MATHEMATICS

V. N. LATYSHEV

ON ALGEBRAS WITH IDENTICAL RELATIONS

(Presented by Academician P. S. Aleksandrov, 5 V 1962)

In § 1 of the present note we shall assume known the concepts and results of the work of Specht \((^1)\). Let \(T\) be a \(T\)-ideal of the free algebra with identity over a field \(K\) of characteristic zero, and let \(\Gamma_n\) be the module of all proper \(n\)-linear forms over \(K\). Denote \(T_n=\Gamma_n\cap T\). The modules \(T_n\) completely determine the \(T\)-ideal \(T\) \((^1)\). In each module \(T_n\), as in a linear space over \(K\), one can choose a basis; the union of the bases of the modules \(T_n\) will be called a basis of the \(T\)-ideal \(T\). Denote by \(T^2\) the \(T\)-ideal generated by the commutator \([u,v]=uv-vu\); then \(T_n^2=\Gamma_n\cap T^2=\Gamma_n\). It is clear that the union of the Specht bases \((^1)\) is a basis of the \(T\)-ideal \(T^2\). Denote by \(T^3\) the \(T\)-ideal generated by the commutator of the third degree \([u,v,w]=[[u,v],w]\). In § 1 a certain basis of the \(T\)-ideal \(T^3\) is indicated. The result obtained makes it possible, “from general considerations,” to give a negative solution of Kaplansky’s hypothesis \((^2)\) on the possibility of embedding a \(PI\)-algebra in a full matrix algebra over a commutative algebra. Earlier this problem was solved by Cohn in an unpublished work (see \((^3)\)). In § 2 special Lie algebras with identical relations are studied (\(SPI\)-Lie algebras), i.e. Lie algebras admitting an exact embedding in an associative \(PI\)-algebra. A positive solution is given in the class of \(SPI\)-Lie algebras to a problem analogous to A. G. Kurosh’s problem in associative algebras.

1. Introduce the notation: \(G_n\) is the submodule in \(\Gamma_n\) generated by products of commutators in the variables \(x_i\), in which, as a factor, at least once there necessarily occurs a commutator in \(x_i\) of degree higher than two, \(u_1=[x_1,x_2][x_3,x_4]\cdots[x_{2k-1},x_{2k}]\). It is clear that \(G_n\subset T^3\). Let

\[ w=[x_{i_1},x_{i_2}][x_{i_3},x_{i_4}]\cdots[x_{i_{2k-1}},x_{i_{2k}}]\in\Gamma_{2k}. \]

We shall call the composition of the element \(w\) the set of unordered pairs of indices \((i_l,i_{l+1})\) such that the commutator \([x_{i_l},x_{i_{l+1}}]\) occurs in the writing of \(w\). There is one and only one element of the Specht basis \((^1)\) among the elements of the form \(w\) of a given composition. Number all compositions in an arbitrary order, taking the composition of the element \(u_1\) as the first. From the set of elements of the form \(w\) of composition with number \(t\), choose the elements of the Specht basis

\[ u_t=[x_{\xi_1},x_{\xi_2}][x_{\xi_3},x_{\xi_4}]\cdots[x_{\xi_{2k-1}},x_{\xi_{2k}}] \]

and form the element \(z_t=u_1-(-1)^{\varepsilon_t}u_t,\ t>1\), where \(\varepsilon_t\) is the number of inversions in the permutation \(\xi_1\ldots\xi_{2k}\) of the numbers \(1,\ldots,2k\). Since the \(u_t\) are elements of the Specht basis, the \(z_t\) are linearly independent.

Theorem 1. The elements of the Specht basis in \(\Gamma_{2k+1}\) form a basis of the module \(T^3_{2k+1}\). In the module \(T^3_{2k}\), a basis is formed by the elements \(z_t\) and the elements of the Specht basis contained in \(G_{2k}\).

