Reports of the Academy of Sciences of the USSR
1967. Volume 172, No. 5

UDC 519.48

MATHEMATICS

E. N. KUZ’MIN

DIVISION ALGEBRAS OVER THE FIELD OF REAL NUMBERS

(Presented by Academician A. I. Mal’tsev on 14 IV 1966)

The problem of describing division algebras of finite rank over the field of real numbers goes back to Frobenius, who gave a description of such algebras under the additional assumption of associativity. A generalized Frobenius theorem is also known, covering the case of alternative division algebras \((^{1})\). In the present paper certain classes of division algebras with associative powers are considered. With the aid of topological considerations it is possible to prove the existence, in the algebras under consideration, of a certain special basis; under some additional assumptions this makes it possible to solve the isomorphism problem for these algebras.

1. Definition. An algebra \(A\) over a field \(F\) is called a division algebra if \(A\) is an algebra with identity and the equations

\[ ax=b,\qquad ya=b \]

are uniquely solvable in \(A\) for arbitrary \(a\ne 0,\ b\in A\).

Below, the field of real numbers \(R\) will be considered as \(F\). The set of all division algebras having finite dimension (rank) \(n\) over \(R\) will be denoted by \(D_n\). Classical examples of division algebras are the field \(R\) itself, the field of complex numbers, the algebra of quaternions, and the Cayley–Dickson algebra, whose dimensions over \(R\) are respectively \(1, 2, 4, 8\). It is known \((^{2-4})\) that these dimensions are the only ones for which the set \(D_n\) is nonempty. Thus the problem arises of describing the sets \(D_n\) for \(n=1,2,4,8\). The problem is trivial for \(n=1,2\), since the only division algebras of the indicated dimensions over \(R\) are, respectively, the field of real numbers and the field of complex numbers. The sets \(D_4, D_8\) are infinite and require parametrization; the parameters may be the structure constants of the algebras under consideration. It is convenient to regard \(D_n\) as a subset of the set \(\mathfrak A_n\) of all \(n\)-dimensional algebras with identity over \(R\). Let \(A\in\mathfrak A_n\); \(e_0=1, e_1, e_2,\ldots,e_{n-1}\) be some basis of \(A\). Then multiplication in \(A\) is determined by formulas of the form

\[ e_i e_j=\sum_{k=0}^{n-1} c_{ij}^{k} e_k,\qquad i,j=1,2,\ldots,n-1. \tag{1} \]

The structure constants \(c_{ij}^{k}\) will be regarded as the coordinates of a certain point in a \(p\)-dimensional Euclidean space \(R_p\), \(p=n(n-1)^2\); then to each point of \(R_p\) there corresponds a certain algebra from \(\mathfrak A_n\). Introduce in \(R_p\) an equivalence relation \(\theta\), considering two points of \(R_p\) equivalent if the corresponding algebras from \(\mathfrak A_n\) are isomorphic. The set \(\mathfrak A_n\) is in one-to-one correspondence with the quotient space \(R_p/\theta\). In \(R_p/\theta\) (and hence also in \(\mathfrak A_n\)) the structure of a topological space is defined in the natural way: a subset of \(R_p/\theta\) is open if and only if it is the image of an open subset of \(R_p\) under the canonical mapping \(R_p\to R_p/\theta\). The space \(\mathfrak A_n\) repre-

is a union of a finite number of subsets homeomorphic to Euclidean spaces of various dimensions; the greatest of these dimensions will be called the dimension of \(\mathfrak A_n\). (Heuristic considerations show that the dimension of \(\mathfrak A_n\) is equal to \(n(n-1)(n-2)\).)

Theorem 1. The set \(D_n\) is open in \(\mathfrak A_n\), and the dimension of \(D_n\) coincides with the dimension of \(\mathfrak A_n\) \((n=1,2,4,8)\).

For the proof of the theorem one uses deformations \((^5)\) of the algebra \(A \in D_n\). Under arbitrary sufficiently small changes of the structural constants \(c_{ij}^k\) of the algebra \(A\) with multiplication table (1), the property of the algebra \(A\) of being a division algebra is preserved.

