Coradical

Coradical is a concept in coalgebra theory. For a coalgebra C over a field k, the coradical, denoted C0, is the sum of all simple subcoalgebras of C. It is the largest cosemisimple subcoalgebra of C and serves as the first layer in the coradical filtration of C.

The coradical filtration is an ascending chain of subspaces C0 ⊆ C1 ⊆ C2 ⊆ ... of C, defined inductively by C0 being the coradical and, for n ≥ 0, Cn+1 = { c ∈ C | Δ(c) ∈ C ⊗ Cn + Cn ⊗ C }. Equivalently, Cn+1 can be described as the set of elements whose comultiplication lands in the previous layer combined with the entire coalgebra. The union of all Ci equals C, and C0 is the starting point of this filtration.

In the finite-dimensional case, the coradical is the direct sum of all simple subcoalgebras. In general, it is the sum (not always a direct sum) of all simple subcoalgebras. The coradical filtration provides a framework for analyzing how a coalgebra is built from its simple components, and the associated graded coalgebra gr C = ⊕n (Cn/Cn−1) is often studied to understand the structure of C.

The coradical is conceptually dual to the radical in algebra theory. It plays a central role in the study of cosemisimple and pointed coalgebras, and in various decomposition and structural results within coalgebra and Hopf algebra theory.