Weighted infinitesimal unitary bialgebras, pre-Lie, matrix algebras, and polynomial algebras. (English) Zbl 1436.16043
The algebra of \(n\times n\) matrices, with its usual coproduct, is given a coassociative coproduct \(\Delta\) making it an infinitesimal bialgebra, in the sense that for any matrices \(A\) and \(B\),
\[
\Delta(AB)=A\Delta(B)+\Delta(A)B.
\]
The existence of an antipode is proved, in Aguiar’s sense, using a conilpotency argument. Consequently, a prelie product on matrices is obtained, giving a Lie bracket, different from the usual one. The last section of this article is devoted to the construction of a weighted infinitesimal bialgebra structure on polynomial rings.
Reviewer: Loïc Foissy (Calais)
MSC:
| 16T10 | Bialgebras |
| 17D25 | Lie-admissible algebras |
| 16W99 | Associative rings and algebras with additional structure |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
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