Corollary. If any algebra from \(D_n\) satisfies a certain identical relation, then every algebra from \(\mathfrak A_n\) satisfies the same identical relation \((n=1,2,4,8)\).

1. An important class of algebras is formed by algebras with associative powers. Denote by \(\mathfrak B_n\) the set of algebras from \(D_n\) satisfying the additional condition of associativity of powers. Each algebra from \(\mathfrak B_n\) is quadratic, i.e., for any element \(x\) of an algebra \(A \in \mathfrak B_n\), the degree of its minimal polynomial does not exceed 2. Let us denote by \(C_n\) the set of all quadratic algebras from \(\mathfrak A_n\). Let \(A \in C_n\). As Dickson showed \((^6)\), the linear space \(A\) decomposes into a direct sum of subspaces \(R+V\), where \(R\) is a one-dimensional subspace containing the identity of the algebra \(A\), and the elements of \(V\) are characterized by the property that \(x^2 \in R\), \(x \notin R\) for every \(x \in V\). Let \(x,y \in V\); then

\[ xy=-(x,y)+x\cdot y, \tag{2} \]

where \((x,y) \in R,\ x\cdot y \in V\). The composition \(x\cdot y\) is anticommutative and defines a linear mapping \(V \otimes V \to V\); thus \(V\) has the structure of an anticommutative algebra \(A^{(-)}=\langle V,+,\cdot\rangle\). Conversely, if on the linear space \(V\) a bilinear form \((x,y)\) and an anticommutative operation \(x\cdot y\) are defined, then formula (2) makes it possible to define on the direct sum of spaces \(R+V\) the structure of a quadratic algebra. As in § 1, the set \(C_n\) (and \(\mathfrak B_n\)) is provided with a parametrization and becomes a topological space.

Theorem 2 \((^7)\). In order that an algebra \(A \in C_n\) be a division algebra, it is necessary and sufficient that the quadratic form \((x,x)\) be positive definite and that the algebra \(A^{(-)}\) satisfy condition \((Q)\): for any linearly independent \(x,y \in A^{(-)}\), the elements \(x,y,x\cdot y\) also form a linearly independent system.

An anticommutative algebra satisfying condition \((Q)\) will be called a \(Q\)-algebra.

Theorem 3. The set \(\mathfrak B_n\) is open in \(C_n\). The dimension of \(\mathfrak B_n\) coincides with the dimension of \(C_n\) \((n=1,2,4,8)\) and is equal to \(r_n=\frac12(n-1)^2(n-2)\).

In the case \(n=4\) the theorem follows easily from the results of \((^7)\). For \(n=8\) the proof of the theorem is based on the above description of algebras from \(\mathfrak B_n\) and uses a certain class of deformations of the Cayley–Dickson algebra. From Theorem 3 it follows, in particular, that the set of algebras from \(\mathfrak B_8\) depends on 147 “essential” parameters.

1. It is said that an algebra \(A \in \mathfrak A_n\) is an algebra with \(k\) generators if \(k\) is the least number for which there exist such elements \(a_1,a_2,\ldots,a_k \in A\) that the set \(\{1,a_1,a_2,\ldots,a_k\}\) generates the algebra \(A\). From the theorem on the dimension of division algebras over the field of real numbers it follows that every algebra from \(D_8\) has at most three generators. If an algebra \(A \in D_8\) has exactly three generators, then such an algebra is necessarily quadratic, and any linearly independent system of elements of the form \(\{1,x,y\}\) generates in \(A\) a subalgebra of dimension 4. The corresponding algebra \(A^{(-)}\) has an analogous property: it is generated by some triple of elements, while any two linearly independent elements generate in \(A^{(-)}\) a three-dimensional subalgebra. Below

by \(V\) we shall denote an arbitrary seven-dimensional \(Q\)-algebra possessing this property.

