Cyclotomic identity explained

In mathematics, the cyclotomic identity states that

{ 1 \over 1- \alpha z} = \prod ^{inf ty\left({ 1} _{j =1} \over 1- z ^{j}\right)} ^{M(\alpha,j)}

where M is Moreau's necklace-counting function,

M(\alpha,n) = { 1 \over n}\sum _{d|n} \mu\left({n \over d}\right)\alpha ^{d,}

and μ is the classic Möbius function of number theory.

The name comes from the denominator, 1 - z ^j, which is the product of cyclotomic polynomials.

The left hand side of the cyclotomic identity is the generating function for the free associative algebra on α generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra on α generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic.

There is also a symmetric generalization of the cyclotomic identity found by Strehl:

\prod ^{inf ty\left({ 1} _{j =1} \over 1- \alpha z ^{j}\right)} ^{M(\beta,j)} = \prod ^{inf ty\left({ 1} _{j =1} \over 1- \beta z ^{j}\right)} ^{M(\alpha,j)}