UDC 512.934+513.836

A. BLOKH

THE CARTAN—EILENBERG HOMOLOGY THEORY FOR A GENERALIZATION OF THE CLASS OF LIE ALGEBRAS

(Presented by Academician P. S. Novikov on 19 IX 1966)

In the note \((^1)\) the concept of a \(D\)-algebra was introduced, generalizing the concept of a Lie algebra. We define, for \(D\)-algebras, the concept of a representation, in connection with which it becomes possible to construct for \(D\)-algebras a homology theory by the Cartan—Eilenberg method \((({}^2),\) Chap. XIII). On the other hand, the theory of extensions, which in the case of Lie algebras leads to the coincidence \(\Sigma(\mathscr L,\mathfrak M)=H^2(\mathscr L,\mathfrak M)\) (see \(({}^2),\) Chap. XIV), for \(D\)-algebras differs substantially, in its results, from the theory based on the study of the corresponding cohomology group. For \(D\)-algebras the latter corresponds to a proper part of the set of all extensions. By an extension we shall everywhere in this paper mean an extension with abelian kernel. In general, one may call a homology theory for a certain class of algebras classical if for it there holds an analogue of the equality \(\Sigma(\mathscr L,\mathfrak M)=H^2(\mathscr L,\mathfrak M)\). Thus, the main assertion of the present paper is that the Cartan—Eilenberg homology theory for \(D\)-algebras is not classical.

We shall consider here only left \(D\)-algebras; the symbol \(\mathfrak Z\) in a left \(D\)-algebra \(\mathscr D\) over a field \(K\) (of characteristic zero) will denote the minimal \(L\)-ideal in \(\mathscr D\), i.e. the least of those ideals in \(\mathscr D\) whose quotient algebras are Lie algebras. The definition of associativity of a distributive algebra \(\mathscr D\) and of an associative algebra \(\mathfrak A\), used in the present paper, differs from that introduced in \((^1)\) in that condition 1) is replaced by the following condition: the vector space of the algebra \(\mathscr D\) is a subspace of \(\mathfrak A\). It can be proved that the multiplication law in a distributive algebra polynomially associated with \(\mathfrak A\) is determined by the formula \(a*b=\xi_1ab+\xi_2ba\), where \(\xi_i\in K\) are certain fixed constants.

Theorem 1. Let \(\mathfrak A\) be an associative algebra, and let \(\mathscr D\) be a \(D\)-algebra polynomially associated with it. If \(\mathscr D\) contains at least one element \(a\) for which \(a^3\ne0\), then \(\mathscr D\) is a Lie algebra and \(a*b=\xi(ab-ba)\), \(\xi\in K\) a fixed constant.

Proof. In a (left) \(D\)-algebra the equality \((a*a)*a=0\) must hold; passing to the above-indicated form of the multiplication formula in \(\mathscr D\), we obtain the assertion of the theorem.

This theorem contains Theorem 2 of \((^1)\). In what follows it will be assumed that the condition of Theorem 1 is satisfied and that \(\xi=1\). Let us consider the application of this theorem to the representation theory of \(D\)-algebras. Let \(\mathfrak M\) be a vector space over \(K\); let \(\mathscr D\) be a \(D\)-algebra. A mapping \(Q:\mathscr D\to\mathscr E(\mathfrak M)\), where \(\mathscr E(\mathfrak M)\) denotes the algebra of linear transformations of \(\mathfrak M\), is called a representation of \(\mathscr D\) if there exists a bilinear function \(f:\mathscr E\times\mathscr E\to\mathscr E\) such that for all \(a,b\in\mathscr D\) the equality
\[ Q(a*b)=f(Q(a),Q(b)) \]
holds. A representation is called polynomial if \(f(x,y)\) is a polynomial in the (noncommuting) unknowns \(x,y\) over \(K\). We shall also say that by means of the representation \(Q\) a \(D\)-module \(\mathfrak M\) over \(\mathscr D\) is given.

