MATHEMATICS

A. I. KOSTRIKIN

ON THE HEIGHT OF SIMPLE LIE ALGEBRAS

(Presented by Academician P. S. Novikov, December 10, 1964)

1. Let \(\Omega\) be a simple finite-dimensional Lie algebra over an algebraically closed field \(\Phi\); let \(x\) be a nonzero element of \(\Omega\). Define on \(\Omega\) an increasing \(x\)-filtration \(\bigl(\Omega^{\langle i\rangle}\bigr)\), where \(\Omega^{\langle i\rangle}\) is the subspace spanned by all possible products of the form \([\cdots [[xy_1]y_2]\cdots y_k]\), \(y_1,\ldots,y_k\in\Omega\), \(0\leq k\leq i\); \(\Omega^{\langle -1\rangle}=0\). Obviously, \(\Omega^{\langle 0\rangle}=\{x\}\), \(\Omega^{\langle i\rangle}\subseteq \Omega^{\langle i+1\rangle}\), and \([\Omega^{\langle i\rangle},\Omega^{\langle j\rangle}]\subseteq \Omega^{\langle \min(i,j)+1\rangle}\subseteq \Omega^{\langle i+j\rangle}\).

The minimal index \(n\) for which \(\Omega^{\langle n\rangle}=\Omega\) will be called the height \(\Delta(\Omega,x)\) of the algebra \(\Omega\) relative to the element \(x\). The existence of such an index is ensured by the finite-dimensionality and simplicity of the algebra \(\Omega\). In other words, \(\Delta(\Omega,x)+1\) is the number of nonzero homogeneous components \(\overline{\Omega}^{\langle i\rangle}\) of the graded \(\Phi\)-module associated in the usual way with the algebra \(\Omega\) with the \(x\)-filtration. The multiplicative structure of this \(\Phi\)-module, incidentally, is trivial. The integer
\[ \Delta(\Omega)=\max_{x\in\Omega}\Delta(\Omega,x) \]
will be called the height of the simple algebra \(\Omega\).

The concept of height eliminates the need for a lengthy verbal description of various situations arising in degenerate simple \(p\)-Lie algebras. Let us first note the following, directly verifiable

Proposition 1. There exists an absolute constant \(\rho\) such that \(\Delta(\Omega)\leq \rho\) for every simple Lie algebra \(\Omega\) over a field \(\Phi\) of characteristic zero.

The same is true for algebras classical in the sense of Seligman–Mills \((^1)\). On the other hand, the Witt algebra \(\mathfrak W_1\) of dimension \(p\) has height \(p-1\) (i.e., it may be arbitrarily large), while the height of the Witt–Jacobson algebra \(\mathfrak W_n\) \((^2)\) grows without bound together with \(n\) for fixed \(p\). Thus the degeneracy of the simple algebra \(\Omega\) has a decisive effect on the behavior of its height.

2. From the facts known about Lie algebras with strong degeneracy (see \((^3,^4)\)) it follows directly that their height \(\Delta\geq p-2\). More precisely, \(\Delta(\Omega,c)\geq p-2\) for every element \(c\in\Omega^{\left(\frac{p-3}{2}\right)}\), defined by the condition
\[ cX^ic=0,\qquad i=0,1,2,\ldots,p-4. \tag{1} \]
The notation is taken from the paper \((^3)\), where, incidentally, it was noted (in somewhat different terms) that if \(\Delta(\Omega,c)=p-2\), then the dimension of the algebra \(\Omega\) must be sufficiently large. In fact, the following is true.

Theorem 1. In a simple Lie algebra \(\Omega\) with strong degeneracy, the inequality
\[ \Delta(\Omega,c)\geq p-1 \]
holds for every nonzero element \(c\in\Omega^{\left(\frac{p-3}{2}\right)}\).

It is convenient to prove this by contradiction. Denote by \(\mathfrak M\) the subspace in \(\Omega\) spanned by all elements of the form \([cx^{p-2}c]\), \(x\in\Omega\). Since, by assumption, \(\Delta(\Omega,c)=p-2\), it follows that \([ca]\in\mathfrak M\) for every ele-

element \(a \in \mathfrak L\). In this case
\[ [cx^{p-2}c]=[[cx]\cdot x\ldots xc]\in [\mathfrak M x^{p-3}c]. \]
Using this observation a sufficient number of times, we express the product \([ca]\) as a linear combination of elements
\[ [ca_0^{p-2}ca_1^{p-3}ca_2^{p-3}c\ldots ca_m^{p-3}c], \tag{2} \]
where \(m\) is a prescribed positive integer. Lemma 1, formulated below (which is also of independent interest), shows that elements of type (2) with large indices \(m\) are equal to zero. Consequently, \([ca]=0\) or \(c\) is an element of the center of the algebra \(\mathfrak L\), which is impossible.

