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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2112.06085 |
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| _version_ | 1866929466856439808 |
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| author | Terwilliger, Paul |
| author_facet | Terwilliger, Paul |
| contents | We consider the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$ and its basic module $V(Λ_0)$. This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe $V(Λ_0)$ using a $q$-shuffle algebra in the following way. Start with the free associative algebra $\mathbb V$ on two generators $x,y$. The standard basis for $\mathbb V$ consists of the words in $x,y$. In 1995 M. Rosso introduced an associative algebra structure on $\mathbb V$, called a $q$-shuffle algebra. For $u,v\in \lbrace x,y\rbrace$ their $q$-shuffle product is $u\star v = uv+q^{(u,v)}vu$, where $( u,v) =2$ (resp. $(u,v) =-2$) if $u=v$ (resp.
$u\not=v$).
Let $\mathbb U$ denote the subalgebra of the $q$-shuffle algebra $\mathbb V$
that is generated by $x, y$. Rosso showed that the algebra $\mathbb U$ is isomorphic to the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$.
In our first main result, we turn $\mathbb U$ into a $U_q(\widehat{\mathfrak{sl}}_2)$-module. Let $\bf U$ denote the $U_q(\widehat{\mathfrak{sl}}_2)$-submodule of $\mathbb U$ generated by the empty word. In our second main result, we show that the $U_q(\widehat{\mathfrak{sl}}_2)$-modules $\bf U$ and $V(Λ_0)$ are isomorphic. Let $\bf V$ denote the subspace of $\mathbb V$ spanned by the words that do not begin with $y$ or $xx$. In our third main result, we show that $\bf U = \mathbb U \cap {\bf V}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_06085 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Using a $q$-shuffle algebra to describe the basic module $V(Λ_0)$ for the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$ Terwilliger, Paul Quantum Algebra Combinatorics 17B37 We consider the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$ and its basic module $V(Λ_0)$. This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe $V(Λ_0)$ using a $q$-shuffle algebra in the following way. Start with the free associative algebra $\mathbb V$ on two generators $x,y$. The standard basis for $\mathbb V$ consists of the words in $x,y$. In 1995 M. Rosso introduced an associative algebra structure on $\mathbb V$, called a $q$-shuffle algebra. For $u,v\in \lbrace x,y\rbrace$ their $q$-shuffle product is $u\star v = uv+q^{(u,v)}vu$, where $( u,v) =2$ (resp. $(u,v) =-2$) if $u=v$ (resp. $u\not=v$). Let $\mathbb U$ denote the subalgebra of the $q$-shuffle algebra $\mathbb V$ that is generated by $x, y$. Rosso showed that the algebra $\mathbb U$ is isomorphic to the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$. In our first main result, we turn $\mathbb U$ into a $U_q(\widehat{\mathfrak{sl}}_2)$-module. Let $\bf U$ denote the $U_q(\widehat{\mathfrak{sl}}_2)$-submodule of $\mathbb U$ generated by the empty word. In our second main result, we show that the $U_q(\widehat{\mathfrak{sl}}_2)$-modules $\bf U$ and $V(Λ_0)$ are isomorphic. Let $\bf V$ denote the subspace of $\mathbb V$ spanned by the words that do not begin with $y$ or $xx$. In our third main result, we show that $\bf U = \mathbb U \cap {\bf V}$. |
| title | Using a $q$-shuffle algebra to describe the basic module $V(Λ_0)$ for the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$ |
| topic | Quantum Algebra Combinatorics 17B37 |
| url | https://arxiv.org/abs/2112.06085 |