Varieties of Lie algebras with the identity \([[x_1,x_2,x_3], [x_4,x_5,x_6]] = 0\) over a field of characteristic zero. (English. Russian original) Zbl 0575.17006
Sib. Math. J. 25, 370-382 (1984); translation from Sib. Mat. Zh. 25, No. 3(145), 40-54 (1984).
The paper under review contains the complete proofs of the author’s results announced earlier [Dokl. Akad. Nauk BSSR 25, 1063–1066 (1981; Zbl 0472.17008)]. Let \(\mathfrak{AN}_2\) be the variety of Lie algebras over a field of characteristic 0, defined by the polynomial identity \([[x_1,x_2,x_3], [x_4,x_5,x_6]] = 0.\)
The main result of this interesting paper claims that \(\mathfrak{AN}_2\) satisfies the Specht property, i.e. every subvariety of \(\mathfrak{AN}_2\) is defined by a finite system of identities. The proof is based on the technique of representation theory of symmetry groups. In order to achieve the result the author proves the Specht property for the ideal of all weak identities generated by \([x_1,x_2,x_3]\) in the free associative algebra with 1. He establishes a series of extremal properties for \(\mathfrak{AN}_2\) as well.
Note that the Specht problem for \(\mathfrak{AN}_k\), \(k>2\), is still open and over a field of characteristic two there exists an infinitely based subvariety of \(\mathfrak{AN}_2\) [M. R. Vaughan-Lee, J. Lond. Math. Soc., II. Ser. 11, 263–266 (1975; Zbl 0316.17007)].
The main result of this interesting paper claims that \(\mathfrak{AN}_2\) satisfies the Specht property, i.e. every subvariety of \(\mathfrak{AN}_2\) is defined by a finite system of identities. The proof is based on the technique of representation theory of symmetry groups. In order to achieve the result the author proves the Specht property for the ideal of all weak identities generated by \([x_1,x_2,x_3]\) in the free associative algebra with 1. He establishes a series of extremal properties for \(\mathfrak{AN}_2\) as well.
Note that the Specht problem for \(\mathfrak{AN}_k\), \(k>2\), is still open and over a field of characteristic two there exists an infinitely based subvariety of \(\mathfrak{AN}_2\) [M. R. Vaughan-Lee, J. Lond. Math. Soc., II. Ser. 11, 263–266 (1975; Zbl 0316.17007)].
Reviewer: Vesselin Drensky (Sofia)
MSC:
| 17B01 | Identities, free Lie (super)algebras |
| 17B30 | Solvable, nilpotent (super)algebras |
| 16Rxx | Rings with polynomial identity |
| 20C30 | Representations of finite symmetric groups |
Keywords:
weak polynomial identities characteristic zero; variety of Lie algebras; Specht property; finite system of identities; representation theory of symmetry groups; free associative algebraReferences:
| [1] | Yu. P. Razmyslov, ?Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero,? Algebra Logika,12, No. 1, 83-113 (1973). |
| [2] | G. Higman, ?Ordering by divisibility in abstract algebras,? Proc. London Math. Soc.,2, No. 2, 326-336 (1952). · Zbl 0047.03402 · doi:10.1112/plms/s3-2.1.326 |
| [3] | M. R. Vaughan-Lee, ?Abelian-by-nilpotent varieties of Lie algebras,? J. London Math. Soc.,11, No. 3, 263-266 (1975). · Zbl 0316.17007 · doi:10.1112/jlms/s2-11.3.263 |
| [4] | S. A. Amitsur, ?The identities of PI-rings,? Proc. Am. Math. Soc.,4, No. 1, 27-34 (1953). · Zbl 0050.02902 |
| [5] | A. Regev, ?The representations of Sn and explicit identities for PI-algebras,? J. Algebra,51, 25-40 (1978). · Zbl 0374.16009 · doi:10.1016/0021-8693(78)90133-3 |
| [6] | A. Regev, ?Existence of polynomial identities in A ? B? Bull. Am. Math. Soc.,77, No. 6, 1067-1069 (1971). · Zbl 0225.16012 · doi:10.1090/S0002-9904-1971-12869-0 |
| [7] | V. N. Latyshev, ?Regev’s theorem on identities in the tensor product of PI-algebras,? Usp. Mat. Nauk,27, No. 4, 213-214 (1972). · Zbl 0254.16013 |
| [8] | I. B. Volichenko, ?Bases of a free Lie algebra modulo certain T-ideals,? Dokl. Akad. Nauk BSSR,24, No. 5, 400-403 (1980). · Zbl 0434.17010 |
| [9] | I. B. Volichenko, ?Varieties of representations of Lie algebras,? in: Proceedings of the 16th All-Union Algebra Conference, Part 1, Leningrad (1981), p. 32. · Zbl 0472.17008 |
| [10] | G. de B. Robinson, Representation Theory of the Symmetric Group, University of Toronto Press, Toronto (1961). · Zbl 0102.02002 |
| [11] | C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York, London (1952). |
| [12] | V. N. Latyshev, ?On the choice of basis in a T-ideal,? Sib. Mat. Zh.,4, No. 5, 1122-1127 (1963). · Zbl 0199.07501 |
| [13] | J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, New York-Heidelberg-Berlin (1970). |
| [14] | L. A. Bokut’, ?A basis for free polynilpotent Lie algebras,? Algebra Logika,2, No. 4, 13-20 (1963). |
| [15] | V. N. Latyshev, ?The Spechtianness of certain varieties of associative algebras,? Algebra Logika,8, No. 6, 600-673 (1969). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
