Quantum Affine Ortho-Symplectic Superalgebras

• Quantum affine ortho-symplectic superalgebras are quantized enveloping superalgebras unifying affine, super, and ortho-symplectic structures through distinct algebraic presentations.
• They exhibit explicit isomorphisms between Chevalley–Serre, Drinfeld, and RTT realizations via braid group actions and quantum root vector constructions.
• These algebras underpin integrable supersymmetric models and facilitate precise classification of oscillator modules and tensor product fusion rules.

Quantum affine ortho-symplectic superalgebras constitute a distinguished class of quantized enveloping superalgebras associated to the affinization of $\mathfrak{osp}(M|N)$, unifying the structural features of quantum affine algebras, superalgebras, and the intricate combinatorics of ortho-symplectic (type B/C/D) series. These objects play a central role in the paper of quantum integrability, representation theory, and categorification corresponding to super-symmetric systems. Their definition, structure theory, and representation theory expand the quantum group formalism into the superalgebraic domain, exhibiting highly nontrivial behaviors arising from parity, especially relating to the existence of a single odd simple root and its associated fermionic relations.

1. Foundational Data and Algebraic Presentations

The quantum affine ortho-symplectic superalgebra $U_q(\mathfrak{osp}(2m+1|2n)^{(1)})$ is realized in several, provably equivalent, presentations, each tailored to a particular aspect of the theory.

• Drinfeld–Jimbo (Chevalley-Serre) presentation. Generators are $X_i^\pm$ and $K_i^{\pm1}$ for $i=0,1,\dots,m+n$, with $[X_i^\pm]=[\alpha_i]$ (parity of the corresponding simple root; all even except $\alpha_n$). Defining relations comprise standard Cartan, $q$-commutator, and $q$-Serre relations; the latter include higher-order constraints for links adjacent to the parity-odd node and are displayed in full in (Wu et al., 29 Oct 2024). The algebra admits a triangular decomposition $U_q(\osp^{(1)})\cong U_q^-\otimes U_q^0\otimes U_q^+$.
• Drinfeld (current) realization. The generators are $x_{i,r}^\pm$, $h_{i,r}$ ($r\in\mathbb{Z}, i=1,\dots,m+n$), and central/degree generators $\gamma^{\pm1/2}$ and $d$; $[x_{i,r}^{\pm}]=[\alpha_i]$ and $[h_{i,r}]=0$. The current relations (Cartan, root–root, $q$-Serre) encode the quantum affinization explicitly, with generating functions $x_i^\pm(u)=\sum_r x_{i,r}^{\pm}u^{-r-1}$ and $\psi_i^{\pm}(u)$.
• $R$-matrix (RTT) realization. The algebra $U(R)$ is generated by entries $l_{ij}^{\pm}[r]$ of two $\mathbb{Z}_2$-graded matrix power series $L^\pm(z)$ acting on $V\cong\mathbb{C}^{2m+1|2n}$. The defining relations are encoded by the RTT relation with the explicit ortho-symplectic $R$-matrix $R(z)$ and additional "unitarity" conditions. The Gaussian decomposition of $L^\pm(z)$ relates this realization directly to the Drinfeld currents (Wu et al., 14 Jun 2024).

A remarkable feature is the existence of explicit isomorphisms between these presentations for all $m\geq 1$, constructed via a combination of braid group action and root vector calculus, providing a unified algebraic framework (Wu et al., 29 Oct 2024, Wu et al., 14 Jun 2024).

2. Braid Group Action and Quantum Root Vector Construction

The affine Weyl group $W$ associated to $\mathfrak{osp}(2m+1|2n)^{(1)}$ acts through a family of Lusztig-type braid operators $T_i$ on the Chevalley generators, enabling the systematic construction of all quantum root vectors. The explicit action is: $$T_i(X_i^+)= -d_i^{-1}X_i^-K_i,\quad T_i(X_i^-)= -K_i^{-1}d_iX_i^+,\quad T_i(K_j)=K_{s_i(\alpha_j)}$$ with recursive formulas for $T_i(X_j^\pm)$ for $j\neq i$ (see (Wu et al., 29 Oct 2024), eq. 3.12).