It follows from Theorem 1 that if \(Q:\mathscr D\to\mathscr E(\mathfrak M)\) is a \(D\)-module such that \(Q(a)^3\ne0\) for at least one \(a\in\mathscr D\), then \(Q(\mathscr D)\) is a Lie algebra. Thus \(Q\) narrows to a homomorphism \(Q:\mathscr D\to\mathscr E_L(\mathfrak M)\), where \(\mathscr E_L(\mathfrak M)\) ...

denotes the Lie algebra adjoined to \(\mathscr E(\mathfrak M)\). A homomorphism
\(Q:\mathscr D \to \mathscr E_L(\mathfrak M)\) will be called an \(L\)-representation of the \(D\)-algebra \(\mathscr D\). It is obvious that the kernel of every \(L\)-representation contains \(\mathfrak Z\). In what follows only polynomial \(L\)-representations will be considered. Under this assumption, every \(D\)-module \(\mathfrak M\) may be regarded as an \(L\)-module over the Lie algebra \(\mathcal L=\mathscr D/\mathfrak Z\).

Definition. The Cartan–Eilenberg homology groups of a \(D\)-algebra \(\mathscr D\) with coefficients in a \(D\)-module \(\mathfrak M\) are the homology groups of the Lie algebra \(\mathcal L=\mathscr D/\mathfrak Z\) with coefficients in \(\mathfrak M\), regarded as an \(\mathcal L\)-module.

Let us now consider the relation of the homology theory thus defined to the theory of abelian extensions in the class of \(D\)-algebras. Let \(\mathfrak F\) be a vector space; \(\overline{\mathscr D}\) a \(D\)-algebra, and
\(0\to\mathfrak F\to\mathscr D\to\overline{\mathscr D}\to0\) an extension of \(\overline{\mathscr D}\) in the class of \(D\)-algebras. The mapping
\(P:\overline{\mathscr D}\to\mathscr E_L(\mathfrak F)\), defined by the formula
\(P(\overline l)f=l*f\), endows \(\mathfrak F\) with the structure of a (left) \(D\)-module. Identifying now the vector spaces
\(\mathscr D=\mathfrak F+\overline{\mathscr D}\), the law of multiplication in \(\mathscr D\) (denoted by \(*\)) may be expressed in the following way:

\[ f_1*f_2=0;\qquad l_1*f_1=P(l_1)f_1;\qquad f_1*l_1=-P(l_1)f_1+z(f_1,l_1); \]

\[ l_1*l_2=l_1\overline{*}l_2+g(l_1,l_2), \]

where \(f_i\in\mathfrak F\), \(l_i\in\overline{\mathscr D}\), and \(z(f,l)\), \(g(l_1,l_2)\) are bilinear functions, defined respectively on
\(\mathfrak F\times\overline{\mathscr D}\), \(\overline{\mathscr D}\times\overline{\mathscr D}\), with values in \(\mathfrak F\); \(\overline{*}\) is the symbol for multiplication in \(\overline{\mathscr D}\). It can be shown that the fulfillment of the system of relations

\[ z(f,l_1)*l_2=0; \]

\[ l_1*(f*l_2)=(l_1*f)*l_2+f*(l_1*l_2); \]

\[ l_1*(l_2*l_3)=(l_1*l_2)*l_3+l_2*(l_1*l_3) \]

for all \(f\in\mathfrak F,\ l_i\in\overline{\mathscr D}\) is the condition necessary and sufficient for \(\mathscr D\) to be a \(D\)-algebra and an extension of \(\overline{\mathscr D}\) with abelian kernel \(\mathfrak F\) relative to the representation \(P\). On the other hand, equivalent extensions may differ only by the different manner of representing \(\mathscr D\) as the direct sum of \(\mathfrak F\) and \(\overline{\mathscr D}\). Passing from these conditions to conditions on the system of functions \(\{z(f,l);g(l_1,l_2)\}\) (the system of factors), we obtain the theorem:

