A quantum shuffle approach to quantum affine super algebra of type $C(2)^{(2)}$ and its equitable presentation

In this study, we focus on the positive part $U_q^{+}$ of the quantum affine superalgebra $U_q(C(2)^{(2)})$. This algebra admits a presentation with two two generators $e_α$ and $e_{δ-α}$, which satisfy the cubic $q$-Serre relations. According to the work of Khoroshkin-Lukierski-Tolstoy, the Damiani and the Beck $PBW$ bases exist for this superalgebra. In this paper, we utilize the $q$-shuffle superalgebra and Catalan words to present these two bases in a closed-form expression. Ultimately, we present the bosonization of $U_q(C(2)^{(2)})$.

💡 Research Summary

The paper investigates the positive part U⁺₍q₎ of the quantum affine superalgebra U₍q₎(C(2)^{(2)}), which is generated by two odd elements e₍α₎ and e₍δ−α₎ satisfying cubic q‑Serre relations. The authors’ main goal is to embed U⁺₍q₎ into a suitable q‑shuffle superalgebra, to express known PBW bases (Damiani‑type and Beck‑type) in closed combinatorial form, and to obtain a minimal bosonization together with an equitable (balanced) presentation.

First, the algebraic background is reviewed. The superalgebra U₍q₎(C(2)^{(2)}) is defined by Chevalley generators k_{±1}^γ, e_{±α}, e_{±(δ−α)} together with the usual q‑commutation relations, parity function ϑ, and the q‑super‑commutators. Its positive subalgebra U⁺₍q₎ is generated by A:=e₍α₎ and B:=e₍δ−α₎, which obey the cubic q‑Serre identities A⋆A⋆A⋆B + {3}q A⋆A⋆B⋆A − {3}q A⋆B⋆A⋆A − B⋆A⋆A⋆A = 0, and the analogous one with A and B interchanged. The Damiani PBW generators E{nδ+α₀}, E{nδ+α₁}, E_{nδ} are introduced recursively (formulas (8)–(10)) and shown to commute for different n. Theorem 5 establishes a PBW basis consisting of ordered monomials in these generators.

The core of the work is the construction of a q‑shuffle superalgebra V. Starting from the free associative algebra on two non‑commuting letters x and y, the authors define a graded bilinear form and a q‑shuffle product “⋆” that extends the Rosso–Green shuffle to the super setting by inserting a sign factor (−1)^{deg x deg y}. Explicit formulas (11)–(14) give the recursive definition of the product for arbitrary words. Proposition 10 verifies that the cubic q‑Serre relations hold in V when A↦x and B↦y, which yields an injective superalgebra homomorphism φ: U⁺₍q₎ → V (Corollary 11). The injectivity is proved by pairing the non‑degenerate bilinear form on U⁺₍q₎ with the canonical inner product on V, showing that the kernel of φ is trivial.

To obtain closed expressions for the PBW bases, the authors introduce Catalan words. A word in x and y of even length 2n is called Catalan if, after assigning weights \bar{x}=+1, \bar{y}=−1, the partial sums never become negative. The set Catₙ of all Catalan words of length 2n has cardinality the Catalan number Cₙ. For each n they define Cₙ = ∑{w∈Catₙ} w. Using the q‑shuffle product, they prove that the Damiani generators admit the compact formulas E{nδ+α₀}=q^{-2n}