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Classification of left octonionic modules. (English) Zbl 1472.17001

The article is devoted to left \(\mathbb{O}\)-modules, where \(\mathbb{O}\) denotes the classical octonion (Cayley) algebra over the real field \(\mathbb{R}\). This is a particular case of modules over alternative algebras. A review of some previous publications is given. The octonion algebra contains the classical quaternion skew field \(\mathbb{H}\) of Hamilton. The octonion algebra considered as the vector space over \(\mathbb{R}\) has the basis \(i_0\), \(i_1,\ldots,i_7\) such that \(i_0=1\), \(i_k^2=-1\) for each \(k=1,\ldots,7\), \(i_ki_l=-i_li_k\) for each \(k\ne l\) such that \(k\ge 1\) and \(l\ge 1\). The subsequent doubling procedures are:
\(i_1\) is the doubling generator of the complex field \(\mathbb{C}\) over the real field \(\mathbb{R}\), \(i_2\) is the doubling generator of \(\mathbb{H}\) generated from \(\mathbb{C}\) and \(\mathbb{C}i_2\) such that \(i_3=i_1i_2\), then \(i_4\) denotes the doubling generator of \(\mathbf{O}\) generated from \(\mathbb{H}\) and \(\mathbb{H}i_4\) by the smashed product, where \(i_5\), \(i_6\), \(i_7\) are obtained by multiplication of \(i_1\), \(i_2\), \(i_3\) respectively on \(i_4\) with the corresponding order up to a notation choice and an automorphism of \(\mathbb{O}\). It is nonassociative, for example, \((i_1i_2)i_4=-i_1(i_2i_4)\).
The commutator \((i_k,i_j)\) and the associator \((i_k,i_j,i_l)\) belong to \(\mathbb{Z}_2\) for each \(k\), \(j\), \(l\), where \(ab=(ba)(a,b)\) and \((ab)c=(a(bc))(a,b,c)\) for each \(a\), \(b\), \(c\) in \(\mathbf{O}\setminus \{ 0 \} \), \(\mathbb{Z}_2= \{ -1, 1 \} \). There is an involution \(\mathbb{O}\ni z\mapsto \bar{z}\in \mathbb{O}\) such that \(\overline{ab}=\bar{b} \bar{a}\) for each \(a\), \(b\) in \(\mathbb{O}\), \(|b|^2=b\bar{b}\). The octonion division algebra is nonassociative alternative with center \(\mathbb{ R}\) and the multiplicative norm. It is shown in the article that left \(\mathbb{O}\)-modules are of the type \(M=\mathbb{O}^n\bigoplus \bar{\mathbb{O}}^m\). This induces the algebra structure on \(\mathbb{O}^n\). This matter is also described in:
\([1]\) [N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 1 à 3. Reprint of the 1970 original. Berlin: Springer (2007; Zbl 1111.00001)].
\([2]\) [R. D. Schafer, An introduction to nonassociative algebras. New York and London: Academic Press (1966; Zbl 0145.25601)].
\([3]\) [R. H. Bruck, A survey of binary systems. Berlin: Springer-Verlag (1958; Zbl 0081.01704)].
Using the opposite algebra \(\mathbb{O}_o\), or \(\overline{\mathbb{O}}\) obtained by the involution from \(\mathbb{O}\), one gets the standard correspondence between left and right modules, the left module over the enveloping algebra \(\mathbb{O}_e\) also corresponds to the two-sided \(\mathbb{O}\)-module as in [N. Bourbaki, (loc. cit.); R. D. Schafer, (loc. cit.)]. The algebra \(L(\mathbb{O})\) generated by left multipliers \(L_b\) on \(\mathbb{O}\), \(b\in \mathbb{O}\), with the associative composition obtained by the set-theoretic composition of maps, is isomorphic to the proper subalgebra of the matrix algebra \(Mat_{8\times 8}(\mathbb{R})\) (see [R. D. Schafer, (loc. cit.)]) satisfying relations \((5.14)\)-\((5.20)\) in [R. H. Bruck, (loc. cit.)] implying particularly that \(i_lL_{i_j}L_{i_k}\mathbb{Z}_2=i_lL_{i_ki_j}\mathbb{Z}_2\) for each \(l\), \(j\), \(k\) in \(\{ 0,...,7 \} \). On the other hand, \(\mathbb{Z}_2\) is the normal subgroup in the Moufang multiplicative loop \(G= \{ \pm i_k: k=0,...,7 \} \) such that its quotient by \(\mathbb{Z}_2\) is the commutative group \(G/\mathbb{Z}_2\) by Theorem IV.1.1 in [loc. cit.].
It is proposed in the article to use the Clifford algebra \(Cl_7=Cl(0,7,\mathbb{R})\) over \(\mathbb{R}\) for studying the octonion modules by using \(\hat{e}_1, ...,\hat{e_7}\) as generators of \(Cl_7\) with Clifford multiplication \(\hat{e}_{k_1}\cdot ... \cdot \hat{e}_{k_m}\) instead of \(L_{i_1},...,L_{i_7}\). The Clifford algebra is associative semisimple and isomorphic to \(Mat_{8\times 8}(\mathbb{R})\bigoplus Mat_{8\times 8}(\mathbb{R})\). The octonion algebra is simple. There is no any nontrivial homomorphism from \(Cl_7\) to \(L(\mathbb{O})\), or from \(Cl_7\) to \(\mathbf{O}\). For comparison \(L(\mathbb{H})=\mathbb{H}\), since \(\mathbb{H}\) is associative. Then Theorem 4.1 of Huo, Li, Ren contradicts to [N. Bourbaki, (loc. cit.)] and the Cartan-Jacobson Theorem 3.28 and Corollary 3.29 in [R. D. Schafer, (loc. cit.)].
In the article under review there is wrongly cited reference \([12]\) in Russian. It is available also in English translation: [S. V. Ludkovsky, J. Math. Sci., New York 144, No. 4, 4301–4366 (2007; Zbl 1178.47057); translation from Sovrem. Mat. Prilozh. 35 (2005)]. In the latter paper were considered vector spaces \(X\) over \(\mathbb{O}\), which also have the structure of the two-sided octonion modules, such that \(X=X_0i_0\oplus X_1i_1\oplus ... \oplus X_7i_7\), where \(X_0\), ...,\(X_7\) are real vector spaces such that \(X_l\) is isomorphic to \(X_k\) for each \(l\), \(k\), \((ab)x_k=a(bx_k)\), \(bx_k=x_kb\), \(x_k(ab)=(x_ka)b\) for each \(a\),\( b\) in \(\mathbb{O}\), \(x_k\in X_k\), \(k\), \(l\) in \(\{ 0,...,7 \} \). It has properties: \((bb)x=b(bx)\), \(x(bb)=(xb)b\), \([a,b,x_ki_k]=[a,b,i_k]x_k\) for each \(x_k\in X_k\), \(k\in \{ 0,...,7 \} \), \(a\), \(b\) in \(\mathbb{O}\), implying by the \(\mathbb{R}\)-linearity in \(X\) and the corresponding identities in \(\mathbb{O}\) that \([a,b,x]=[b,x,a]=[x,a,b]\) for each \(x\in X\), \(a\), \(b\) in \(\mathbb{O}\), where \([a,b,x]=(ab)x-a(bx)\).

MSC:

17A05 Power-associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
13C05 Structure, classification theorems for modules and ideals in commutative rings
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References:

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