Proof. The first assertion is trivial, since

\[ T^3_{2k+1}=\Gamma_{2k+1}\cap T^3=\Gamma_{2k+1}=G_{2k+1}\subset T^3. \]

For the proof of the second assertion we introduce notation: \(F_n\) is the module of all \(n\)-linear forms, \(n=2k\); \(Q_n\) is the submodule in \(F_n\) generated by elements of the form \(p\tau q\), where \(p\) and \(q\) are words in \(x_i\) and \(1\), and \(\tau\) is an element of one of the following forms:

\[ \begin{gathered} [x_i,x_j][x_k,x_l]+[x_i,x_k][x_j,x_l],\\ [x_i,x_j][x_k,x_l]+[x_l,x_j][x_k,x_i],\\ [x_i,x_j][x_k,x_l]+[x_k,x_j][x_i,x_l],\\ [x_i,x_j][x_k,x_l]+[x_i,x_l][x_k,x_j]. \end{gathered} \tag{1} \]

Put \(M_n=\Gamma_n\cap Q_n,\ \mathfrak{M}_n=M_n+G_n\).

Lemma 1. The equality \(\mathfrak M_n=T_n^3\) holds.

Proof. We have:
\([uv,r,s]=[[u,r]v+u[v,r],s]=[u,r,s]v+u[v,r,s]+[u,r][v,s]+[u,s][v,r]\), whence
\([u,r][v,s]+[u,s][v,r]\equiv 0\pmod {T^3}\), and elements of the form (1) lie in \(T^3\); but this means that \(M_n\subset \Gamma_n\cap T^3=T_n^3\), and since \(G_n\subset T_n^3\), we have \(\mathfrak M_n=M_n+G_n\subset T_n^3\).

Let us prove the reverse inclusion. Let \(a\in F_n\cap T^3\); then there is a representation of the form
\(a=\sum \alpha p[u,v,r]q\), where \(\alpha\in K\); \(p,q,u,v,r\) are words in the \(x_i\) and \(1\). For definiteness put
\(r=x_{\alpha_1}x_{\alpha_2}\cdots x_{\alpha_k}\), \(v=x_{\beta_1}x_{\beta_2}\cdots x_{\beta_l}\),
\(u=x_{\gamma_1}x_{\gamma_2}\cdots x_{\gamma_m}\). We shall have

\[ p[u,v,r]q=\sum_{\nu=1}^{k}px_{\alpha_1}\cdots x_{\alpha_{\nu-1}}[u,v,x_{\alpha_\nu}]x_{\alpha_{\nu+1}}\cdots x_{\alpha_k}q= \]

\[ =\sum_{\nu=1}^{k}px_{\alpha_1}\cdots x_{\alpha_{\nu-1}}[u,x_{\alpha_\nu},v]x_{\alpha_{\nu+1}}\cdots x_{\alpha_k}q- \]

\[ -\sum_{\nu=1}^{k}px_{\alpha_1}\cdots x_{\alpha_{\nu-1}}[v,x_{\alpha_\nu},u]x_{\alpha_{\nu+1}}\cdots x_{\alpha_k}q= \]

\[ =\sum_{\nu=1}^{k}\sum_{\mu=1}^{l} px_{\alpha_1}\cdots x_{\alpha_{\nu-1}}x_{\beta_1}\cdots x_{\beta_{\mu-1}} [u,x_{\alpha_\nu},x_{\beta_\mu}] x_{\beta_{\mu+1}}\cdots x_{\beta_l}x_{\alpha_{\nu+1}}\cdots x_{\alpha_k}q- \]

\[ -\sum_{\nu=1}^{k}\sum_{\pi=1}^{m} px_{\alpha_1}\cdots x_{\alpha_{\nu-1}}x_{\gamma_1}\cdots x_{\gamma_{\pi-1}} [v,x_{\alpha_\nu},x_{\gamma_\pi}] x_{\gamma_{\pi+1}}\cdots x_{\gamma_m}x_{\alpha_{\nu+1}}\cdots x_{\alpha_k}q. \]

Denote by \(G_n^*\) the submodule in \(F_n\) generated by elements of the form
\(p[x_i,x_j,x_k]q\); clearly, \(G_n^*\cap \Gamma_n=G_n\). Since the commutator of generators in the elements \(F_n\) is commutable with any generator modulo \(G_n^*\), we easily obtain

\[ p[v,x_i,x_j]q=\sum\bigl([x_{\alpha_s},x_i][x_{\alpha_t},x_j]+[x_{\alpha_s},x_j][x_{\alpha_t},x_i]\bigr)R \pmod {G_n^*}. \tag{3} \]

Let \(a\in \Gamma_n\cap T^3=T_n^3\); from (2) and (3) it follows that \(a\in Q_n+G_n^*\), i.e.
\(a=b+c,\ b\in Q_n,\ c\in G_n^*\). To the right-hand side of the equality \(a=b+c\) we apply the Specht algorithm: “moving” the variables \(x_1,x_2,\ldots\) to the right with successive formation of commutators (1); we shall have:
\(a=b^*+c^*,\ b^*\in M_n,\ c^*\in G_n\), i.e. \(a\in M_n+G_n=\mathfrak M_n\), and \(T_n^3\subset \mathfrak M_n\). Lemma 1 is proved.

