Cyclotomic identity - Alchetron, The Free Social Encyclopedia

In mathematics, the cyclotomic identity states that

1 1 − α z = ∏ j = 1 ∞ ( 1 1 − z j ) M ( α , j )

where M is Moreau's necklace-counting function,

M ( α , n ) = 1 n ∑ d | n μ ( n d ) α 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:

∏ j = 1 ∞ ( 1 1 − α z j ) M ( β , j ) = ∏ j = 1 ∞ ( 1 1 − β z j ) M ( α , j )

References

Cyclotomic identity Wikipedia

(Text) CC BY-SA