Cyclotomic identity

In mathematics, the cyclotomic identity states that

${\displaystyle {1 \over 1-\alpha z}=\prod _{j=1}^{\infty }\left({1 \over 1-z^{j}}\right)^{M(\alpha ,j)}}$

where M is

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

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:

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

References

• Metropolis, N.;
  MR 0777692