Lemma 1. The subspace \(\mathfrak M\) is invariant under the endomorphisms \(X^{p-3}C\) of the vector space \(\mathfrak L\). The associative algebra generated by all endomorphisms \(\sigma_x=X^{p-3}C|_{\mathfrak M}\), \(x\in\mathfrak L\), is nilpotent.

The first assertion of the lemma is a simple consequence of identity (*) in paper (5). From the identities given in § 1 of the same paper there follow the following three properties of the endomorphisms \(\sigma_x\): 1) \(\sigma_x^2=0\); 2) \(\sigma_x\sigma_y\sigma_x=0\); 3) the vector space \(\mathfrak S\), spanned by all endomorphisms \(\sigma_x\), is closed under the operation \(\circ\): \(\sigma_x\circ\sigma_y=\sigma_x\sigma_y-\sigma_y\sigma_x\). Thus \(\mathfrak S\) is a Lie algebra of linear transformations acting on \(\mathfrak M\). In view of properties 1) and 2), the elements \(\sigma_x\) have nilpotency index two in \(\mathfrak S\): \(((\mathfrak S\circ\sigma_x)\circ\sigma_x)=0\). All such elements form a weakly closed system (see (6), Ch. II, § 1, 2). The enveloping algebra of this system coincides with the enveloping algebra \(\overline{\mathfrak S}\) of the algebra \(\mathfrak S\). Its nilpotency is ensured by Jacobson’s theorem.

3. Proposition 2. In a simple Lie algebra \(\mathfrak L\) with a strong degeneration there exists a nonzero element \(c_0\in \mathfrak L^{\left(\frac{p-3}{2}\right)}\) whose associated endomorphism \(C_0\) satisfies the identity
\[ C_0X^{p-3}C_0Y^{p-3}C_0=0,\qquad x,y\in\mathfrak L. \tag{3} \]

This assertion, in view of Lemma 1, is proved almost in the same way as Lemma 3.2 in paper (5). The property of the element \(c_0\) expressed by identity (3) is very important for the description of simple \(p\)-Lie algebras with relative height \(\Delta(\mathfrak L,c_0)=p-1\). Besides \(\mathfrak W_1\), there are less trivial examples of simple \(p\)-Lie algebras in which all these conditions are satisfied (one of the most interesting examples is the Block algebra (7)). We note further that identity (3) is in a certain sense optimal. Indeed, let \(c\in \mathfrak L^{\left(\frac{p-3}{2}\right)}\) and \(\Delta(\mathfrak L,c)=p-1\). The identity \(CX^{p-3}C=0\), for obvious reasons, cannot hold. On the other hand, in many algebras the subspace \(\mathfrak M\) associated with the element \(c\) contains this element, and it is quite natural to consider the case when \(\mathfrak M=\{c\}\). In other words, let
\[ [cx^{p-2}c]=\Gamma(x)\cdot c,\qquad x\in\mathfrak L,\quad \Gamma(x)\in\Phi, \tag{4} \]
where \(\Gamma(x)\ne0\) for at least one element \(a\in\mathfrak L\).

An element \(c\in \mathfrak L^{\left(\frac{p-3}{2}\right)}\) satisfying condition (4) will be called stable. It is easy to see that the requirement of stability is less restrictive than the identity \(CX^{p-3}C=0\), but more stringent than (3).

Theorem 2. A simple \(p\)-Lie algebra \(\mathfrak L\) with a strong degeneration, distinct from the Witt algebra \(\mathfrak W_1\), cannot have a stable element \(c\) with respect to which \(\Delta(\mathfrak L,c)=p-1\).

The proof, based on the additive properties of the multiplier \(\Gamma(x)\), is rather long. As an intermediate stage of the argument it is established that a simple \(p\)-Lie algebra \(\mathfrak T\) with a stable element \(c\), \(\Delta(\mathfrak L,c)=p-1\), must have a basis of the form \(c;\ [ce_1],\ldots,[ce_r];\) some-

the second part of the products \([ce_i e_j]\), \([ce_1^i]\), ..., \([ce_r^i]\); \(r \geqslant 1\); \(i=3,\ldots,p-1\). Here \(e_k=[ce_k^{p-2}]\), \([ce_k^{p-2}c]=2c\), \(1\leqslant k\leqslant r\).

