Abstract
We have recently shown that \(\widehat{\mathfrak{g}\mathfrak{l}}\left (1\vert 1\right )\) admits an infinite family of simple current extensions. Here, we review these findings and add explicit free field realizations of the extended algebras. We use them for the computation of leading contributions of the operator product algebra. Amongst others, we find extensions that contain the Feigin–Semikhatov W N (2) algebra at levels k = N(3 − N) ∕ (N − 2) and k = − N + 1 + N − 1 as subalgebras.
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- 1.
We mention that the typical irreducibles are also projective in the category of finite-dimensional \(\mathfrak{g}\mathfrak{l}\left (1\vert 1\right )\)-modules.
- 2.
It is perhaps also worth pointing out that the adjoint representation of \(\mathfrak{g}\mathfrak{l}\left (1\vert 1\right )\) is isomorphic to \({ \mathcal{P}}_{0}\).
- 3.
More precisely, \(\widehat{{ \mathcal{P}}}_{n,0}\) is the affine counterpart to \({ \mathcal{P}}_{n}\) and the remaining \(\widehat{{ \mathcal{P}}}_{n,\ell }\) are obtained by spectral flow.
- 4.
Here, H and Z should be associated with the matrices diag {1, − 1, 0} and diag {1, 1, 2} in the defining representation of \(\mathfrak{s}\mathfrak{l}\left (2\vert 1\right )\).
- 5.
There is a third solution, Δ n, ℓ + ℓ + 1 = 0, but this is invalid as we require ℓ, Δ n, ℓ > 0.
- 6.
Taking n = −½(ℓ + 1) also satisfies these requirements, but then 2nℓ is necessarily even. Moreover, there is again a solution of the form Δ n, ℓ − ℓ + 1 = 0, but it is easy to check that it leads to the wrong operator product expansion of \({ \mathcal{T}}_{N}\) with itself.
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Creutzig, T., Ridout, D. (2013). W-Algebras Extending \(\widehat{\mathfrak{g}\mathfrak{l}}(1\vert 1)\) . In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 36. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54270-4_24
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DOI: https://doi.org/10.1007/978-4-431-54270-4_24
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