Let \(H\) be an arbitrary two-dimensional subspace of \(V\), and let \(\{x,y\}\) be any basis of \(H\). Then the elements \(x,y\) generate a three-dimensional subalgebra \(\mathscr H \subset V\); this subalgebra, obviously, does not depend on the choice of basis in \(H\). As shown in (7), any three-dimensional \(Q\)-algebra is characterized, up to isomorphism, by a single parameter \(\beta\), taking values in the semi-interval \([0,1)\); with a suitable choice of basis \(\{u,v,w\}\), the multiplication table of such an algebra has the form

\[ uv=w,\qquad vw=u+2\beta v,\qquad wu=v. \]

Thus, to the subalgebra \(\mathscr H\) there corresponds a certain number \(\beta\), which we shall denote by \(\beta(H)\). On the set \(M\) of all two-dimensional subspaces of the space \(V\) there is the structure of a real Grassmann manifold (of dimension 10). The function \(\beta(H)\) is continuous on \(M\) and, by compactness of \(M\), attains at some point \(H_0\) a maximal value, which we shall again denote by \(\beta\). Denote by \(\mathscr H_0\) the subalgebra in \(V\) generated by the subspace \(H_0\). Extending a basis of \(\mathscr H_0\) to a basis of \(V\), we arrive at the following theorem.

Theorem 4. In the \(Q\)-algebra \(V\) one can choose a basis \(u_i, v_i, w\) \((i=1,2,3)\) such that the multiplication table of \(V\) in this basis has the form

\[ u_i v_i=w,\qquad v_i w=u_i+2\beta v_i,\qquad w u_i=v_i\quad (i=1,2,3); \tag{3} \]

\[ u_i u_j=-v_k,\qquad u_j v_k=-u_i,\qquad v_k u_i=-u_j,\qquad v_i v_j=v_k+2\beta u_k, \]

where \((i,j,k)=(1,2,3),(2,3,1),(3,1,2)\); \(\beta\) is a uniquely determined parameter taking values in the semi-interval \([0,1)\).

The computation of the function \(\beta(H)\) is based on lemmas that make it possible to give this function an explicit algebraic expression in terms of the structure constants of the algebra \(V\) and the coordinates of the vectors \(x,y\) spanning the subspace \(H\). In what follows the \(Q\)-algebra \(V\) with multiplication table (3) will be denoted by \(V_\beta\). We shall denote by \(\mathscr K_8\) the set of all algebras with three generators from \(\mathfrak B_8\). Adjoining to the \(Q\)-algebra \(V_\beta\) an arbitrary bilinear form \((x,y)\), for which the form \((x,x)\) is positive definite, we obtain a certain algebra from \(\mathscr K_8\). Different values of \(\beta\) correspond, of course, to nonisomorphic algebras from \(\mathscr K_8\). On the other hand, for each individual algebra \(A\) we shall obtain a certain set of algebras isomorphic to it, if we subject the basis \(A^{(-)}\) to all possible transformations that do not change the multiplication table of the algebra \(A^{(-)}\), i.e. the multiplication table (3). The indicated transformations determine the automorphisms of the algebra \(V_\beta\). Therefore, for the classification of algebras from \(\mathscr K_8\) up to isomorphism, one must find the group \(\Gamma_\beta\) of all automorphisms of the algebra \(V_\beta\). For \(\beta=0\) the group \(\Gamma_\beta\) coincides with the automorphism group of the Cayley–Dickson algebra; this is the well-known simple Lie group \(G_2\) of dimension 14. The groups \(\Gamma_\beta\) for \(\beta\ne0\) can also be found. They are all isomorphic to one another and isomorphic to a certain 8-dimensional subgroup of the group \(\Gamma_0\). Hence one may conclude that the dimension of \(\mathscr K_8\) is equal to 42.

