## What is it about?

Several fields of mathematics have developed in total isolation, using their own ‘undecipherable’ coded languages. Mathematicians have long sought effective ways to bridge these isolated mathematical continents. With the present study, a new tool called “big algebras” works as a two-way ‘dictionary’ between symmetry, algebra, and geometry. With big algebras, mathematicians can visualize abstract concepts about symmetries with the help of algebraic geometry, making them more tangible. In a broader context, big algebras find direct applications in the math behind quantum physics and number theory, and could thus strengthen the connection between these distant mathematical worlds

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## Why is it important?

Continuous symmetry groups, such as the rotation of a circle or a sphere, are mathematically represented by matrices—rectangular arrays of numbers—which can convert the properties of the mathematical object into linear algebra. While some continuous symmetry groups are ‘commutative,’ meaning that the order of symmetric transformations always results in the same outcome, others are ‘non-commutative,’ leading to different outcomes when the order of the symmetric transformations is changed. So far, non-commutative symmetry groups have been represented by non-commutative matrices, i.e., matrices in which the order of operations influences the end result. However, this approach does not allow a geometric interpretation as the geometry of non-commutative algebras is not yet well understood. On the other hand, commutative algebras can be well understood through their geometry. A “big algebra” is a commutative ‘translation’ of a non-commutative matrix algebra. Thus, it allows using algebraic geometry techniques to visualize abstract concepts about non-commutative continuous symmetry groups. The math behind quantum physics makes extensive use of matrices. However, these matrices are typically ‘non-commutative,’ which poses a problem in algebra and algebraic geometry. With big algebras, the information initially enclosed within non-commutative matrices can be decoded and represented geometrically to reveal their hidden properties. In addition, big algebras not only reveal relationships between related symmetry groups but also when their so-called “Langlands duals” are related—a central concept in the purely mathematical world of number theory. Thus, big algebras could relate the Langlands duality in number theory with quantum physics.

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## Read the Original

This page is a summary of: Commutative avatars of representations of semisimple Lie groups, Proceedings of the National Academy of Sciences, September 2024, Proceedings of the National Academy of Sciences,

DOI: 10.1073/pnas.2319341121.

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