Lemma 2. The elements \(z_t\in M_n\). Moreover, the residue classes of the elements \(z_t,\ t>1\), form a basis of the quotient module \(M_n/M_n\cap G_n\).

Proof. On the set of linear combinations of elements of the form
\(w=[x_{i_1},x_{i_2}][x_{i_3},x_{i_4}]\cdots [x_{i_{n-1}},x_{i_n}]\)
define linear operators \(\sigma_{ij}\): the element \(w\sigma_{ij}\) is obtained from \(w\) by changing the signs and transposing the variables \(x_i\) and \(x_j\). It can be shown that \(w-w\sigma_{ij}\in M_n\), whence it follows that if
\(\Sigma=\sigma_{i_1j_1}\sigma_{i_2j_2}\cdots\sigma_{i_mj_m}\), then
\(w-w\Sigma\in M_n\). For any \(t>1\) one can indicate such a product \(\Sigma\) of the operators \(\sigma_{ij}\) that
\(u_1\Sigma=(-1)^{\varepsilon_t}u_t\) and
\(z_t=u_1-(-1)^{\varepsilon_t}u_t=u_1-u_1\Sigma\in M_n\).
Now let us show that every element of \(M_n\) is linearly expressible modulo \(G_n\) in terms of the elements \(z_t,\ t>1\). We shall call the commutator \([x_i,x_j]\) improper if \(i>j\). Denote by \(\varepsilon(w)\) the number of improper commutators occurring in the notation of \(w\); then
\(w\equiv(-1)^{\varepsilon(w)}u_t\pmod {G_n}\), where \(u_t\) is the element of the Specht basis with the same composition as \(w\). Denote by \(\pi(w)\) the permutation of the indices \(i_1,i_2,\ldots,i_n\); \(\pi(u_t)\) has the analogous meaning. The parities of \(\pi(w)\) and \(\pi(u_t)\) coincide if and only if \(\varepsilon(w)\) is even. From the form of the elements (1) it easily follows that the elements of the module \(M_n\) are linearly expressed through elements of the form
\(\theta=w-w\sigma_{ij}=w+w'\), where \(w'=-w\sigma_{ij}\). Let the composition number of \(w\) be \(t\) and

the number of the component \(w'\) is \(s\). We shall have:
\[ u_t \equiv (-1)^{\varepsilon(w)} w \pmod {G_n}, \qquad w' \equiv (-1)^{\varepsilon(w')} u_s \pmod {G_n}, \]
\[ z_t=u_1-(-1)^{\varepsilon_t}u_t \equiv u_1-(-1)^{\varepsilon(w)+\varepsilon_t}w \pmod {G_n}, \]
\[ z_s=u_1-(-1)^{\varepsilon_s}u_s \equiv u_1-(-1)^{\varepsilon(w')+\varepsilon_s}w' \pmod {G_n}, \]
whence
\[ z_s-z_t \equiv (-1)^{\varepsilon(w)+\varepsilon_t}w -(-1)^{\varepsilon(w')+\varepsilon_s}w' \pmod {G_n}. \]

Obviously, \(\pi(w)\) and \(\pi(w')\) have different parities; therefore the numbers \(\varepsilon(w)+\varepsilon_t\) and \(\varepsilon(w')+\varepsilon_s\) also have different parities, and \(\theta \equiv \pm (z_s-z_t)\pmod {G_n}\), while the elements \(z_t,\ t>1\), form a basis of \(M_n\) modulo \(G_n\). Lemma 2 is proved.

From Lemmas 1 and 2 Theorem 1 follows trivially, as was required to prove.

