Bialgebra

Definition
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.

Notation

 * 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:

$$\epsilon \circ \eta = id$$

Stronger notions

 * Hopf algebra
 * Group bialgebra
 * Quantum group

Weaker notions

 * Associative algebra
 * Coalgebra