Noncommutative Geometry and Representation Theory in Mathematical Physics

Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.

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Loop groups and noncommutative geometry

Reviews in Mathematical Physics ◽

10.1142/s0129055x17500295 ◽

2017 ◽

Vol 29 (09) ◽

pp. 1750029

Author(s):

Sebastiano Carpi ◽

Robin Hillier

Keyword(s):

Field Theory ◽

Conformal Field Theory ◽

Representation Theory ◽

Noncommutative Geometry ◽

Loop Group ◽

Positive Energy ◽

Loop Groups ◽

Spectral Triples ◽

Conformal Field ◽

Coset Models

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level [Formula: see text] projective unitary positive-energy representations of any given loop group [Formula: see text]. The construction is based on certain supersymmetric conformal field theory models associated with [Formula: see text] in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

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Approximations, ghosts and derived equivalences

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.120 ◽

2019 ◽

Vol 150 (2) ◽

pp. 813-840

Author(s):

Yiping Chen ◽

Wei Hu

Keyword(s):

Algebraic Geometry ◽

Representation Theory ◽

Noncommutative Geometry ◽

Endomorphism Rings ◽

Derived Equivalences ◽

Symmetric Approximation ◽

Quotient Rings ◽

Tilting Objects

AbstractApproximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly n-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.

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