Corollary 1. A \(T\)-ideal containing the commutator \([u,v,w]\) has a finite number of generators.

The assertion follows easily from a comparison of the dimensions of \(\Gamma_n\) and \(T_n^3\).

Corollary 2. The universal algebra \(A\) corresponding to the identity \([x_1,x_2,x_3]=0\) cannot be embedded in a full matrix algebra over a commutative algebra.

Indeed, for the standard identity one can establish the relation:
\[ S_{n+2}(x_1,\ldots,x_{n+2})= \]
\[ = \sum_{1\le i<k\le n+2} (-1)^{k+i-1}[x_i,x_k]\, S_n(x_1,\ldots,\hat{x}_i,\ldots,\hat{x}_k,\ldots,x_{n+2}), \]
where the sign \(\wedge\) denotes omission of the corresponding argument. Hence it is not difficult to obtain the congruence:
\[ S_{2k}(x_1,\ldots,x_{2k}) \equiv C_2^2 C_4^2 C_6^2\cdots C_{2k}^2 [x_1,x_2][x_3,x_4]\cdots[x_{2k-1},x_{2k}] \pmod {T^3}; \]
but this means that no standard identity is satisfied in \(A\).

1. A Lie algebra \(L\) over a field \(K\) is called a special \(PI\)-algebra of Lie (an \(SPI\)-algebra of Lie) if it has a faithful representation in an associative \(PI\)-algebra. It is easy to show that an \(SPI\)-algebra of Lie is a \(PI\)-algebra of Lie. Examples of \(SPI\)-algebras of Lie are: a subalgebra and a direct sum of \(SPI\)-algebras of Lie, a nilpotent Lie algebra, a solvable Lie algebra of index 2 without center, and a finite-dimensional Lie algebra. A free solvable Lie algebra of index greater than 2 is not an \(SPI\)-algebra of Lie. In Lie algebras there is a known Burnside-type problem, analogous to the problem of A. G. Kurosh in associative algebras: will an algebraic Lie algebra be locally finite? Algebraicity of a Lie algebra is understood here in the sense of Liu Shao-xue \((^4)\). The analogous question for algebraic Lie algebras of bounded index was solved only in the special case of Engel algebras by A. I. Kostrikin \((^5)\). In the class of solvable Lie algebras the problem was solved by Liu Shao-xue. In the class of \(SPI\)-algebras of Lie this problem is solved affirmatively.

Theorem 2. An algebraic \(SPI\)-algebra of Lie is locally finite.

Lemma 3. If a Lie algebra is faithfully embeddable in an associative \(PI\)-algebra that is semisimple in the sense of Jacobson, then the adjoint algebra of the Lie algebra is an associative \(PI\)-algebra.

Indeed, let \(L\) be embedded in an associative \(PI\)-algebra \(A\) of degree \(d\), and suppose \(A\) is semisimple in the sense of Jacobson, i.e. \(A\) is isomorphic to a subdirect sum of primitive algebras \(P_\alpha\), each of which over its center \(F_\alpha\) has finite dimension \(\le [d/2]^2\). Let \(\varphi_\alpha\) denote a homomorphism:
\[ A\to P_\alpha;\quad L^{\varphi_\alpha}\text{ the image of the algebra }L\text{ in }P_\alpha; \]
its linear span \(\overline{L}^{\varphi_\alpha}\) over \(F_\alpha\) has finite dimension \(\le [d/2]^2\), and therefore the adjoint algebra \(\mathfrak A_\alpha\) of the algebra \(L^{\varphi_\alpha}\) has over \(F_\alpha\) finite dimension \(\le k=[d/2]^4\). Consequently, in \(\mathfrak A_\alpha\) the standard identity \(S_k(x)=0\) of degree \(k\) is satisfied. The satisfaction of the identity \(S_k(x)=0\) in \(\mathfrak A_\alpha\) is equivalent to a certain set

\(\{f_i(x_{j_1}^{\varphi_\alpha}, \ldots, x_{j_t}^{\varphi_\alpha})=0\}\), defining the relations of the algebra \(\overline{L}^{\varphi_\alpha}\); here the \(x_{j_s}\) are generators of the algebra \(L\). Since the coefficients of the identity \(S_k(x)=0\) lie in the field \(K\), the \(f_i\) are polynomials in the \(x_{j_s}\) with coefficients in the field \(K\). In view of the decomposition of \(A\) into the direct sum of the algebras \(P_\alpha\), in \(L\) there holds a system of defining relations \(\{f_i(x_{j_1}, \ldots, x_{j_t})=0\}\), i.e. in the adjoint algebra \(\mathfrak A\) of the Lie algebra \(L\) the identity \(S_k(x)=0\) is satisfied. Lemma 3 is proved.

