Fuzzy sphere

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. 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(2). They satisfy the relations [ J a , J b ] = i ϵ a b c J c {displaystyle =iepsilon _{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(2) 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 ( j 2 − 1 ) {displaystyle 4r^{4}=k^{2}(j^{2}-1)} , 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.