Abstract
In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n, n). Our choice of these algebras is motivated by the fact that for n = 2 this is the conformal algebra of 4-dimensional Minkowski space-time. Furthermore for general n these algebras belong to a narrow class of algebras, which we call “conformal Lie algebras”, which have very similar properties to the conformal algebras of n 2-dimensional Minkowski space-time. We give the main multiplets of indecomposable elementary representations for n = 2, 3, 4, including the necessary data for all relevant invariant differential operators.
Highlights
Consider a Lie group G, e.g., the Lorentz, Poincare, conformal groups, and differential equationsIf = j which are G-invariant
The main multiplets R6 contain 64(= 26) ERs/generalized Verma modules (GVMs) whose signatures can be given in the following pair-wise manner: χ±0 =
The reduced multiplets of type R16 contain 48 ERs/GVMs whose signatures can be given in the following pair-wise manner: χ±0 =
Summary
Consider a Lie group G, e.g., the Lorentz, Poincare, conformal groups, and differential equations. In the present paper we focus on the groups Sp(n, IR), which are very interesting for several reasons First of all, they belong to the class of Hermitian symmetric spaces, i.e., the pair (G, K) is a Hermitian symmetric pair (K is the maximal compact subgroup of the noncompact semisimple group G). Sp(n, IR) belong to a narrower class of groups/algebras, which we call ’conformal Lie groups or algebras’ since they have very similar properties to the canonical conformal algebras so(n, 2) of n-dimensional Minkowski space-time. This class was identified from our point of view in [3].
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