Conformal Galilei groups, Veronese curves and Newton–Hooke spacetimes

Abstract

Finite-dimensional nonrelativistic conformal Lie algebras spanned by polynomial vector fields of Galilei spacetime arise if the dynamical exponent is z = 2/N with N = 1, 2, . . . . Their underlying group structure and matrix representation are constructed (up to a covering) by means of the Veronese map of degree N . Suitable quotients of the conformal Galilei groups provide us with Newton-Hooke nonrelativistic spacetimes with a quantized reduced negative cosmological constant λ = −N .

Unitary representations ofN-conformal Galilei group

“…(B.31) and (B.32) make use of the properties of finite dimensional irreducible representations of SL(2, R). In fact, finitedimensional irreducible representations of SL(2, R) are classified by natural number N and they matrices are of the form (see, e.g., [14]) …”

Обобщения Релятивистской Кинематики

“…Кро-ме того, при a 0 = 0 преобразование (51) совпадает с (35), а при a = 0, a 0 = cτ tg α -с (25). Возведя равенство (51) в квадрат, получаем …”

“…Проведенный анализ показывает, что дискретное (в обычном галилее-вом пространстве-времени) преобразование "четности" (отражение пространствен-ных координат) содержится в непрерывной группе SL(2, R). Таким образом, ясно, что построенные выше преобразования симметрии реализуются не на обычном нере-лятивистском пространстве-времени R d ⊗ R, а скорее на пространстве Мёбиуса [25].…”

Higher-derivative generalization of conformal mechanics

“…The free higher-derivative particle 13,14 and the Pais-Uhlenbeck (PU) oscillator 15 can be viewed as higher-derivative analogues of the free particle and the harmonic oscillator, re-spectively. Recently, symmetries of these higher-derivative models have been extensively studied 13,14,[16][17][18][19][20][21][22][23] .…”

“…Recently, symmetries of these higher-derivative models have been extensively studied 13,14,[16][17][18][19][20][21][22][23] . In particular it has been shown that the l-conformal Galilei group is the maximal symmetry group of the free (2l + 1)-order particle 14 .…”