A rigidity theorem for pre-Lie algebras. (English) Zbl 1134.17001
The starting point of the paper under review is the classical Leray theorem that any graded connected commutative and cocommutative Hopf algebra is a free commutative algebra and a free cocommutative coalgebra and its analogues for unital infinitesimal bialgebras [J.-L. Loday and M. Ronco, J. Reine Angew. Math. 592, 123–155 (2006; Zbl 1096.16019)] and dendriform and codendriform bialgebras [L. Foissy, Bull. Sci. Math. 126, No. 3, 193–239 (2002; Zbl 1013.16026), ibid. 126, No. 4, 249–288 (2002; Zbl 1013.16027)]. These three results are viewed from the operadic point of view: Given the operad \(\mathcal P\) (where \(\mathcal P\) is the commutative, associative or dendriform operad), then any graded connected \(\mathcal P\)-algebra which is also a \(\mathcal P\)-coalgebra equipped with a suitable relation between the algebra and coalgebra structures, is rigid in the sense that it is free as a \(\mathcal P\)-algebra and as a \(\mathcal P\)-coalgebra. In the present paper the author obtains a result of this kind for pre-Lie algebras. The new moment is that the involved costructure is not pre-Lie, but nonassociative permutative. Recall that pre-Lie algebras (or right-symmetric algebras, or Vinberg algebras) with multiplication \(\circ\) satisfy the identity \((x\circ y)\circ z - x\circ (y\circ z)=(x\circ z)\circ y - x\circ (z\circ y)\) and permutative algebras are defined by the identity \((ab)c=(ac)b\). Several years ago F. Chapoton and M. Livernet [Int. Math. Res. Not. 2001, No. 8, 395–408 (2001; Zbl 1053.17001)] described pre-Lie algebras in terms of rooted trees. Now the author establishes that nonassociative permutative algebras can also be described in terms of rooted trees. This allows to prove the main result of the paper:
Any pre-Lie algebra together with a non-associative permutative connected coproduct \(\Delta\) satisfying the compatibility relation \(\Delta(a\circ b)=\Delta(a)\circ b + a\otimes b\) is a free pre-Lie algebra and a free nonassociative permutative coalgebra.
The author also interprets this theorem in terms of cogroups in the category of pre-Lie algebras.
Any pre-Lie algebra together with a non-associative permutative connected coproduct \(\Delta\) satisfying the compatibility relation \(\Delta(a\circ b)=\Delta(a)\circ b + a\otimes b\) is a free pre-Lie algebra and a free nonassociative permutative coalgebra.
The author also interprets this theorem in terms of cogroups in the category of pre-Lie algebras.
Reviewer: Vesselin Drensky (Sofia)
MSC:
| 17A30 | Nonassociative algebras satisfying other identities |
| 17A50 | Free nonassociative algebras |
| 18D50 | Operads (MSC2010) |
Keywords:
pre-Lie algebras; or right-symmetric algebras; Vinberg algebras; permutative algebras; Leray theoremReferences:
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