Without dwelling on the other details, let us show how the conclusion of the theorem already follows from this, if one additionally assumes that on \(\mathfrak L\) there is defined some invariant bilinear form \(f(x,y)\) (not connected with any representation). From the identities (1), which define the element \(c\), it follows that \(f(c,x)\ne 0 \Rightarrow f(c,[ce_k^{p-2}])\ne 0\) or \(f(c,e_k)\ne 0\) for some \(k\). But
\[ f(c,[ce_k^{p-2}])=\frac12 f([[ce_k^{p-2}]c],[ce_k^{p-2}])=0 \]
by the invariance of \(f\). Further,
\[ f(c,e_k)=\frac12 f([ce_k^{p-2}c],e_k) =-\frac12 f(c,[[ce_k^{p-2})e_k])=-\frac12 f(c,e_k), \]
i.e. also \(f(c,e_k)=0\). The theorem in this case is proved. It is worth recording the lemma used in the construction, indicated above, of a basis of the algebra \(\mathfrak L\).

Lemma 2. Let \(\mathfrak L\) be an arbitrary \(p\)-algebra of Lie, \(c\in \mathfrak L^{(p-3)/2}\), and let \([ca^{p-2}c]=c\) for some element \(a\in\mathfrak L\). Put \(e=[ca^{p-1}]\). Then \([ce^{p-2}c]=2c\). If, in addition, \(c\) is a stable element, then \([ce^{p-1}]=e\).

Proof. First,
\[ [cx^{p-1}c]=-\frac12[cx^{p-2}cx] \]
for every \(x\in\mathfrak L\). Therefore
\[ [ce]=-[ca^{p-1}c]=\frac12[ca^{p-2}ca]=\frac12[ca]. \]
Suppose that the equality \([ce^k]=(1/2)^k[ca^k]\) has already been proved. Applying the known relations (see (6), p. 49, formula (6)), we find:
\[ [ce^{k+1}]=(1/2)^k[ca^k e]=(1/2)^k\sum_{i=0}^k(-1)^{k-i}\binom{k}{i}[c[ea^{k-i}]a^i]. \]
But \([ea]=[ca^p]\). Since \(\mathfrak L\) is a \(p\)-algebra, \([c[ea^{k-i}]]=-[ea^{k-i}c]=-[ca^p\cdot a^{k-i-1}c]=0\) for \(k-i<p-2\) (see (1)). If \(k<p-2\), then also \(k-i<p-2\), and then
\[ [ce^{k+1}]=(1/2)^k[ce a^k]=(1/2)^k(1/2)[ca\cdot a^k]=(1/2)^{k+1}[ca^{k+1}]. \]
In particular, \([ce^{p-2}]=2[ca^{p-2}]\) and \([ce^{p-2}c]=2c\). For \(k=p-2\) one obtains a somewhat different expression:
\[ [ce^{p-1}]=(1/2)^{p-2}[ca^{p-2}e] =(1/2)^{p-2}[ce a^{p-2}]-(1/2)^{p-2}[c[ea^{p-2}]] =[ca^{p-1}]+2[ca^p\cdot a^{p-3}c]. \]
In view of the fact that now \(c\) is a stable element, we have
\[ e=[ce^{p-1}]+\varepsilon\cdot c. \tag{5} \]
Using (5) and the expression for \(\Lambda(a,b)\) in the formula
\[ (a+b)^p=a^p+b^p+\Lambda(a,b), \]
we find
\[ e^p=([ce^{p-1}]+\varepsilon c)^p=[ce^{p-1}]^p+\varepsilon[c[ce^{p-1}]^{p-1}] +\varepsilon^2[c[ce^{p-1}]^{p-2}c] \]
\[ =[ce^{p-1}]^p+\varepsilon[c(e-\varepsilon c)^{p-1}] +\varepsilon^2[c(e-\varepsilon c)^{p-2}c] =[ce^{p-1}]^p+\varepsilon([ce^{p-1}]-\varepsilon[ce^{p-2}c])+\varepsilon^2[ce^{p-2}c], \]
or
\[ e^p=[ce^{p-1}]^p+\varepsilon[ce^{p-1}]. \]
Hence, also from relation (5),
\[ (e-\varepsilon c)^p=[ce^{p-1}]^p=e^p-\varepsilon[ce^{p-1}]. \]
But, on the other hand, direct computations give
\[ (e-\varepsilon c)^p=e^p-\varepsilon[ce^{p-1}]+\varepsilon^2[ce^{p-2}c]. \]
Comparing the expressions obtained, we find \(\varepsilon^2[ce^{p-2}c]=0\), i.e. \(\varepsilon=0\). Thus relation (5) coincides with what is asserted in the lemma.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
2 XII 1964

CITED LITERATURE

1. W. H. Mills, G. B. Seligman, J. Math., Mech., 6, 519 (1957).
2. N. Jacobson, Duke Math. J., 10, 107 (1943).
3. A. I. Kostrikin, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 64, 79 (1961).
4. A. I. Kostrikin, DAN, 150, No. 2 (1963).
5. A. I. Kostrikin, Izv. AN SSSR, ser. matem., 23, 3 (1959).
6. N. Jacobson, Lie Algebras, Moscow, 1964.
7. R. Block, Trans. Am. Math. Soc., 89, 421 (1958).