Are we making finally more progress on the general theory of everything?
Lustig or Lusztig? Spelling error? (Hint: In German lustig means funny)
"Several fields of mathematics have developed in total isolation, using their own "undecipherable" coded languages. In a new study ... presents "big algebras," a two-way mathematical 'dictionary' between symmetry, algebra, and geometry, that could strengthen the connection between the distant worlds of quantum physics and number theory. ...
This new theorem is comparable to a "dictionary" that deciphers the most abstract aspects of mathematical symmetry using algebraic geometry. By operating at the intersection of symmetry, abstract algebra, and geometry, big algebras use more tangible geometric information to recapitulate sophisticated mathematical information about symmetries. ..."
From the significance and abstract:
"Significance
"Significance
Representations of continuous symmetry groups by matrices are fundamental to mathematical models of quantum physics and also to the Langlands program in number theory. Here, we attach a commutative matrix algebra, called big algebra, to a noncommutative irreducible matrix representation of a bounded continuous symmetry group. We show that the geometry of our commutative algebras captures sophisticated information of the representation, for example, its weight multiplicities. We have, and expect more, applications to polynomial identities between quantum numbers of baryon multiplets in particle physics, to mathematical problems related to Higgs fields in quantum physics and also to compatibility with Langlands duality in number theory.
Abstract
Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig’s q-weight multiplicity polynomials i.e., certain affine Kazhdan–Lusztig polynomials."
Fig 1 (untitled)
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