Symmetric polynomials in the free metabelian Lie algebras. (English) Zbl 1475.17010
The symmetric group \(S_n\) of degree \(n\) is a subgroup of the automorphism group of the free metabelian Lie algebra \(F_n\) of rank \(n\), acting via permutation of the free generators. The Lie subalgebra \(F_n^{S_n}\) of invariants is not finitely generated as a Lie algebra. However, its intersection with the commutator ideal of \(F_n\) is naturally a module over \(K[X_n]^{S_n}\), the commutative algebra of symmetric polynomials in \(n\) indeterminates. In the paper the authors find explicit generators for this module. The case \(n=2\) was settled earlier in [Ş. Fındık and N. Ş. Öğüşlü, Int. J. Algebra Comput. 29, No. 5, 885–891 (2019; Zbl 1429.17008)].
Reviewer: Matyas Domokos (Budapest)
MSC:
| 17B01 | Identities, free Lie (super)algebras |
| 17B30 | Solvable, nilpotent (super)algebras |
| 05E05 | Symmetric functions and generalizations |
Citations:
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