Zhelyabin, V.N., Kolesnikov, P.S. Dual coalgebra of the differential polinomial algebra in one variable and related coalgebras

V. N. Zhelyabin ; P. S. Kolesnikov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 792-803

Résumé

We show that the dual coalgebra of the polynomial algebra in one variable is the space of linearly recursive sequences over an arbitrary field. Moreover, this coalgebra is a differential one relative to the dual standard derivation and does not contain nonzero finite-dimensional differentially closed subcoalgebras if the characteristic of the ground field is zero. We construct a Novikov coalgebra which is the dual coalgebra of the left-symmetric Witt algebra of index one. Also, we construct a Jordan supercoalgebra which is dual to the Jordan superalgebra of vector type of the polynomial algebra in one variable. All these coalgebras do not contain non-zero finite-dimensional subcoalgebras if the characteristic of ground field is zero. It is shown that over a field of characteristic different from 2 the adjoint Lie coalgebra of the dual coalgebra of the left-symmetric Witt algebra of index one is isomorphic to the dual coalgebra of the Witt algebra of index one.

Détail

Mots-clés : coalgebra
Keywords: coderivation, associative commutative algebra, differential algebra, Novikov algebra, Lie algebra, Witt algebra, Jordan superalgebra, locally finite coalgebra.