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Dual coalgebras of algebras over commutative rings

Summary: Dual coalgebras of algebras over
commutative rings
Jawad Y. Abuhlail, Jos´e G´omez-Torrecillas
and Robert Wisbauer
In the study of algebraic groups the representative functions related to monoid
algebras over fields provide an important tool which also yields the finite dual coal-
gebra of any algebra over a field. The purpose of this note is to transfer this basic
construction to monoid algebras over commutative rings R. As an application we
obtain a bialgebra (Hopf algebra) structure on the finite dual of the polynomial ring
R[x] over a noetherian ring R. Moreover we give a sufficient condition for the fi-
nite dual of any R-algebra A to become a coalgebra. In particular this condition is
satisfied provided R is noetherian and hereditary.
Let k be a field and consider a group G. The commutative Hopf algebra Rk(G) of all
k-valued representative functions over G plays a prominent role in the finite dimensional
representation theory of G (e.g., [4]). From the point of view of the algebraic theory
of Hopf algebras, Rk(G) can be considered as the dual Hopf algebra k[G]
of the group
algebra k[G]. In fact, the construction of the dual coalgebra A


Source: Abuhlail, Jawad Younes - Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals


Collections: Mathematics