In mathematics, the noncommutative symmetric functions form a

Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...

NSymm analogous to the Hopf algebra of symmetric functions. The Hopf algebra NSymm was introduced by Israel M. Gelfand, Daniel Krob,

Alain Lascoux Alain Lascoux (17 October 1944 – 20 October 2013) was a French mathematician at Université de Paris VII, University of Marne la Vallée and Nankai University. His research was primarily in algebraic combinatorics, particularly Affine Hecke alge ...

, Bernard Leclerc, Vladimir Retakh, and Jean-Yves Thibon. It is noncommutative but cocommutative graded Hopf algebra. It has the Hopf algebra of

symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...

s as a quotient, and is a subalgebra of the

Hopf algebra of permutations In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf algebra is a Hopf algebra with a basis of all elements of all the finite symmetric groups ''S'n'', and is a non-commutative analogue of the Hopf algebra o ...

, and is the graded dual of the Hopf algebra of quasisymmetric function. Over the rational numbers it is isomorphic as a Hopf algebra to the

universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representa ...

of the free Lie algebra on countably many variables.

Definition

The underlying algebra of the Hopf algebra of noncommutative symmetric functions is the free ring Z⟨''Z''₁, ''Z''₂,...⟩ generated by non-commuting variables ''Z''₁, ''Z''₂, ... The coproduct takes ''Z''_''n'' to Σ ''Z''_''i'' ⊗ ''Z''_{''n''–''i''}, where ''Z''₀ = 1 is the identity. The counit takes ''Z''_''i'' to 0 for ''i'' > 0 and takes ''Z''₀ = 1 to 1.

Related notions

Michiel Hazewinkel showed that a

Hasse–Schmidt derivation In mathematics, a Hasse–Schmidt derivation is an extension of the notion of a derivation. The concept was introduced by . Definition For a (not necessarily commutative nor associative) ring ''B'' and a ''B''-algebra ''A'', a Hasse–Schmidt de ...

D: A \to A t

on a ring ''A'' is equivalent to an action of NSymm on ''A'': the part

D_i : A \to A

of ''D'' which picks the coefficient of

t^i

, is the action of the indeterminate ''Z''_''i''.

Relation to free Lie algebra

The element Σ ''Z''_''n''''t''^''n'' is a group-like element of the Hopf algebra of formal

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

over NSymm, so over the rationals its logarithm is primitive. The coefficients of its logarithm generate the free Lie algebra on a countable set of generators over the rationals. Over the rationals this identifies the Hopf algebra NSYmm with the universal enveloping algebra of the free Lie algebra.

References

{{reflist Hopf algebras