**Formal Definition**

Formally, a Hopf algebra is a (associative and coassociative) bialgebra *H* over a field *K* together with a *K*-linear map *S*: *H* → *H* (called the **antipode**) such that the following diagram commutes:

Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless Sweedler notation, this property can also be expressed as

As for algebras, one can replace the underlying field *K* with a commutative ring *R* in the above definition.

The definition of Hopf algebra is self-dual (as reflected in the symmetry of the above diagram), so if one can define a dual of *H* (which is always possible if *H* is finite-dimensional), then it is automatically a Hopf algebra.

Read more about this topic: Hopf Algebra

### Other articles related to "formal definition":

**Formal Definition**

... The above controller for crosswalk lights can be modeled by an atomic SP-DEVS model ... Formally, an atomic SP-DEVS is a 7-tuple where is a finite set of input events is a finite set of output events is a finite set of states is the initial state is the time advanced function which defines the lifespan of a state where is the set of non-negative rational numbers plus infinity ...

... In typical usage, the

**formal definition**of O notation is not used directly rather, the O notation for a function f(x) is derived by the following simplification rules If f(x) is a sum of several terms ... One may confirm this calculation using the

**formal definition**let f(x) = 6x4 − 2x3 + 5 and g(x) = x4 ... Applying the

**formal definition**from above, the statement that f(x) = O(x4) is equivalent to its expansion, for some suitable choice of x0 and M and ...

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