Constructing the quantum queer supergroup using Hecke-Clifford superalgebras
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arXiv
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2024
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| author | Du, Jie Gu, Haixia Li, Zhenhua Wan, Jinkui |
| author_facet | Du, Jie Gu, Haixia Li, Zhenhua Wan, Jinkui |
| contents | In [DGLW], we use certain special elements and their commutation relations in the Hecke-Clifford algebras $H^c_{r,R}$ to derive some fundamental multiplication formulas associated with the natural bases in queer $q$-Schur superalgebras $Q_q(n,r;R)$ introduced in [DW2]. Here a natural basis element is defined by a special element $T_{A^{\star}}$ in $H^c_{r,R}$ associated with a pair of certain $n\times n$ matrices $A^{\star}=(A^{\bar0}|A^{\bar1})$ over $\mathbb{N}$ with entries sum to $r$. The definition of $T_{A^\star}$ consists of an element $c_{A^{\star}}$ in the Clifford superalgebra and an element $T_A$ in the Hecke algebra, where $A=A^{\bar0}+A^{\bar1}$. Note that all $T_A$ can be used to define the natural basis for the corresponding $q$-Schur algebra $S_q(n,r)$.
This paper is a continuation of [DGLW]. We start with standardized queer $v$-Schur superalgebras $ Q^s_v(n,r)$, for $R=\mathbb{Z}[v,v^{-1}]$ and $q=v^2$, and their natural bases. With the $v$-Schur algebra ${ S}_v(n,r)$ at the background, the first key ingredient is a standardisation of the natural basis for $Q^s_v(n,r)$ and their associated standard multiplication formulas. By introducing some long elements of finite sums, we then extend the formulas to these long elements which allow us to explicitly define $\mathbb{Q}(v)$-superalgebra homomorphisms $ξ_{n,r}$ from the quantum queer supergroup $\boldsymbol{U}_v(\mathfrak{q}_n)$ to queer $q$-Schur superalgebras $\boldsymbol{Q}^s_v(n,r)$, for all $r\geq1$. Finally, taking limits of long elements yields certain infinitely long elements as formal infinite series which eventually lead to a new construction for $\boldsymbol{U}_v(\mathfrak{q}_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14764 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing the quantum queer supergroup using Hecke-Clifford superalgebras Du, Jie Gu, Haixia Li, Zhenhua Wan, Jinkui Quantum Algebra 17B37, 17A70, 20G42, 20C08 In [DGLW], we use certain special elements and their commutation relations in the Hecke-Clifford algebras $H^c_{r,R}$ to derive some fundamental multiplication formulas associated with the natural bases in queer $q$-Schur superalgebras $Q_q(n,r;R)$ introduced in [DW2]. Here a natural basis element is defined by a special element $T_{A^{\star}}$ in $H^c_{r,R}$ associated with a pair of certain $n\times n$ matrices $A^{\star}=(A^{\bar0}|A^{\bar1})$ over $\mathbb{N}$ with entries sum to $r$. The definition of $T_{A^\star}$ consists of an element $c_{A^{\star}}$ in the Clifford superalgebra and an element $T_A$ in the Hecke algebra, where $A=A^{\bar0}+A^{\bar1}$. Note that all $T_A$ can be used to define the natural basis for the corresponding $q$-Schur algebra $S_q(n,r)$. This paper is a continuation of [DGLW]. We start with standardized queer $v$-Schur superalgebras $ Q^s_v(n,r)$, for $R=\mathbb{Z}[v,v^{-1}]$ and $q=v^2$, and their natural bases. With the $v$-Schur algebra ${ S}_v(n,r)$ at the background, the first key ingredient is a standardisation of the natural basis for $Q^s_v(n,r)$ and their associated standard multiplication formulas. By introducing some long elements of finite sums, we then extend the formulas to these long elements which allow us to explicitly define $\mathbb{Q}(v)$-superalgebra homomorphisms $ξ_{n,r}$ from the quantum queer supergroup $\boldsymbol{U}_v(\mathfrak{q}_n)$ to queer $q$-Schur superalgebras $\boldsymbol{Q}^s_v(n,r)$, for all $r\geq1$. Finally, taking limits of long elements yields certain infinitely long elements as formal infinite series which eventually lead to a new construction for $\boldsymbol{U}_v(\mathfrak{q}_n)$. |
| title | Constructing the quantum queer supergroup using Hecke-Clifford superalgebras |
| topic | Quantum Algebra 17B37, 17A70, 20G42, 20C08 |
| url | https://arxiv.org/abs/2411.14764 |