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The E2 algebra has three elements, J, u, and v, which satisfy the commutationrelations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the HamiltonianH=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian isHermitian and consequently it has real eigenvalues. However, we can alsoconstruct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where againg is real. As in the case of PT-symmetric Hamiltonians constructed from theelements x and p of the Heisenberg algebra, there are two regions in parameterspace for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry inwhich all the eigenvalues are real and a region of broken PT symmetry in whichsome of the eigenvalues are complex. The two regions are separated by acritical value of g.

Extending PT symmetry from Heisenberg algebra to E2 algebra