English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The universal Askey-Wilson algebra $AW(3)$ can be obtained as the commutantof $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1))^{\otimes3}$. Weanalyze the commutant of$\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)$in $q$-oscillator representations of $\mathfrak{o}_{q^{1/2}}(6)$ and show thatit also realizes $AW(3)$. These two pictures of $AW(3)$ are shown to be dual inthe sense of Howe; this is made clear by highlighting the role of theintermediate Casimir elements of each members of the dual pair$\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(6)\big)$. We alsogeneralize these results. A higher rank extension of the Askey-Wilson algebradenoted $AW(n)$ can be defined as the commutant of $U_q(\mathfrak{su}(1,1))$ in$U_q(\mathfrak{su}(1,1))^{\otimes n}$ and a dual description of $AW(n)$ as thecommutant of $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ in $q$-oscillatorrepresentations of $\mathfrak{o}_{q^{1/2}}(2n)$ is offered by calling upon thedual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$.

The dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$, $q$-oscillators and Askey-Wilson algebras