Given any real positive root $\beta=s_{i_1}\cdots s_{i_k}(\alpha_j)$, the positive root vector $E_\beta$ is defined as the successive action of braid operators on $X_j^+$. Invariance under the $q$-commutator structure yields a PBW-type basis for $U_q^+$, and the construction extends via Chevalley involution to $U_q^-$. In the Drinfeld picture, these root vectors correspond to Fourier modes of iterated $q$-commutators of current generators.

The braid-group action is also crucial for proving the isomorphism between the Drinfeld–Jimbo and Drinfeld realizations, as it facilitates the explicit translation between root vectors expressed in either presentation (Wu et al., 29 Oct 2024).

3. Isomorphism of Presentations

A central result is the existence of a unique superalgebra isomorphism between the Drinfeld (new/current) and the Drinfeld–Jimbo (Chevalley) presentations: $$U_q[\mathfrak{osp}(2m+1|2n)^{(1)}]_{\text{Drinfeld new}} \cong U_q[\mathfrak{osp}(2m+1|2n)^{(1)}]_{\text{Drinfeld–Jimbo}}$$ This isomorphism is constructed on generators as

$$\Phi(x_{i,r}^+) = \gamma^{r/2} \sum_{s\ge0}(-1)^{s[\alpha_i]} E_{\alpha_i + (r-s)\delta} K_i^{-1-r_a}$$

(similar expressions for $x_{i,r}^-$ and $h_{i,r}$, with $r_a=(\alpha_i,\alpha_i)r/2$ and $\delta$ the primitive imaginary root).

The map $\Phi$ preserves all current relations (Drinfeld–Cartan, current–current, Serre), and a straightforward computation yields its inverse on the Chevalley generators. This explicit identification degree-by-degree validates the full equivalence of the two presentations, unifying the Chevalley–Serre and current-type approaches (Wu et al., 29 Oct 2024, Wu et al., 14 Jun 2024).

Further, the $R$-matrix (RTT) realization is also shown to be isomorphic to the Drinfeld realization via identification of Drinfeld currents with local matrix elements in the Gaussian decomposition of $L^\pm(z)$, confirming a triad of mutually isomorphic algebraic frameworks for $\mathfrak{osp}(2m+1|2n)^{(1)}$ (Wu et al., 14 Jun 2024).

4. Structure of Representations and Oscillator Modules

Finite-dimensional representations of quantum affine ortho-symplectic superalgebras are classified as level-0 highest $\ell$-weight modules, determined by rational Drinfeld polynomials $P_i(x)$ via: $$\Psi_i^+(x) = q_i^{\deg P_i} \frac{P_i(xq_i^{-1})}{P_i(xq_i)},\quad \Psi_i^-(x) = q_i^{-\deg P_i}\frac{P_i(xq_i)}{P_i(xq_i^{-1})}$$ (Xu et al., 2018). All finite-dimensional simples are tensor products of evaluation modules — images of finite-type $\mathfrak{osp}(M|N)$-modules under the evaluation homomorphism, up to parity automorphisms (Xu et al., 2018, Xu et al., 2016).

A significant development is the construction of $q$-oscillator representations and their fusion products for quantum affine D/C-type superalgebras, yielding a new family of irreducibles whose highest weights correspond to explicit combinatorial labels (partitions, hook partitions). The oscillator modules interpolate between infinite-dimensional representations and finite-dimensional irreducibles under categorically exact monoidal functors, paralleling classical Howe duality at the quantum affine level (Kwon et al., 2023). The category of such modules is closed under tensor/fusion product, and their $q$-characters admit closed combinatorial expressions.

The equivalence of strict tensor categories between $U_q(\mathfrak{osp}(1|2n)^{(1)})$ and $U_{-q}(\mathfrak{so}(2n+1)^{(1)})$ — a quantum manifestation of Rittenberg–Scheunert duality — implies that all fusion rules, $q$-characters, and even R-matrices coincide under $q\rightarrow -q$ (Xu et al., 2016).

