Locally finite coalgebras and the locally nilpotent radical II

Abstract

In this article, we describe a criterion for an element of the dual space of an algebra to belong to the finite dual. This result is used to study when a certain subspace of the dual space is a subcoalgebra of the finite dual. We further apply it to find a right alternative coalgebra that is not locally finite. This work is motivated by a conjecture from I. Shestakov, which states that all coalgebras of a given variety are locally finite if, and only if, this variety admits locally nilpotent radical.

Keywords:

• Finite dual
• locally finite coalgebras
• right alternative algebras

2020 Mathematics Subject Classification:

• 17A99
• 17D15
• 17A65

Acknowledgements

When the paper was already submitted, V. N. Zhelyabin has informed us on the paper [2] by D. Kh. Kozybaev, where a non-locally finite Novikov coalgebra was constructed, showing that an analogue for the Fundamental Theorem of Coalgebras is not true for Novikov coalgebras. The same result was claimed for the class of right alternative coalgebras. However, the example of non-locally finite right alternative coalgebra constructed contained a mistake, which was corrected by U. Umirbaev in a recent personal communication. For the previously described reasons, the authors would like to thank V. N. Zhelyabin and U. Umirbaev respectively.

Additional information

Funding

G. Santos Filho is supported by grant CAPES 88882.386845/2019-01. L. Murakami is supported by grant FAPESP 2018/23690-6. I. Shestakov is supported by grants FAPESP 2018/23690-6 and CNPq 303916/2014-1. Conselho Nacional de Desenvolvimento Científico e Tecnológico; Coordenação de Aperfeiçoamento de Pessoal de Nível Superior; Fundação de Amparo à Pesquisa do Estado de São Paulo.