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A note on the zeroth products of Frenkel–Jing operators

Kožić, Slaven
A note on the zeroth products of Frenkel–Jing operators // Journal of algebra and its applications, 16 (2017), 2; 1750053-1 doi:10.1142/S0219498817500530 (međunarodna recenzija, članak, znanstveni)

Naslov
A note on the zeroth products of Frenkel–Jing operators

Autori
Kožić, Slaven

Izvornik
Journal of algebra and its applications (0219-4988) 16 (2017), 2; 1750053-1

Ključne riječi
Affine Lie algebra ; Quantum affine algebra ; Quantum vertex algebra ; Highest weight module

Sažetak
Let $\widehat{; ; \mathfrak{; ; g}; ; }; ;$ be an untwisted affine Kac-Moody Lie algebra. The top of every irreducible highest weight integrable $\widehat{; ; \mathfrak{; ; g}; ; }; ;$-module is the finite-dimensional irreducible $\mathfrak{; ; g}; ;$-module, where the action of the simple Lie algebra $\mathfrak{; ; g}; ;$ is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level $1$ Frenkel-Jing operators corresponding to Drinfeld realization of the quantum affine algebra $U_q (\widehat{; ; \mathfrak{; ; sl}; ; }; ; _{; ; n+1}; ; )$. By applying these products, which originate from the quantum vertex algebra theory developed by H.-S. Li, on the extension of Koyama vertex operator $\mathcal{; ; Y}; ; _i (z)$, we obtain an infinite-dimensional vector space $\left<\mathcal{; ; Y}; ; _i(z)\right>$. Next, we introduce an associative algebra $U_q (\mathfrak{; ; sl}; ; _{; ; n+1}; ; )_z$, a certain quantum analogue of the universal enveloping algebra $U(\mathfrak{; ; sl}; ; _{; ; n+1}; ; )$, and construct some infinite-dimensional $U_{; ; q(\mathfrak{; ; sl}; ; _{; ; n+1}; ; )_z$-modules $L(\lambda_i)_z$ corresponding to the finite- dimensional irreducible $U_{; ; q}; ; (\mathfrak{; ; sl}; ; _{; ; n+1}; ; )$-modules $L(\lambda_i)$. We show that the space $\left<\mathcal{; ; Y}; ; _i (z)\right>$ carries a structure of an $U_{; ; q}; ; (\mathfrak{; ; sl}; ; _{; ; n+1}; ; )_z$-module and, furthermore, we prove that the $U_{; ; q}; ; (\mathfrak{; ; sl}; ; _{; ; n+1}; ; )_z$-module $\left<\mathcal{; ; Y}; ; _i (z)\right>$ is isomorphic to the $U_{; ; q}; ; (\mathfrak{; ; sl}; ; _{; ; n+1}; ; )_z$-module $L(\lambda_i)_z$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

Projekt / tema
037-0372794-2806 - Algebre verteks-operatora i beskonačno dimenzionalne Liejeve algebre (Mirko Primc, )
HRZZ-IP-2013-11-2634 - Algebarske i kombinatorne metode u teoriji verteks algebri (Dražen Adamović, )

Ustanove
Prirodoslovno-matematički fakultet, Matematički odjel, Zagreb

Autor s matičnim brojem:
Slaven Kožić, (314194)

Časopis indeksira:

• Current Contents Connect (CCC)
• Web of Science Core Collection (WoSCC)
• Science Citation Index Expanded (SCI-EXP)
• SCI-EXP, SSCI i/ili A&HCI
• Scopus

Uključenost u ostale bibliografske baze podataka:

• Zentrallblatt für Mathematik/Mathematical Abstracts
• CNKI
• Current Contents®/Physical Chemical and Earth Sciences
• Ebsco Discovery Service
• ExLibris Primo Central