[Paper Review] Drinfeld realisations of quantum affine superalgebras

Ying Xu, R. B. Zhang|arXiv (Cornell University)|Nov 19, 2016

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper constructs Drinfeld realizations for quantum affine superalgebras of types $ m ilde{osp}(1|2n)^{(1)}$, $ m ilde{sl}(1|2n)^{(2)}$, and $ m ilde{osp}(2|2n)^{(2)}$, establishing a superalgebra isomorphism between the Drinfeld realization and the standard Drinfeld-Jimbo presentation via quantum correspondences and reduction to ordinary quantum affine algebras. The key contribution is the first systematic construction of Drinfeld realizations for these non-simply-laced, non-isotropic-odd-root quantum affine superalgebras.

ABSTRACT

We construct Drinfeld realisations for the quantum affine superalgebras associated with the osp(1|2n)^{(1)}, Sl(1|2n)^{(2)} and osp(2|2n)^{(2)} series of affine Lie superalgebras.

Motivation & Objective

• To extend Drinfeld realizations—previously known only for untwisted quantum affine superalgebras of type $A$ and $D(2,1;\alpha)$—to a broader class of quantum affine superalgebras.
• To address the lack of Drinfeld realizations for quantum affine superalgebras without isotropic odd roots, particularly for twisted and non-simply-laced types.
• To establish a superalgebra isomorphism between the Drinfeld realization and the standard Drinfeld-Jimbo presentation for these algebras.
• To utilize quantum correspondences and known isomorphisms between ordinary quantum affine algebras to lift the construction to the superalgebra setting.
• To provide a Hopf superalgebra isomorphism that preserves grading and algebraic structure, enabling further study of representations and integrable models.

Proposed method

• Construct the Drinfeld realization $\mathcal{U}^D_q(\mathfrak{g})$ for each quantum affine superalgebra $\mathcal{U}_q(\mathfrak{g})$ associated with $\mathfrak{g} = \tilde{osp}(1|2n)^{(1)}$, $\tilde{sl}(1|2n)^{(2)}$, and $\tilde{osp}(2|2n)^{(2)}$.
• Employ quantum correspondences between affine Lie superalgebras and their ordinary affine Lie algebra counterparts to relate the superalgebra structures to known quantum affine algebras.
• Use the isomorphism between Drinfeld realizations and standard presentations in the ordinary quantum affine algebra case (via Drinfeld's theorem) to transfer the isomorphism to the superalgebra setting.
• Define generators $\xi^{\pm}_{i,r}$, $\kappa_{i,r}$, and $\gamma_i$ in the Drinfeld realization and verify the defining relations, including Serre relations and commutation relations.
• Verify that the map $\Phi = \varphi \circ \rho \circ \psi$ induces a superalgebra isomorphism by checking preservation of grading, relations, and Hopf superalgebra structure.
• Use the parity functor and supercommutator relations $[x,y]_a = xy - (-1)^{[x][y]} a yx$ to handle graded commutation and superalgebra consistency.

Experimental results

Research questions

• RQ1 Can Drinfeld realizations be systematically constructed for quantum affine superalgebras of types $\tilde{osp}(1|2n)^{(1)}$, $\tilde{sl}(1|2n)^{(2)}$, and $\tilde{osp}(2|2n)^{(2)}$?
• RQ2 Is there a superalgebra isomorphism between the Drinfeld realization and the standard Drinfeld-Jimbo presentation for these quantum affine superalgebras?
• RQ3 Can quantum correspondences between Lie superalgebras and ordinary Lie algebras be used to lift known isomorphisms from the non-super to the super setting?
• RQ4 How do the Serre relations and commutation relations behave in the Drinfeld realization of these quantum affine superalgebras?
• RQ5 Can the Hopf superalgebra structure be preserved under the constructed isomorphism?

Key findings

• The paper constructs Drinfeld realizations $\mathcal{U}^D_q(\mathfrak{g})$ for quantum affine superalgebras of types $\tilde{osp}(1|2n)^{(1)}$, $\tilde{sl}(1|2n)^{(2)}$, and $\tilde{osp}(2|2n)^{(2)}$.
• A superalgebra isomorphism $\Phi: \mathcal{U}_q(\mathfrak{g}) \to \mathcal{U}^D_q(\mathfrak{g})$ is established in Theorem 2.5, proving equivalence between the standard and Drinfeld presentations.
• The isomorphism is shown to preserve the Hopf superalgebra structure, as confirmed in Remark 2.6.
• The proof relies on quantum correspondences and reduction to known isomorphisms in ordinary quantum affine algebras, leveraging Drinfeld's theorem.
• The Serre relations in the Drinfeld realization are verified using twisted commutators and the structure of the quantum correspondences.
• The construction provides a foundation for studying integrable representations and vertex operator realizations in these quantum affine superalgebras.