On Reducible Verma Modules over
Jacobi Algebra
V. K. Dobrev
Abstract With this paper we start the study of reducible representations of the
Jacobi algebra with the ultimate goal of constructing differential operators invariant
w.r.t. the Jacobi algebra. In this first paper we show examples of the low level
singular vectors of Verma modules over the Jacobi algebra. According to our
methodology these will produce the invariant differential operators.
Keywords Jacobi algebra · Verma modules · Singular vectors
1 Introduction
The role of nonrelativistic symmetries in theoretical physics was always important.
Currently one of the most popular fields in theoretical physics—string theory,
pretending to be a universal theory—encompasses together relativistic quantum
field theory, classical gravity, and certainly, nonrelativistic quantum mechanics, in
such a way that it is not even necessary to separate these components.
Since the cornerstone of quantum mechanics is the Schrödinger equation then
it is not a surprise that the Schrödinger group—the group that is the maximal
group of symmetry of the Schrödinger equation—was the first to play a prominent
role in theoretical physics. The latter is natural since originally the Schrödinger
group, actually the Schrödinger algebra, was introduced in [1, 2] as a nonrelativistic
limit of the vector-field realization of the conformal algebra. For a review on these
developments we refer to [3].
Another interesting nonrelativistic example is the Jacobi algebra [4, 5] which is
the semi-direct sum of the Heisenberg algebra and the sp(n) algebra. Actually the
lowest case of the Jacobi algebra coincides with the lowest case of the Schrödinger
algebra which makes it interesting to apply to the Jacobi algebra the methods we
V. K. Dobrev ()
Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia,
Bulgaria
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_20
217
Jacobi Algebra
V. K. Dobrev
Abstract With this paper we start the study of reducible representations of the
Jacobi algebra with the ultimate goal of constructing differential operators invariant
w.r.t. the Jacobi algebra. In this first paper we show examples of the low level
singular vectors of Verma modules over the Jacobi algebra. According to our
methodology these will produce the invariant differential operators.
Keywords Jacobi algebra · Verma modules · Singular vectors
1 Introduction
The role of nonrelativistic symmetries in theoretical physics was always important.
Currently one of the most popular fields in theoretical physics—string theory,
pretending to be a universal theory—encompasses together relativistic quantum
field theory, classical gravity, and certainly, nonrelativistic quantum mechanics, in
such a way that it is not even necessary to separate these components.
Since the cornerstone of quantum mechanics is the Schrödinger equation then
it is not a surprise that the Schrödinger group—the group that is the maximal
group of symmetry of the Schrödinger equation—was the first to play a prominent
role in theoretical physics. The latter is natural since originally the Schrödinger
group, actually the Schrödinger algebra, was introduced in [1, 2] as a nonrelativistic
limit of the vector-field realization of the conformal algebra. For a review on these
developments we refer to [3].
Another interesting nonrelativistic example is the Jacobi algebra [4, 5] which is
the semi-direct sum of the Heisenberg algebra and the sp(n) algebra. Actually the
lowest case of the Jacobi algebra coincides with the lowest case of the Schrödinger
algebra which makes it interesting to apply to the Jacobi algebra the methods we
V. K. Dobrev ()
Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia,
Bulgaria
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_20
217