Continual Lie algebras determined by chain complexes
Continual Lie algebras determined by chain complexes
Continual Lie algebras are infinite-dimensional generalizations of Lie algebras with discrete root system by considering continual root systems. In this paper we establish a general relation between chain complexes and continual Lie algebras. The natural orthogonality condition with respect to a product among elements of a chain complex $\mathcal C$ spaces brings about to $\mathcal C$ the structure of a graded algebra with differential relations. We prove the main result of this paper: a chain complex endowed with an appropriate Leibniz-property product of elements of its spaces brings about the structure of a continual Lie algebra with the root space determined by parameters for the complex. That provides a new source of examples of continual Lie algebras. Finally, as an example, we consider the case of Äech-de Rham complex associated to a foliation of a smooth manifold. In a particular case of this chain complex, we derive explicitly the commutation relations for the corresponding continual Lie algebra.
A. Zuevsky
数学
A. Zuevsky.Continual Lie algebras determined by chain complexes[EB/OL].(2025-07-22)[2025-08-18].https://arxiv.org/abs/2012.07431.点此复制
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