Symmetric polynomials in the free metabelian Poisson algebras. (English) Zbl 1520.17031
The metabelian variety of Poisson algebras is defined by the polynomial identity \([[x_1,x_2],[x_3,x_4]]=0\). In the paper under review the authors study the canonical action of the symmetric group \(S_n\) of degree \(n\) on the free \(n\)-generated metabelian Poisson algebra \(P_n\) over a field \(K\) of characteristic zero. The main result gives a complete description of the elements of \(P_n\) which are fixed under the action of \(S_n\). The proofs are based on the descriptions of \(P_n\) given in the preprint [Z. Z. Zhang et al., “Word problem for finitely presented metabelian Poisson algebras”, Preprint, arXiv:1907.05953] and of the symmetric polynomials in the free metabelian Lie algebra given in [V. Drensky et al., Mediterr. J. Math. 17, No. 5, Paper No. 151, 11 p. (2020; Zbl 1475.17010)].
Reviewer: Vesselin Drensky (Sofia)
MSC:
| 17B63 | Poisson algebras |
| 17B01 | Identities, free Lie (super)algebras |
| 17B30 | Solvable, nilpotent (super)algebras |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
| 17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |
| 16W22 | Actions of groups and semigroups; invariant theory (associative rings and algebras) |
Citations:
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