariant in \(\Omega\).

In a manner analogous to that set out above, one proves:

Theorem 3. The sum of all \(SB\)-ideals of an algebra is again an \(SB\)-ideal.

A derivation \(D\) of an algebra \(\Omega\) is called stable if in \(\Omega\) there exists an increasing normal series of subalgebras

\[ 0=\Omega_0\subset \Omega_1\subset \cdots \subset \Omega_\alpha\subset \Omega_{\alpha+1}\subset \cdots \subset \Omega_\gamma=\Omega \]

such that \(xD\in\Omega_\alpha\) for every \(x\in\Omega_{\alpha+1}\).

Lemma 8. Every nil-derivation of a commutative algebra is stable.

An algebra is called an \(RN^*\)-algebra if in it there exists an increasing normal series with commutative factors. The \(RN^*\)-radical of an algebra \(\Omega\) will be denoted by \(RN^*(\Omega)\).

With the aid of Lemma 8 it is proved.

Theorem 4. \(B(\Omega)=LN(\Omega)\cap RN^{*}(\Omega)\).

The author expresses his deep gratitude to Prof. B. I. Plotkin for posing the problems and discussing this work, and to Sh. S. Kemkhadze for providing the opportunity to become acquainted with his unpublished work on analogous questions in group theory.

Received
12 II 1964

REFERENCES

¹ R. Baer, Math. Zs., 62, 4, 402 (1955). ² B. I. Plotkin, UMN, 13, 6, 133 (1958).