5. Parity Effects, Braid Relations, and Root System Phenomena

Unlike non-super quantum affine algebras, the presence of a fermionic simple root (the so-called "spin" node) introduces novel algebraic phenomena:

• Parity-dependent $q$-Serre relations. If node $i$ is odd, the square of the associated generator vanishes: $x_i^\pm(z)^2=0$. All commutators and $q$-Serre relations are modified by parity sign factors $(-1)^{[\alpha_i][\alpha_j]}$.
• Distinguished Dynkin diagram. The super case uses a diagram of type $B^{(1)}(0,n)$, with exactly one parity-odd node. This is essential for delineating the precise $q$-commutator structure and extra fermionic Serre constraints (Wu et al., 14 Jun 2024, Wu et al., 29 Oct 2024).
• Braid group relations. The braid operators satisfy the usual Artin and affine-braid relations (e.g., $T_iT_j=T_jT_i$ for $|i-j|>1$), with parity corrections dictated by the underlying root parities.
• Root vector organization. The PBW bases constructed by root vectors — recursively built via braid group action — have graded structure and parity-dependent $q$-commutation.

These features yield nontrivial constraints for both the structure of the algebra and its representation category, especially in the classification of integrable and spinor-type modules (Xu et al., 2018).

6. Character Formulas and Folding Phenomena

Recent advances provide closed character formulas for Kirillov-Reshetikhin (KR) modules of quantum affine ortho-symplectic superalgebras via folding of supercharacters from $\mathfrak{gl}(M|N)$.

• Supersymmetric Schur functions and Cauchy identities. Supercharacters are expressed by supersymmetric Schur functions $s_\lambda(X|Y)$, determinantal formulas built from generating series of complete supersymmetric functions. For ortho-symplectic types, Jacobi–Trudi-like determinant modifications yield B-type and C-type supercharacters $s_{[\lambda]}(X|Y)$ and $s_{\langle\lambda\rangle}(X|Y)$ (Tsuboi, 9 Dec 2025).
• Folding procedure. Algebraic folding (reduction) of $\mathfrak{gl}(M|N)$ supercharacters — by identification of variables and diagram automorphisms — gives rise to supercharacters and KR characters for the quantum affine ortho-symplectic and twisted types. The resulting character decompositions obey explicit combinatorial laws with multiplicity one.
• KR modules. The character of a KR module $W_m^{(a)}$ for untwisted B-type $U_q(\mathfrak{osp}(2r+1|2s)^{(1)})$ is

$$\mathrm{ch}\, W^{(a)}_{(1+\delta_{ar})m} = S_{(m^a)}(x_1,\dots,x_r,x_1^{-1},\dots,x_r^{-1}\mid \{-1\})$$

(similar formulas for other types), confirming conjectures from Bethe ansatz and providing closed combinatorial formulae for all such modules (Tsuboi, 9 Dec 2025).

7. Applications and Open Directions

Quantum affine ortho-symplectic superalgebras underpin the algebraic theory of integrable supersymmetric lattice models and serve as symmetry algebras in quantum statistical mechanics with "super" symmetry. The $R$-matrix presentations guarantee solutions to graded Yang–Baxter equations and explicit universal $R$-matrices (modulo orderings).

Evaluation homomorphisms and explicit $R$-matrices facilitate the construction of integrable representations and ensure compatibility with physical transfer matrices. The formalism robustly generalizes the ordinary quantum affine BCD-theory to superanalogues, with all fusion and spectral properties preserved modulo the $q\mapsto -q$ correspondence for the $\mathfrak{osp}(1|2n)$ family (Xu et al., 2016).

Emerging directions include further combinatorial and categorification results, the extension to twisted and twisted-parity types, and applications to supersymmetric gauge theory and quantum Schur–Weyl duality.

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Key references: (Wu et al., 29 Oct 2024, Wu et al., 14 Jun 2024, Tsuboi, 9 Dec 2025, Kwon et al., 2023, Xu et al., 2018, Xu et al., 2016, Xu et al., 2016).