arXiv subjects

Sin-Myung Lee

Publications and source records attributed to Sin-Myung Lee.

4 recordsLinked to original sources

arXiv2026

On the theory of $q$-characters for quantum affine superalgebras of type $A$

We develop the theory of $q$-characters for quantum affine superalgebras of type $A$ in connection with deformed Cartan matrices. To achieve this, we establish a Khoroshkin-Tolstoy-type multiplicative formula of the universal $R$-matrix of the associated generalized quantum group, from which one can read off a 2-parameter deformation of Cartan matrices of super type $A$. We also propose a Frenkel-Mukhin-type algorithm for $q$-characters of finite-dimensional simple modules with integral highest $\ell$-weights.

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arXiv2024

Oscillator representations of quantum affine orthosymplectic superalgebras

We introduce a category of $q$-oscillator representations over the quantum affine superalgebras of type $D$ and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible $q$-oscillator representations of type $X_n^{(1)}$ and the finite-dimensional irreducible representations of type $Y_n^{(1)}$ for $(X,Y)=(C,D),(D,C)$ under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finite-dimensional representations of classical Lie algebras arising from Howe's reductive dual pairs $(\mathfrak{g},G)$, where $\mathfrak{g}=\mathfrak{sp}_{2n}, \mathfrak{so}_{2n}$ and $G=O_\ell, Sp_{2\ell}$.

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arXiv2023

Affinization of $q$-oscillator representations of $U_q(\mathfrak{gl}_n)$

We introduce a category $\widehat{\mathcal{O}}_{\rm osc}$ of $q$-oscillator representations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$. We show that $\widehat{\mathcal{O}}_{\rm osc}$ has a family of irreducible representations, which naturally corresponds to finite-dimensional irreducible representations of quantum affine algebra of untwisted affine type $A$. It is done by constructing a category of $q$-oscillator representations of the quantum affine superalgebra of type $A$, which interpolates these two family of irreducible representations. The category $\widehat{\mathcal{O}}_{\rm osc}$ can be viewed as a quantum affine analogue of the semisimple tensor category generated by unitarizable highest weight representations of $\mathfrak{gl}_{u+v}$ ($n=u+v$) appearing in the $(\mathfrak{gl}_{u+v},\mathfrak{gl}_\ell)$-duality on a bosonic Fock space.

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arXiv2021

Super duality for quantum affine algebras of type $A$

We introduce a new approach to the study of finite-dimensional representations of the quantum group of the affine Lie superalgebra $\mathrm{L}\mathfrak{gl}_{M|N}=\mathbb{C}[t,t^{-1}]\otimes\mathfrak{gl}_{M|N}$ ($M\neq N$). We explain how the representations of the quantum group of $\mathrm{L}\mathfrak{gl}_{M|N}$ are directly related to those of the quantum affine algebra of type $A$, using an exact monoidal functor called truncation. This can be viewed as an affine analogue of super duality of type $A$.

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