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In this paper, we continue the project of systematic construction of invariant differential operators (IDOs) on the example of the noncompact algebras s p (n , 1). Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which are of split rank one, of which class the other cases were studied, some long time ago. We concentrate on the case n = 2. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations (ER) for, including the necessary data for all relevant invariant differential operators. We also present explicit expressions for the singular vectors and the intertwining differential operators. [ABSTRACT FROM AUTHOR]

In this paper, we construct explicitly the intertwining differential operators for the Jacobi algebra 2 . For the construction, we use the singular vectors of the Verma modules over 2 which we have constructed earlier. We construct the function spaces on which the operators act. We find two versions of the left (representation) action and the right action. These actions are combined with the singular vectors to provide the intertwining differential operators. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to lay the foundations of generalizing the AdS/CFT (holography) idea beyond the conformal setting. The main tool is to find suitable realizations of the bulk and boundary via group theory. We use all ten families of classical real semisimple Lie groups G and Lie algebras 𝒢. For this are used several group and algebra decompositions: the global Iwasawa decomposition and the local Bruhat and Sekiguchi-like decompositions. The same analysis is applied to the exceptional real semisimple Lie algebras. [ABSTRACT FROM AUTHOR]

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed. [ABSTRACT FROM AUTHOR]

In the present paper, we start the systematic explicit construction of invariant differential operators by giving explicit description of one of the main ingredients — the cuspidal parabolic subalgebras. We explicate also the maximal parabolic subalgebras, since these are also important even when they are not cuspidal. Our approach is easily generalized to the supersymmetric and quantum group settings and is necessary for applications to string theory and integrable models. [ABSTRACT FROM AUTHOR]

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev–Reshetikhin–Takhtajan RTT prescription with non-triangular R-matrices which are denoted R[sub 03] and R[sub 14] in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1|1). We give the spectral decomposition which involves two, respectively, three, projectors. These projectors are then used to provide the baxterization procedure with one, respectively, two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess–Zumino prescription. In all these constructions there appear nonstandard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators. [ABSTRACT FROM AUTHOR]

We propose an infinite family of difference equations, which are derived from the first principle that they are invariant with respect to the Schrödinger algebra. The first member of this family is a difference analogue of the free Schrödinger equation. These equations are obtained via a purely algebraic construction from a corresponding family of singular vectors in Verma modules over the Schrödinger algebra. The crucial moment in the construction is the realization of the Schrödinger algebra through additive difference vector fields, i.e. vector fields with difference operators instead of differential operators. Our method produces also differential-difference equations in which only space- or time-differentiation is replaced with the corresponding difference operators. [ABSTRACT FROM AUTHOR]