Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 071, 6 pages      arXiv:2504.15597      https://doi.org/10.3842/SIGMA.2025.071
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

Linear Independence for $A_1^{(1)}$ by Using $C_{2}^{(1)}$

Mirko Primc ^a and Goran Trupčević ^b
a) Faculty of Science, University of Zagreb, Zagreb, Croatia
^b) Faculty of Teacher Education, University of Zagreb, Zagreb, Croatia

Received May 02, 2025, in final form August 12, 2025; Published online August 19, 2025

Abstract
In the previous paper, the authors proved linear independence of the combinatorial spanning set for standard $C_\ell^{(1)}$-module $L(k\Lambda_0)$ by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky's type subspace $W(k\Lambda_0)$ of $C_{2\ell}^{(1)}$-module $L(k\Lambda_0)$. In this note we extend this argument for $C_{1}^{(1)}\cong A_{1}^{(1)}$ to all standard $A_{1}^{(1)}$-modules $L(\Lambda)$. In the proof we use a coefficient of an intertwining operator of the type $\binom{L(\Lambda_2)}{L(\Lambda_1)\ L(\Lambda_1)}$ for standard $C_{2}^{(1)}$-modules.

Key words: affine Lie algebras; standard modules; Feigin-Stoyanovsky's type subspace; combinatorial basis.

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References

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