The Schrödinger algebra is the Lie algebra of the Schrödinger group. It is not semi-simple. In one space dimension, it can be obtained as a semi-direct sum of the Lie algebra sl and the Heisenberg algebra; similar constructions apply to higher spatial dimensions. It contains a Galilei algebra with central extension. Where are generators of rotations, spatial translations, Galilean boosts and time translation correspondingly. The central extension M has an interpretation as non-relativistic mass and corresponds to the symmetry of Schrödinger equation under phase transformation. There are two more generators which we shall denote by D and C. They have the following commutation relations: The generators H, C and D form the sl algebra. A more systematic notation allows to cast these generators into the four families and, where n ∈ ℤ is an integer and m ∈ ℤ+1/2 is a half-integer and j,k=1,...,dlabel the spatial direction, in d spatial dimensions. The non-vanishing commutators of the Schrödinger algebra become The Schrödinger algebra is finite-dimensional and contains the generators. In particular, the three generators span the sl sub-algebra. Space-translations are generated by and the Galilei-transformations by. In the chosen notation, one clearly sees that an infinite-dimensional extension exists, which is called the Schrödinger–Virasoro algebra. Then, the generators with n integer span a loop-Virasoro algebra. An explicit representation as time-space transformations is given by, with n ∈ ℤ and m ∈ ℤ+1/2 This shows how the central extension of the non-semi-simple and finite-dimensional Schrödinger algebra becomes a component of an infinite family in the Schrödinger–Virasoro algebra. In addition, and in analogy with either the Virasoro algebra or the Kac–Moody algebra, further central extensions are possible. However, a non-vanishing result only exists for the commutator , where it must be of the familiar Virasoro form, namely or for the commutator between the rotations, where it must have a Kac-Moody form. Any other possible central extension can be absorbed into the Lie algebra generators.
Though the Schrödinger group is defined as symmetry group of the free particle Schrödinger equation, it is realised in some interacting non-relativistic systems. The Schrödinger group in d spatial dimensions can be embedded into relativistic conformal group in d+1 dimensions SO. This embedding is connected with the fact that one can get the Schrödinger equation from the masslessKlein–Gordon equation through Kaluza–Klein compactification along null-like dimensions and Bargmann lift of Newton–Cartan theory. This embedding can also be viewed as the extension of the Schrödinger algebra to the maximal parabolic sub-algebra of SO. The Schrödinger group also arises as dynamical symmetry in condensed-matter applications: it is the dynamical symmetry of the Edwards–Wilkinson model of kinetic interface growth. It also describes the kinetics of phase-ordering, after a temperature quench from the disordered to the ordered phase, in magnetic systems.
