In mathematics, the noncommutative symmetric functions form a

Hopf algebra Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedis ...

NSymm analogous to the

Hopf algebra of symmetric functions In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which ...

. 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 the University of Marne la Vallée and Nankai University. His research was primarily in algebraic combinatorics, particularly Hecke algebras and Young tableaux. L ...

, Bernard Leclerc,

Vladimir Retakh Vladimir Solomonovich Retakh (russian: Ретах Владимир Соломонович; 20 May 1948) is a Russian-American mathematician who made important contributions to Noncommutative algebra and combinatorics among other areas. Biogr ...

, 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\l ...

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 of ...

, and is the graded dual of the Hopf algebra of

quasisymmetric function In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring ge ...

. 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 representati ...

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 :

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 In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. ...

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 ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...

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