Yin-Yang of Hopf Algebra

Prof Heinz Hopf invented this new Hopf Algebra during WW II, it remained unnoticed for years after he & his wife's tragic death by the Nazi.

The essence of Hopf Algebra is a weird 'Yin-Yang' view of every algebra concept.
Examples:
Associative co-Associative
Commutative co-Commutative
Algebra co-Algebra
Multiplication co-Multiplication
Unit co-Unit
Homomorphism co-Homomorphism
Monoïd co-Monoïd
...
the 'co-' notion represents the 'Yin' opposite of the 'Yang' notion.

[Analogy: Set & coSet 倍集， l'ensemble à gauche ou à droite]

Hopf Algebra is applied in Quantum Group for Quantum Physics, Yang-Baxtor equation, Combinatorics and Topology, etc.

If you find a structure which has a combined properties of Group, Ring and Vector Space, then it could be a Hopf Algebra.

Applied it in Graphs, we can re-discover the Euler's (E-V+F = 2 ) formula, Möbius function, and Symmetric Group (Sn).

Note: Excellent 30-lecture series by Prof Federico Ardila (San Francisco University) on Hopf Algebra:

http://math.sfsu.edu/federico/Clase/Hopf/lectures.html