# Group rings in Physics

In my institute a few mathematicians work on Group Ring. Since it has close connection with representation theory, I thought that there must be some interesting connections of it with Physics. However I did not manage to find any information from our local experts; perhaps for one reason, they are more interested on finite groups and its structure. Hence I am asking the same question to the community. Does there exists any connection between group rings and physics? (I am having a feeling, that it can be used as a tool for solving representation theoretic problems. But does its usefulness (if there is any) end there only?)

Advanced thanks for any reply. Also please provide some references (paper or books), which describes such connection(s), if exists,

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Well, you can do physics over a ring (See arXiv:0809.0086 for example)...but I don't know whether this is exactly what a physicist might mean by the term "group ring"... –  Alex Nelson Jun 30 '12 at 18:49

Well I suppose you are not expecting to see physics over a group ring, so I'll tell the only example I know. One can find it in Chamseddine-Connes-Marcolli noncommutative geometrical derivation of the Standard Model. This model is based on the concept of almost commutative spectral triples (which are, by definition, a product of an ordinari 4-dimensional spin manifold and a finite geometry.) The algebra they use as part of the finite geometry is $$\mathcal{A}_f=\mathbb{C}\oplus \mathbb{H}\oplus \mathbb{H}\oplus M_3(\mathbb{C}),$$ whose complexification is the same of the truncation $\bigoplus_{i=1}^3 M_i(\mathbb{C})$ of the group ring of $SU(2)$.

The reference is $\S 13$ of "Noncommutative Geometry, Quantum Fields and Motives" by A.Connes and M.Marcolli.

I hope this helps; if totally unrelated, then point out specificaly the direction of physics, where you want to find that group ring structure. Tangentially as well, if the ring $R$ is commutaive $R[G]$ is a group algebra. I am not well-versed on algebraic geometry, but I am sure you can find such structures there (in the construction of toric varieties perhaps [see Fultons book on the subject]), and perhaps you might find physical applications, e.g. http://arxiv.org/abs/hep-th/9711036.

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The group ring and the associated $C^*$-algebra is a basic tool for the analysis of representations of groups.

But physics if done on this abstract level is phrased in terms of $C^*$-algebras (e.g. algebraic quantum field theory) and not in terms of group rings, which only serve as simple examples.

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