Jacobi Algebra
223
Weight 3δ 2
The only possible singular vector is:
v
δ 2
s = μb
+
2 a
+
2 v 0 + ν(a
+
2 )
3 v 0 .
(27)
Imposing (11a) on (27) we obtain:
μ = ν = 0.
(28)
Thus, there is no singular vector of weight 3δ 2 .
Acknowledgment The author acknowledges partial support from Bulgarian NSF Grant DN-18/1.
References
1. U. Niederer, The maximal kinematical invariance group of the free Schrodinger equation. Helv.
Phys. Acta 45, 802–810 (1972). https://doi.org/10.5169/seals-114417
2. C.R. Hagen, Scale and conformal transformations in Galilean-covariant field theory. Phys. Rev.
D5, 377–388 (1972). https://doi.org/10.1103/PhysRevD.5.377
3. V.K. Dobrev, Invariant Differential Operators, Volume 4: AdS/CFT, (Super-)Virasoro and Affine
(Super-)Algebras. De Gruyter Studies in Mathematical Physics, vol. 53 (De Gruyter, Berlin,
2019)
4. M. Eichler, D. Zagier, The Theory of Jacobi Forms. Program in Mathematics, vol. 55
(Birkhäuser, Boston, 1985)
5. R. Berndt, R. Schmidt, Elements of the Representation Theory of the Jacobi Group. Program in
Mathematics, vol. 163 (Birkhäuser, Basel, 1998)
6. S. Berceanu, A holomorphic representation of the semidirect sum of symplectic and Heisenberg
Lie algebras. J. Geom. Symmetry Phys. 5, 5–13 (2006). https://doi.org/10.7546/jgsp-5-2006-513
7. V.K. Dobrev, H.D. Doebner, C. Mrugalla, Lowest weight representations of the Schrödinger
algebra and generalized heat equations. Rep. Math. Phys. 39, 201–218 (1997). https://doi.org/
10.1016/S0034-4877(97)88001-9
8. N. Aizawa, V.K. Dobrev, Intertwining operator realization of non-relativistic holography. Nucl.
Phys. B828, 581–593 (2010). https://doi.org/10.1016/j.nuclphysb.2009.10.019
9. B. Dubsky, R. Lue, V. Mazorchuk, K. Zhao, Category O for the Schrödinger algebra. Linear
Algebra Appl. 460, 17–50 (2014). https://doi.org/10.1016/j.laa.2014.07.030
223
Weight 3δ 2
The only possible singular vector is:
v
δ 2
s = μb
+
2 a
+
2 v 0 + ν(a
+
2 )
3 v 0 .
(27)
Imposing (11a) on (27) we obtain:
μ = ν = 0.
(28)
Thus, there is no singular vector of weight 3δ 2 .
Acknowledgment The author acknowledges partial support from Bulgarian NSF Grant DN-18/1.
References
1. U. Niederer, The maximal kinematical invariance group of the free Schrodinger equation. Helv.
Phys. Acta 45, 802–810 (1972). https://doi.org/10.5169/seals-114417
2. C.R. Hagen, Scale and conformal transformations in Galilean-covariant field theory. Phys. Rev.
D5, 377–388 (1972). https://doi.org/10.1103/PhysRevD.5.377
3. V.K. Dobrev, Invariant Differential Operators, Volume 4: AdS/CFT, (Super-)Virasoro and Affine
(Super-)Algebras. De Gruyter Studies in Mathematical Physics, vol. 53 (De Gruyter, Berlin,
2019)
4. M. Eichler, D. Zagier, The Theory of Jacobi Forms. Program in Mathematics, vol. 55
(Birkhäuser, Boston, 1985)
5. R. Berndt, R. Schmidt, Elements of the Representation Theory of the Jacobi Group. Program in
Mathematics, vol. 163 (Birkhäuser, Basel, 1998)
6. S. Berceanu, A holomorphic representation of the semidirect sum of symplectic and Heisenberg
Lie algebras. J. Geom. Symmetry Phys. 5, 5–13 (2006). https://doi.org/10.7546/jgsp-5-2006-513
7. V.K. Dobrev, H.D. Doebner, C. Mrugalla, Lowest weight representations of the Schrödinger
algebra and generalized heat equations. Rep. Math. Phys. 39, 201–218 (1997). https://doi.org/
10.1016/S0034-4877(97)88001-9
8. N. Aizawa, V.K. Dobrev, Intertwining operator realization of non-relativistic holography. Nucl.
Phys. B828, 581–593 (2010). https://doi.org/10.1016/j.nuclphysb.2009.10.019
9. B. Dubsky, R. Lue, V. Mazorchuk, K. Zhao, Category O for the Schrödinger algebra. Linear
Algebra Appl. 460, 17–50 (2014). https://doi.org/10.1016/j.laa.2014.07.030