On contact brackets on the tensor product

Abstract

We study the behaviour of contact brackets on the tensor product of two algebras, in particular, address the question of Martínez and Zelmanov about extension of a contact bracket on the tensor product from the brackets on the factors.

Keywords:

• Tensor product
• contact bracket
• polynomial algebra

AMS Classifications:

• 17B60
• 17B63
• 17B66

Acknowledgments

Thanks are due to Ivan Kaygorodov for pointing out some relevant literature. GAP [14] was used to check some of the computations performed in this note.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Of course, this is just a manifestation of the vector space isomorphism C²(A ⊗ B) ≃ C²(A) ⊗ S²(B) + S²(A) ⊗ C²(B), where C² and S² denote the vector space of anticommutative and commutative brackets on the corresponding algebra, respectively. However, we use a similar reasoning in more complicated situations, where the argument based on the above decomposition of C²(A ⊗ B) is not enough.

2 Note that Remark 5.1.1(2) in [5], which claims the contrary, is incorrect.