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## Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions

Adamović, Dražen; Kac, Victor; Moseneder Frajria, Pierluigi; Papi, Paolo; Perše, Ozren
Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions // Japanese Journal of Mathematics, 12 (2017), 2; 261-315 (međunarodna recenzija, članak, znanstveni)

Naslov
Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions

Autori
Adamović, Dražen ; Kac, Victor ; Moseneder Frajria, Pierluigi ; Papi, Paolo ; Perše, Ozren

Izvornik
Japanese Journal of Mathematics (0289-2316) 12 (2017), 2; 261-315

Ključne riječi
Conformal embedding ; vertex algebra ; W -algebra

Sažetak
We present methods for computing the explicit decomposition of the minimal simple affine $W$-algebra $W_k(\g, \theta)$ as a module for its maximal affine subalgebra $\mathcal V_k(\g^{; ; ; \natural}; ; ; )$ at a conformal level $k$, that is, whenever the Virasoro vectors of $W_k(\g, \theta)$ and $\mathcal V_k(\g^\natural)$ coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when $\g^{; ; ; \natural}; ; ;$ is a semisimple Lie algebra, we show that, for a suitable conformal level $k$, $W_k(\g, \theta)$ is isomorphic to an extension of $\mathcal V_k(\g^{; ; ; \natural}; ; ; )$ by its simple module. We are able to prove that in certain cases $W_k(\g, \theta)$ is a simple current extension of $\mathcal V_k(\g^{; ; ; \natural}; ; ; )$. In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple $W$-algebra $W_{; ; ; k}; ; ; (sl(4), \theta)$ at $k=-8/3$. We prove, as conjectured in D. Adamovic, Transformation Groups 2016, that $W_{; ; ; k}; ; ; (sl(4), \theta)$ is isomorphic to the vertex algebra $\mathcal R^{; ; ; (3)}; ; ;$, and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra $V_k (sl(n))$ at certain admissible levels and for $V_k (sl(m \vert n)), m\ne n, m, n\geq 1$ at arbitrary levels.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

Projekt / tema
HRZZ-IP-2013-11-2634 - Algebarske i kombinatorne metode u teoriji verteks algebri (Dražen Adamović, )
ZCI QuantiXLie

Ustanove
Prirodoslovno-matematički fakultet, Matematički odjel, Zagreb

#### 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:

• MathSciNet
• Zentralblatt fur Mathematik