Simple and semisimple structurable algebras. (English. Russian original) Zbl 0738.17002
Algebra Logic 29, No. 5, 377-394 (1990); translation from Algebra Logika 29, No. 5, 571-596 (1990).
A structurable algebra was defined by B. N. Allison [in Math. Ann. 237, 133-156 (1978; Zbl 0368.17001)] as a unital algebra with an involution, satisfying the operator identity
\[
[T_ z,V_{x,y}]=V_{T_ zx,y}-V_{x,T_{\overline z}y},
\]
where \(V_{x,y}(z)=(x\overline y)z+(z\overline y)x-(z\overline x)y\), \(T_ z=V_{z,1}\). In the cited paper simple central finite dimensional structurable algebras over a nonmodular field were classified. Later on R. D. Schafer [in J. Algebra 92, 400-412 (1985; Zbl 0552.17001)] proved that every semisimple finite dimensional structurable algebra over a nonmodular field is isomorphic to a direct sum of simple algebras. The author in his previous paper [Algebra Logika 29, No. 4, 491-499 (1990; see the preceding review)] has shown that Allison’s list of simple structurable algebras is incomplete: there is one more simple structurable algebra \(T(C)\) of dimension 35.
Now the author generalizes the results of Allison and Schafer to algebras over an arbitrary field \(F\) of characteristic \(\neq2,3,5\). Let \(A\) be a finite dimensional structurable algebra over \(F\). If \(A\) is simple, then \(A\) is isomorphic to either one of the algebras from Allison’s list or to the algebra \(T(C)\); if \(A\) is semisimple, then \(A\) is isomorphic to a direct sum of simple algebras.
Now the author generalizes the results of Allison and Schafer to algebras over an arbitrary field \(F\) of characteristic \(\neq2,3,5\). Let \(A\) be a finite dimensional structurable algebra over \(F\). If \(A\) is simple, then \(A\) is isomorphic to either one of the algebras from Allison’s list or to the algebra \(T(C)\); if \(A\) is semisimple, then \(A\) is isomorphic to a direct sum of simple algebras.
Reviewer: I.P.Shestakov (Novosibirsk)
MSC:
| 17A30 | Nonassociative algebras satisfying other identities |
| 17A60 | Structure theory for nonassociative algebras |
| 17B70 | Graded Lie (super)algebras |
References:
| [1] | B. N. Allison, ”A class of nonassociative algebras with involution containing the class of Jordan algebras,” Math. Ann.,237, 133–156 (1978). · doi:10.1007/BF01351677 |
| [2] | O. N. Smirnov, ”An example of a simple structurable algebra,” Algebra Logika,29, No. 4, 491–499 (1990). · Zbl 0738.17001 |
| [3] | R. D. Schaffer, ”On structurable algebras,” J. Algebra,92, 400–412 (1985). · Zbl 0552.17001 · doi:10.1016/0021-8693(85)90131-0 |
| [4] | K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That are Nearly Associative [in Russian], Nauka, Moscow (1978). · Zbl 0445.17001 |
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| [9] | E. I. Zel’manov, ”Absolute zero divisors in Jordan pairs and Lie algebras,” Mat. Sb.,112 (154), 611–629 (1980). · Zbl 0442.17003 |
| [10] | G. Benkart, ”On inner ideals and ad-nilpotents of Lie algebras,” Trans. Am. Math. Soc.,232, No. 1, 61–81 (1977). · Zbl 0373.17003 · doi:10.1090/S0002-9947-1977-0466242-6 |
| [11] | R. Black, ”Trace froms on Lie algebras,” Can. J. Math.,14, 533–564 (1962). |
| [12] | V. T. Filippov, ”Primary Malcev algebras,” Mat. Zametki,31, No. 5, 669–678 (1982). |
| [13] | O. Loos, Lectures on Jordan Triples, University of British Columbia Lecture Notes, Vancouver (1971). · Zbl 0337.17006 |
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