Hopf Algebra - Quantum Groups and Non-commutative Geometry

Quantum Groups and Non-commutative Geometry

All examples above are either commutative (i.e. the multiplication is commutative) or co-commutative (i.e. Δ = T Δ where T: HHHH is defined by T(xy) = yx). Other interesting Hopf algebras are certain "deformations" or "quantizations" of those from example 3 which are neither commutative nor co-commutative. These Hopf algebras are often called quantum groups, a term that is so far only loosely defined. They are important in noncommutative geometry, the idea being the following: a standard algebraic group is well described by its standard Hopf algebra of regular functions; we can then think of the deformed version of this Hopf algebra as describing a certain "non-standard" or "quantized" algebraic group (which is not an algebraic group at all). While there does not seem to be a direct way to define or manipulate these non-standard objects, one can still work with their Hopf algebras, and indeed one identifies them with their Hopf algebras. Hence the name "quantum group".

Read more about this topic:  Hopf Algebra

Other articles related to "quantum groups, geometry, group, quantum group":

Hopf Algebras - Quantum Groups and Non-commutative Geometry
... These Hopf algebras are often called quantum groups, a term that is so far only loosely defined ... They are important in noncommutative geometry, the idea being the following a standard algebraic group is well described by its standard Hopf algebra of ... Hence the name "quantum group" ...

Famous quotes containing the words geometry, quantum and/or groups:

    ... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.
    Louise Bourgeois (b. 1911)

    A personality is an indefinite quantum of traits which is subject to constant flux, change, and growth from the birth of the individual in the world to his death. A character, on the other hand, is a fixed and definite quantum of traits which, though it may be interpreted with slight differences from age to age and actor to actor, is nevertheless in its essentials forever fixed.
    Hubert C. Heffner (1901–1985)

    Some of the greatest and most lasting effects of genuine oratory have gone forth from secluded lecture desks into the hearts of quiet groups of students.
    Woodrow Wilson (1856–1924)