Cyclotomic identity
Encyclopedia
In mathematics
, the cyclotomic identity states that
where M is Moreau's necklace-counting function,
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 polynomial
s.
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:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the cyclotomic identity states that
where M is Moreau's necklace-counting function,
and μ(·) is the classic Möbius function
Möbius function
The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
of number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
The name comes from the denominator, 1 − z j, which is the product of cyclotomic polynomial
Cyclotomic polynomial
In algebra, the nth cyclotomic polynomial, for any positive integer n, is the monic polynomial:\Phi_n = \prod_\omega \,where the product is over all nth primitive roots of unity ω in a field, i.e...
s.
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
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
of the free Lie algebra
Free Lie algebra
In mathematics, a free Lie algebra, over a given field K, is a Lie algebra generated by a set X, without any imposed relations.-Definition:Given a set X, one can show that there exists a unique free Lie algebra L generated by X....
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: