Single sided multiplier Hopf algebras
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913261051445248 |
|---|---|
| author | Van Daele, Alfons |
| author_facet | Van Daele, Alfons |
| contents | Let $A$ be a non-degenerate algebra over the complex numbers and $Δ$ a homomorphism from $A$ to the multiplier algebra $M(A\otimes A)$. Consider the linear maps $T_1$ and $T_2$ from $A\otimes A$ to $M(A\otimes A)$ defined by \begin{equation*} T_1(a\otimes b)=Δ(a)(1\otimes b) \qquad\text{and}\qquad T_2(c\otimes a)=(c\otimes 1)Δ(a). \end{equation*} The pair $(A,Δ)$ is a multiplier Hopf algebra if these two maps have range in $A\otimes A$ and are bijections from $A\otimes A$ to itself. In our recent paper on the Larson-Sweedler theorem, single sided multiplier Hopf algebras emerge in a natural way. For this case, instead of requiring the above for the maps $T_1$ and $T_2$, we now have this property for the maps $T_1$ and $T_4$ or for $T_2$ and $T_3$ where \begin{equation*} T_3(a\otimes b)=(1\otimes b)Δ(a) \qquad\text{and}\qquad T_4(c\otimes a)=Δ(a)(c\otimes 1). \end{equation*} As it turns out, also for these single sided multiplier Hopf algebras, the existence of a unique counit and antipode can be proven. In fact, rather surprisingly, using the properties of the antipode, one can actually show that for a single sided multiplier Hopf algebra all four canonical maps are bijections from $A\otimes A$ to itself. In other words, $(A,Δ)$ is automatically a regular multiplier Hopf algebra. We take the advantage of this approach to reconsider some of the known results for a regular multiplier Hopf algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Single sided multiplier Hopf algebras Van Daele, Alfons Rings and Algebras 16T05 Let $A$ be a non-degenerate algebra over the complex numbers and $Δ$ a homomorphism from $A$ to the multiplier algebra $M(A\otimes A)$. Consider the linear maps $T_1$ and $T_2$ from $A\otimes A$ to $M(A\otimes A)$ defined by \begin{equation*} T_1(a\otimes b)=Δ(a)(1\otimes b) \qquad\text{and}\qquad T_2(c\otimes a)=(c\otimes 1)Δ(a). \end{equation*} The pair $(A,Δ)$ is a multiplier Hopf algebra if these two maps have range in $A\otimes A$ and are bijections from $A\otimes A$ to itself. In our recent paper on the Larson-Sweedler theorem, single sided multiplier Hopf algebras emerge in a natural way. For this case, instead of requiring the above for the maps $T_1$ and $T_2$, we now have this property for the maps $T_1$ and $T_4$ or for $T_2$ and $T_3$ where \begin{equation*} T_3(a\otimes b)=(1\otimes b)Δ(a) \qquad\text{and}\qquad T_4(c\otimes a)=Δ(a)(c\otimes 1). \end{equation*} As it turns out, also for these single sided multiplier Hopf algebras, the existence of a unique counit and antipode can be proven. In fact, rather surprisingly, using the properties of the antipode, one can actually show that for a single sided multiplier Hopf algebra all four canonical maps are bijections from $A\otimes A$ to itself. In other words, $(A,Δ)$ is automatically a regular multiplier Hopf algebra. We take the advantage of this approach to reconsider some of the known results for a regular multiplier Hopf algebra. |
| title | Single sided multiplier Hopf algebras |
| topic | Rings and Algebras 16T05 |
| url | https://arxiv.org/abs/2403.06863 |