Applications of Algebra: Groups, Rings, Ideals

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SUMMARY

The discussion centers on the applications of algebraic structures such as groups, rings, and ideals, particularly in the context of physics. Participants highlight that while groups have clear applications, the relevance of rings and ideals remains less understood. Notably, group rings and Lie algebras are identified as significant in physical theories, with rings of operators on Hilbert spaces being a crucial concept. The conversation emphasizes the need for practical examples to enhance comprehension of these abstract algebraic concepts.

PREREQUISITES
  • Understanding of abstract algebra concepts such as groups, rings, and ideals
  • Familiarity with Lie algebras and their applications in physics
  • Knowledge of Hilbert spaces and operator theory
  • Basic comprehension of topological quantum field theories
NEXT STEPS
  • Research the concept of group rings and their applications in physics
  • Explore the role of Lie algebras in quantum mechanics
  • Study the properties of operators on Hilbert spaces
  • Investigate topological quantum field theories and their mathematical foundations
USEFUL FOR

Mathematicians, physicists, and students interested in the practical applications of abstract algebra, particularly in the fields of theoretical physics and quantum mechanics.

matness
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always abstract?

things are really very abstract in algebra as its name.But it would be nicer to learn some applications especially to physics
apperently there are applications about groups but i could not find anything related to rings ideals etc.
discussing the applications would make it more meaningfull at least for me

thanks in advance
n:bugeye:
 
Physics news on Phys.org
Read jon baez's this weeks finds. there are too many things to talk about here, and we have no idea what you do and do not understand. for instance does the phrase: a topological quantum field theory is esseentially a monoidal category with a functor to a category of riemann surfaces and cobordisms mean anything to you? i suspect not; it means little to me.

the group applications are an application of rings (the group ring) and lie algebras are used in physics, you look at the ring of operators on a hilbert space, rings occur everywhere because matrices are rings.
 

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