ON GENERALIZED CODERIVATIONS

Year 2012, Volume: 12 Issue: 12, 37 - 52, 01.12.2012

Atsushi Nakajima

Abstract

Let C (resp. A) be a coalgebra (resp. algebra) over a commutative
ring R and M (resp. N) a C-bicomodule (resp. an A-bimodule). We define a
dual notion of a generalized derivation from A to N in the sense of the paper
On categorical properties of generalized derivations, Sci. Math., 2(3) (1999),
345-352, by A. Nakajima, which we call a generalized coderivation from M
to C. We give some elementary properties of generalized coderivations and
discuss the relations of the set of generalized coderivations gCoder(M, C) between
the set of generalized derivations gDer(C∗, M∗) for their dual algebra
C∗ and module M∗. Using these coderivations, we define a notion of a weakly
coseparable coalgebra which is a dual notion of a weakly separable algebra
defined in the paper of N. Hamaguchi and A. Nakajima, Weakly separable
polynomials (in preparation), and give related examples of coseparable coalgebras.

Keywords

derivation , generalized derivation , coderivation , generalized coderivation , weakly coseparable coalgebra

References

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• M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. Atsushi Nakajima
• 1, Nakaku, Nakai,
• Okayama 703-8205, Japan
• e-mail: [e-mail]