mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a double affine Hecke algebra, or Cherednik algebra, is an

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

containing the Hecke algebra of an affine Weyl group, given as the quotient of the

group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...

of a double affine braid group. They were introduced by Cherednik, who used them to prove Macdonald's constant term conjecture for Macdonald polynomials. Infinitesimal Cherednik algebras have significant implications in

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

, and therefore have important applications in

particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...

and

chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

References

* * *A. A. Kirillo
Lectures on affine Hecke algebras and Macdonald's conjectures
Bull. Amer. Math. Soc. 34 (1997), 251–292. * Macdonald, I. G. ''Affine Hecke algebras and orthogonal polynomials.'' Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, 2003. x+175 pp. {{mr, 1976581 Algebras Representation theory