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This article or section of article is sourced from:Wikipedia


A bialgebra over a field K is defined as a set equipped with the structure of a unital associative algebra over K as well as a coalgebra over K satisfying certain compatibility conditions.


  • Let B be the set and K be the underlying field
  • Let \nabla denote the multiplication and \eta the unit of B (for its algebra structure)
  • Let \Delta denote the comultiplication and \epsilon the counit of B.
  • Let \tau be the unique linear map from B \otimes B to itself that sends each pure tensor x \otimes y to y \otimes x.

Compatibility conditions

The compatibility conditions are as follows:

  • Compatibility between multiplication and comultiplication:

\Delta \circ \nabla = (\nabla \otimes \nabla) \circ (id \otimes \tau \otimes id) \circ (\Delta \otimes \Delta)

  • Compatibility between multiplication and counit:

\epsilon \otimes \epsilon = \epsilon \circ \nabla under the canonical identification of K \otimes K with K.

  • Compatibility between comultiplication and unit:

\Delta \circ \eta = \eta \otimes \eta

  • Compatibility between unit and counit:

\epsilon \circ \eta = id

Related notions

Stronger notions

Weaker notions