Proof of Theorem 2. Let \(L\) be an algebraic \(SPI\)-Lie algebra with a finite number of generators \(x_1, x_2, \ldots, x_n\) over the field \(K\); we shall prove that \(L\) has finite dimension over \(K\). \(L\) is embedded isomorphically in the associative \(PI\)-algebra \(A\) generated by the elements \(x_i\), \(i=1,\ldots,n\). Let \(D\) be the Jacobson radical of the algebra \(A\); \(D\) is a locally nilpotent algebra \((^{6,7})\), and therefore \(L_1=L\cap D\) is locally nilpotent, and the factor-algebra \(\overline{L}=L/L_1\) is embedded isomorphically in the associative \(PI\)-algebra \(\overline{A}=A/D\). By Lemma 3, the adjoint algebra \(\mathfrak A\) of the Lie algebra \(\overline{L}\) is an associative \(PI\)-algebra. We shall prove that \(\mathfrak A\) has finite dimension over \(K\). The algebra \(\mathfrak A\) has a finite number of generators \(\overline{x}_1,\ldots,\overline{x}_n\); let \(\overline{\Lambda}\) be the Lie algebra generated by this system of generators in the sense of the multiplication operation \([a,b]=ab-ba\). We have: \(\overline{\Lambda}=\overline{L}/\overline{Z}\), where \(\overline{Z}\) is the center of \(\overline{L}\). Every element of \(\overline{\Lambda}\) is algebraic in the associative sense. Order the generators \(\overline{x}_i\) arbitrarily; thereby some ordering is defined on the set of all words in the generators \(\overline{x}_i\) \((^8)\). We shall show that in the algebra \(\mathfrak A\) any word \(v\) of sufficiently large length in the generators \(\overline{x}_i\) has a representation
\[ v=v_1+v_2+\cdots+v_t+\delta, \]
where the words \(v_i\) have the same length and the same composition as \(v\) with respect to the \(\overline{x}_i\), but are lexicographically smaller; \(\delta\) is a sum of words in the \(\overline{x}_i\) shorter than \(v\). Indeed, \(\mathfrak A\) has bounded height, i.e. there exists a finite number of words \(v_1,v_2,\ldots,v_k\) in the \(\overline{x}_i\) and a number \(N\) such that every word \(v\) of length \(\geq N\) in the \(\overline{x}\) has in the algebra \(\mathfrak A\) a representation of the form:
\[ v=\sum a v_{i_1}^{q_1} v_{i_2}^{q_2}\ldots v_{i_k}^{q_k},\qquad a\in K, \tag{4} \]
where the words standing under the summation sign in (4) have the same composition and the same length with respect to the \(\overline{x}_i\) as \(v\), and are lexicographically not greater than \(v\) \((^9)\). For every word \(v\) in the \(\overline{x}_i\) one can indicate such a regular word \(t\) in the \(\overline{x}_i\) \((^8)\) that \(v^{s+1}=at^s b\) for any natural number \(s\). The word \(t\) uniquely determines an element \(\tau\in\overline{\Lambda}\) \((^8)\); from the algebraicity of \(\tau\) in the associative sense it follows that for some \(s\) the word \(w=t^s\) is represented in the form of a linear combination of shorter or lexicographically smaller words. From (4) follows the validity of the last assertion for any sufficiently long word \(v\in\mathfrak A\), and this shows that \(\mathfrak A\) has finite dimension. Consequently, \(\overline{\Lambda}\), and hence also \(\overline{L}\), has finite dimension. The algebra \(L\), being an algebraic extension of the locally finite algebra \(\overline{L}\) by means of the locally finite algebra \(L_1\), also has finite dimension \((^4)\), as was required to prove.

In conclusion I take this opportunity to express my gratitude to A. I. Shirshov for valuable comments made on the work.

Received
27 IV 1962

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