Introducing on the linear space \(V_\beta\) the scalar product \((x,y)\), for which the basis \(u_i, v_i, w\) \((i=1,2,3)\) is orthonormal, we obtain by formula (2) a certain algebra from \(\mathscr K_8\), which we shall denote by \(A_\beta\). The algebras \(A_\beta\) for different \(\beta\in[0,1)\) are not only pairwise nonisomorphic, but not even isotopic; this follows from consideration of the determinant \(\det L_x\), where \(L_x\) is the operator of left multiplication, acting in the linear space \(A_\beta\) by the formula \(a\to xa\) (cf. (8)).

1. Let \(\mathscr E_n\) be the set of all algebras from \(\mathfrak B_n\) satisfying the additional condition of elasticity, and let \(A\) be an arbitrary algebra from \(\mathscr E_8\). Then in \(A^{(-)}\) one can choose a basis \(e_1,\ldots,e_7\) such that the bilinear form

\((x,y)\) and multiplication in \(A^{(-)}\) satisfy the relations (9).

\[ (e_i,e_j)=\delta_{ij}, \qquad e_i\cdot e_j=\sum_{k=1}^{7} c_{ij}^{k} e_k,\quad i,j=1,\ldots,7, \tag{4} \]

where \(\delta_{ij}\) is the Kronecker symbol, \(c_{ij}^{k}=-c_{ji}^{k}=-c_{ik}^{j}\) \((i,j,k=1,\ldots,7)\). Relations (4) remain invariant under arbitrary orthogonal transformations of the basis of \(A^{(-)}\).

For any element \(x\in A^{(-)}\), denote its length by \(|x|\): \(|x|^{2}=(x,x)\). Let \(H\) be an arbitrary two-dimensional subspace of \(A^{(-)}\), and let \(\{x,y\}\) be an arbitrary orthonormal basis of \(H\). Since \(A^{(-)}\) is a \(Q\)-algebra, in particular \(x\cdot y\ne 0\), \(|x\cdot y|>0\). It is easy to see that the number \(\alpha=|x\cdot y|\) does not depend on the choice of the orthonormal basis of \(H\); therefore one may put \(\alpha=\alpha(H)\), thus obtaining a numerical function on the Grassmann manifold \(M\) of two-dimensional subspaces of the space \(A^{(-)}\). The function \(\alpha(H)\) is continuous on \(M\). Using points at which either the maximum of the function \(\alpha(H)\), or its maximum on certain compact submanifolds of \(M\), is attained, we arrive at the construction of an orthonormal basis of \(A^{(-)}\) whose multiplication table depends on only 14 parameters. Under certain restrictions of inequality type these parameters are determined uniquely and have an invariant meaning. Hence the following theorem follows:

Theorem 5. The space \(\mathscr{E}_8\) has dimension 14.

It is interesting to note that every elastic division algebra of dimension \(n>2\) over \(R\) is either commutative or is an algebra with associative powers. But from the result announced in (10) it follows that commutative algebras in \(D_n\) for \(n>2\) do not exist. Therefore the following theorem holds:

Theorem 6. If an algebra \(A\in D_n\) \((n=1,2,4,8)\) is elastic, i.e. satisfies the identity

\[ (xy)x=x(yx), \]

then \(A\) is an algebra with associative powers.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
6 IV 1966

CITED LITERATURE

1. A. G. Kurosh, Lectures on General Algebra, Moscow, 1962.
2. J. E. Adams, Bull. Am. Math. Soc., 64, 279 (1958).
3. R. Bott, I. Milnor, Bull. Am. Math. Soc., 64, 87 (1958).
4. M. A. Kervaire, Proc. Nat. Acad. Sci. U. S. A., 44, 280 (1958).
5. M. Gerstenhaber, Ann. Math., 79, 1, 59 (1964).
6. L. E. Dickson, Linear Algebras, Cambridge, 1914.
7. J. M. Osborn, Trans. Am. Math. Soc., 105, 2, 202 (1962).
8. R. H. Bruck, Trans. Am. Math. Soc., 56, 2, 141 (1944).
9. A. A. Albert, Trans. Am. Math. Soc., 64, 3, 552 (1948).
10. H. Toda, Ann. Math., 78, No. 2, 300 (1963).