Noncommutative symmetric function

In mathematics, the noncommutative symmetric functions form a Hopf algebra NSymm analogous to the Hopf algebra of symmetric functions. The Hopf algebra NSymm was introduced by Israel M. Gelfand, Daniel Krob, and Alain Lascoux et al. (1995). It is noncommutative but cocommutative graded Hopf algebra. It has the Hopf algebra of symmetric functions as a quotient, and is a subalgebra of the Hopf algebra of permutations, 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 of the free Lie algebra on countably many variables.

Definition

The underlying algebra of the Hopf algebra of symmetric functions is the free ring Z⟨Z₁,Z₂,...⟩ generated by non-commuting variables Z₁,Z₂,...

The coproduct takes Z_n to ΣZ_i⊗Z_n–i, where Z₀=1 is the identity.

The counit takes Z_i to 0 for i>0 and takes Z₀=1 to 1.

References

• Gelfand, Israel M.; Krob, Daniel; Lascoux, Alain; Leclerc, Bernard; Retakh, Vladimir S.; Thibon, Jean-Yves (1995), "Noncommutative symmetric functions", Adv. Math., 112 (2): 218–348, doi:10.1006/aima.1995.1032, MR 1327096