Theorem 2. In order that the \(D\)-algebra \(\mathscr D\) be an abelian extension of the \(\overline D\)-algebra \(\overline{\mathscr D}\) relative to the \(L\)-representation
\(P:\overline{\mathscr D}\to\mathscr E_L(\mathfrak F)\) with system of factors \(\{z,g\}\), it is necessary and sufficient that

\[ P(l_2)z(f,l_1)=z(z(f,l_1),l_2); \]

\[ P(l_1)z(f,l_2)=z(P(l_1)f,l_2)+z(f,l_1\overline{*}l_2); \]

\[ g(l_1,l_2\overline{*}l_3)-P(l_3)g(l_1,l_2)+z(g(l_1,l_2),l_3)= \]

\[ = g(l_1,l_2\overline{*}l_3)-g(l_2,l_1\overline{*}l_3)+P(l_1)g(l_2,l_3)-P(l_2)g(l_1,l_3) \]

for all \(f\in\mathfrak F,\ l_i\in\overline{\mathscr D}\). Moreover, two such extensions are equivalent if and only if there exists a linear function
\(h:\overline{\mathscr D}\to\mathfrak F\) such that the systems of factors \(\{z,g\}\), \(\{z',g'\}\) are related by

\[ z'(f,l)=z(f,l); \]

\[ g'(l_1,l_2)=g(l_1,l_2)-h(l_1\overline{*}l_2)+P(l_1)h(l_2)-P(l_2)h(l_1)+z(h(l_1),l_2) \]

for all \(f\in\mathfrak F;\ l,l_i\in\overline{\mathscr D}\).

It follows from this theorem that the set \(S(\overline{\mathscr D},\mathfrak F)\) of all pairwise inequivalent extensions determined by the \(D\)-module \(\mathfrak F\) over \(\overline{\mathscr D}\) can be divided into disjoint classes \(S(\overline{\mathscr D},\mathfrak F;z(f,l))\), each of which contains the extensions having the corresponding factor \(z(f,l)\). Each

one of the classes \(S(\overline{\mathfrak D},\mathcal F;z)\) can in the usual way be turned into an abelian group.

Let \(\overline{\mathfrak Z}\) denote the minimal \(L\)-ideal in \(\overline{\mathfrak D}\), \(\overline{\mathcal L}=\overline{\mathfrak D}/\overline{\mathfrak Z}\), \(\overline P\) the narrowing of \(P\) to \(\overline{\mathcal L}\). We note that the group of extensions \(\Sigma(\overline{\mathcal L},\mathcal F)\), defined in the class of Lie algebras over the \(L\)-module \(\mathcal F\), is a subgroup in \(S(\overline{\mathfrak D},\mathcal F;0)\). Indeed, if \(\overline A\in\Sigma(\overline{\mathcal L},\mathcal F)\) is an extension defined by a system of factors \(\{\overline g(\overline l_1,\overline l_2)\}\), \(\overline l_i\in\overline{\mathcal L}\), then one may set \(A\in S(\overline{\mathfrak D},\mathcal F;0)\)—the extension defined by the system of factors \(\{g(l_1,l_2)\}\), \(l_i\in\overline{\mathfrak D}\), where \(g(l_1,l_2)=\overline g(\overline l_1,\overline l_2)\) for all \(l_i\in\overline l_i\).

However, as follows from example 1 in \((^1)\), for every \(D\)-algebra \(\mathfrak D\) and every one of its \(D\)-modules \(\mathcal F\) there exist nonzero extensions not contained in \(S(\overline{\mathfrak D},\mathcal F;0)\). This completes the proof of the following theorem:

Theorem 3. The Cartan–Eilenberg homology theory for the class of left \(D\)-algebras is not classical.

We do not know whether a classical homology theory exists for \(D\)-algebras.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
16 IX 1966

REFERENCES

\({}^1\) A. Bloch, DAN, 165, No. 3 (1965).
\({}^2\) H. Cartan, S. Eilenberg, Homological Algebra, IL, 1960.