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In mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is generated by spherical harmonics whose spin l is at most equal to some j. The terms in the product of two spherical harmonics that involve spherical harmonics with spin exceeding j are simply omitted in the product. This truncation replaces an infinite-dimensional commutative algebra by a j 2 {\displaystyle j^{2}} -dimensional non-commutative algebra.
The simplest way to see this sphere is to realize this truncated algebra of functions as a matrix algebra on some finite-dimensional vector space.Take the three j-dimensional matrices J a , a = 1 , 2 , 3 {\displaystyle J_{a},~a=1,2,3} that form a basis for the j dimensional irreducible representation of the Lie algebra su. They satisfy the relations = i ϵ a b c J c {\displaystyle =i\epsilon _{abc}J_{c}} , where ϵ a b c {\displaystyle \epsilon _{abc}} is the totally antisymmetric symbol with ϵ 123 = 1 {\displaystyle \epsilon _{123}=1} , and generate via the matrix product the algebra M j {\displaystyle M_{j}} of j dimensional matrices. The value of the su Casimir operator in this representation is
where I is the j-dimensional identity matrix.Thus, if we define the 'coordinates' x a = k r − 1 J a {\displaystyle x_{a}=kr^{-1}J_{a}} where r is the radius of the sphere and k is a parameter, related to r and j by 4 r 4 = k 2 {\displaystyle 4r^{4}=k^{2}} , then the above equation concerning the Casimir operator can be rewritten as
which is the usual relation for the coordinates on a sphere of radius r embedded in three dimensional space.