Quantum Algebra

[xStFtx]

I have been researching Quantum Algebra for a few days. I have a few questions about it in hopes that people can help me on here.

1. How do quantum groups differ from classical groups in mathematics?

2. Can you explain the concept of Hopf algebras and their significance in quantum algebra?

 

[joigus]

Quantum groups are deformations of Lie groups themselves in the space of parameters. They're of concern mainly to mathematicians or very mathematically-minded mathematical physicists. Related to algebraic topology.

I don't know what you mean by "classical groups". Finite groups? Lie groups? Groups relevant to classical mechanics only?

I don't know what you mean by "quantum algebra". Seems to be some kind of umbrella term for all the tinkering tools somehow related to quantum mechanics, quantum field theory, and the like.

For Hopf algebras I would recommend you more specialised forums, like Mathoverflow or MathStackExchange, after you're through with the obvious sources you can find on the internet.

 

[studiot]

1 hour ago, xStFtx said:

I have been researching Quantum Algebra for a few days. I have a few questions about it in hopes that people can help me on here.

1. How do quantum groups differ from classical groups in mathematics?

2. Can you explain the concept of Hopf algebras and their significance in quantum algebra?

 

 

 

1)  There is no difference. Groups are sets with a suitable associative binary operation. Some groups have additional structure, eg abelian groups, which have a commutativity requirement. Non commutativity is very important in QM and leads to the uncertainty principle.

 

Try this postgrad book.

 

 

2)  Hopf algebras also explot non commutativity.

 

https://www.theoremoftheday.org/MathsStudyGroup/SeligHopf.pdf

 

[xStFtx]

Ok thank you for the replies. I have been reading on hopf and so on. And I read a few of the papers daily on arxiv.org under the Quantum Algebra topic to further extend my knowledge.

[joigus]

More info:

Quote

Intuitive meaning[edit]

The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of the noncommutative geometry of Alain Connes. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum Yang–Baxter equation and quantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev, Leon Takhtajan, Evgeny Sklyanin, Nicolai Reshetikhin and Vladimir Korepin) and related work by the Japanese School.^[1] The intuition behind the second, bicrossproduct, class of quantum groups was different and came from the search for self-dual objects as an approach to quantum gravity.^[2]

From: https://en.wikipedia.org/wiki/Quantum_group

BTW, you didn't answer. What do you mean by "classical groups"? As in Hermann Weyl's "classical groups"?

https://en.wikipedia.org/wiki/Classical_group

That's different from "classical" as opposed to "quantum" as used in physics.

In that sense, "classical groups" are "rigid" or "static", while quantum groups "flow" from one to another by varying the parameters.

And that's practically all I can tell you.

[xStFtx]

Thank you for all the resources. Classical in the sense like you said earlier lie groups and so on. But I understand now. I mixed them up in my head. My fault.

[xStFtx]

What does Sedenion-like Associative mean?

 

[joigus]

50 minutes ago, xStFtx said:

What does Sedenion-like Associative mean?

 

Sedenions are non-associative. They're also the first algebra you can build with the Cayley-Dickson construction that is not a division algebra. Ie, it has zero divisors. They're some kind of generalisation of complex numbers. @studiot can probably tell you more.

Meanwhile,

https://en.wikipedia.org/wiki/Sedenion

https://en.wikipedia.org/wiki/Cayley–Dickson_construction

And a nice 30-min video by Michael Penn that I recommend,