Simple Representations of BPS Algebras: the case of $Y(\widehat{\mathfrak{gl}}_2)$
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916296149434368 |
|---|---|
| author | Galakhov, Dmitry Morozov, Alexei Tselousov, Nikita |
| author_facet | Galakhov, Dmitry Morozov, Alexei Tselousov, Nikita |
| contents | BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians -- the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for $Y(\widehat{\mathfrak{gl}}_r)$ these representations are related to Uglov polynomials, whose families are also labeled by natural $r$. They arise in the limit $\hbar\longrightarrow 0$ from Macdonald polynomials, and generalize the well-known Jack polynomials ($β$-deformation of Schur functions), associated with $r=1$. For $r=2$ they approximate Macdonald polynomials with the accuracy $O(\hbar^2)$, so that they are eigenfunctions of {\it two} immediately available commuting operators, arising from the $\hbar$-expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, -- what provides a technically simple way to build an explicit representation of Yangian $Y(\widehat{\mathfrak{gl}}_2)$, where $U^{(2)}$ are associated with the states $|λ\rangle$, parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables $p_{2n+1}$ can be expressed through mutually commuting operators from Yangian, however even time-variables $p_{2n}$ are inexpressible. Implications to higher $r$ become now straightforward, yet we describe them only in a sketchy way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simple Representations of BPS Algebras: the case of $Y(\widehat{\mathfrak{gl}}_2)$ Galakhov, Dmitry Morozov, Alexei Tselousov, Nikita High Energy Physics - Theory Mathematical Physics Quantum Algebra Representation Theory BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians -- the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for $Y(\widehat{\mathfrak{gl}}_r)$ these representations are related to Uglov polynomials, whose families are also labeled by natural $r$. They arise in the limit $\hbar\longrightarrow 0$ from Macdonald polynomials, and generalize the well-known Jack polynomials ($β$-deformation of Schur functions), associated with $r=1$. For $r=2$ they approximate Macdonald polynomials with the accuracy $O(\hbar^2)$, so that they are eigenfunctions of {\it two} immediately available commuting operators, arising from the $\hbar$-expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, -- what provides a technically simple way to build an explicit representation of Yangian $Y(\widehat{\mathfrak{gl}}_2)$, where $U^{(2)}$ are associated with the states $|λ\rangle$, parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables $p_{2n+1}$ can be expressed through mutually commuting operators from Yangian, however even time-variables $p_{2n}$ are inexpressible. Implications to higher $r$ become now straightforward, yet we describe them only in a sketchy way. |
| title | Simple Representations of BPS Algebras: the case of $Y(\widehat{\mathfrak{gl}}_2)$ |
| topic | High Energy Physics - Theory Mathematical Physics Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2402.05920 |