Cyclotomic identity

Template:Short description In mathematics, the cyclotomic identity states that

$\frac{1}{1-kz}=\prod _{j=1}^{\mathrm{\infty }}{\left(\frac{1}{1-z^j}\right)}^{M_j(k)}$

where M is Moreau's necklace-counting function,

$M_n(k)=\frac{1}{n}\sum _{d|n}\mu \left(\frac{n}{d}\right)k^d,$

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

Another interpretation is as the Hasse–Weil zeta function of the affine line over the finite field ${𝔽}_q$ with $k=q$ elements. The exponent $M_n(k)$ counts the number of number of maximal ideals of degree n in the coordinate ring ${𝔽}_q[x]$, i.e. the number of monic irreducible polynomials of degree n.

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

$\prod _{j=1}^{\mathrm{\infty }}{\left(\frac{1}{1-k{z}^j}\right)}^{M_j(\ell )}=\prod _{j=1}^{\mathrm{\infty }}{\left(\frac{1}{1-\ell z^j}\right)}^{M_j(k)}$

References

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