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

SIGMA 12 (2016), 113, 13 pages      arXiv:1605.08608      https://doi.org/10.3842/SIGMA.2016.113

### On Free Field Realizations of $W(2,2)$-Modules

a) Department of Mathematics, University of Zagreb, Bijenička 30, 10 000 Zagreb, Croatia
b) Faculty of Science, University of Split, Rudera Boškovića 33, 21 000 Split, Croatia

Received June 09, 2016, in final form December 03, 2016; Published online December 06, 2016; Several presentational changes made and misprints corrected January 15, 2017

Abstract
The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra $\mathcal H$ at level zero as modules for the $W(2,2)$-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight ${\mathcal H}$-module is irreducible as $W(2,2)$-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly $W(2,2)$ vertex algebra.

Key words: Heisenberg-Virasoro Lie algebra; vertex algebra; $W(2,2)$ algebra; screening-operators.

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References

1. Adamović D., Classification of irreducible modules of certain subalgebras of free boson vertex algebra, J. Algebra 270 (2003), 115-132, math.QA/0207155.
2. Adamović D., Milas A., Logarithmic intertwining operators and $W(2,2p-1)$ algebras, J. Math. Phys. 48 (2007), 073503, 20 pages, math.QA/0702081.
3. Adamović D., Milas A., Lattice construction of logarithmic modules for certain vertex algebras, Selecta Math. (N.S.) 15 (2009), 535-561, arXiv:0902.3417.
4. Adamović D., Radobolja G., Free field realization of the twisted Heisenberg-Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707.
5. Adamović D., Radobolja G., Self-dual and logarithmic representations of the twisted Heisenberg-Virasoro algebra at level zero, in preparation.
6. Arbarello E., De Concini C., Kac V.G., Procesi C., Moduli spaces of curves and representation theory, Comm. Math. Phys. 117 (1988), 1-36.
7. Billig Y., Energy-momentum tensor for the toroidal Lie algebra, math.RT/0201313.
8. Billig Y., Representations of the twisted Heisenberg-Virasoro algebra at level zero, Canad. Math. Bull. 46 (2003), 529-537, math.RT/0201314.
9. Billig Y., A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not. 2006 (2006), 68395, 46 pages, math.RT/0509368.
10. Creutzig T., Milas A., False theta functions and the Verlinde formula, Adv. Math. 262 (2014), 520-545, arXiv:1309.6037.
11. Feigin B.L., Fuchs D.B., Representations of the Virasoro algebra, in Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math., Vol. 7, Gordon and Breach, New York, 1990, 465-554.
12. Frenkel I.B., Huang Y.-Z., Lepowsky J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), viii+64 pages.
13. Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu., Feigin B.L., The Kazhdan-Lusztig correspondence for the representation category of the triplet $W$-algebra in logorithmic conformal field theories, Theoret. Math. Phys. 148 (2006), 1210-1235, math.QA/0512621.
14. Jiang W., Pei Y., On the structure of Verma modules over the $W$-algebra $W(2,2)$, J. Math. Phys. 51 (2010), 022303, 8 pages.
15. Jiang W., Zhang W., Verma modules over the $W(2,2)$ algebras, J. Geom. Phys. 98 (2015), 118-127.
16. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
17. Li H.S., Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994), 279-297.
18. Liu D., Zhu L., Classification of Harish-Chandra modules over the $W$-algebra $W(2,2)$, arXiv:0801.2601.
19. Radobolja G., Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted Heisenberg-Virasoro algebra and $W(2,2)$ algebra, J. Math. Phys. 54 (2013), 071701, 24 pages, arXiv:1302.0801.
20. Zhang W., Dong C., $W$-algebra $W(2,2)$ and the vertex operator algebra $L(\frac 12,0)\otimes L(\frac 12,0)$, Comm. Math. Phys. 285 (2009), 991-1004, arXiv:0711.4624.