N. Aizawa

Weak Hopf algebras corresponding to Uq[sln]

We investigate the weak Hopf algebras of Li based on Uq[sln] and Sweedler’s finite dimensional example. We give weak Hopf algebra isomorphisms between the weak generalizations of Uq[sln] which are “upgraded” automorphisms of Uq[sln] and hence give a classification of these structures as weak Hopf algebras. We also show how to decompose these examples into a direct sum which leads to unexpected isomorphisms between their algebraic structure.

Highest weight representations and Kac determinants for a class of conformal Galilei algebras with central extension

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrodinger algebra.

Intertwining Operator Realization of Non-Relativistic Holography

Abstract We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrodinger algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov–Witten we give the conditions when both boundary fields are physical. Finally, we recover in our formalism earlier non-relativistic results for scalar fields by Son and others.

Intertwining Operator Realization of anti de Sitter Holography

We give a group-theoretic interpretation of relativistic holography as equivalence between representations of the anti-de Sitter algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators for arbitrary integer spin in the framework of representation theory. Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the de Sitter case, we show that each bulk field has two boundary (shadow) fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary.

Intertwining operators for ℓ-conformal Galilei algebras and hierarchy of invariant equations

l-Conformal Galilei algebra, denoted by g{l}{d}, is a non-semisimple Lie algebra specified by a pair of parameters (d,l). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by g{l}{d} with central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of g{l}{d} and vector field representations for d = 1, 2.

Chiral and real N=2 supersymmetric ℓ-conformal Galilei algebras

Inequivalent N=2 supersymmetrizations of the l-conformal Galilei algebra in d-spatial dimensions are constructed from the chiral (2, 2) and the real (1, 2, 1) basic supermultiplets of the N=2 supersymmetry. For non-negative integer and half-integer l, both superalgebras admit a consistent truncation with a (different) finite number of generators. The real N=2 case coincides with the superalgebra introduced by Masterov, while the chiral N=2 case is a new superalgebra. We present D-module representations of both superalgebras. Then we investigate the new superalgebra derived from the chiral supermultiplet. It is shown that it admits two types of central extensions, one is found for any d and half-integer l, and the other only for d = 2 and integer l. For each central extension, the centrally extended l-superconformal Galilei algebra is realized in terms of its super-Heisenberg subalgebra generators.

On irreducible representations of the exotic conformal Galilei algebra

We investigate the representations of the exotic conformal Galilei algebra. This is done by explicitly constructing all singular vectors within the Verma modules, and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite-dimensional irreducible modules is presented.

Galilean conformal algebras in two spatial dimension

A class of infinite dimensional Galilean conformal algebra in (2+1) dimensional spacetime is studied. Each member of the class, denoted by \alg_{\ell}, is labelled by the parameter \ell. The parameter \ell takes a spin value, i.e., 1/2, 1, 3/2, .... We give a classification of all possible central extensions of \alg_{\ell}. Then we consider the highest weight Verma modules over \alg_{\ell} with the central extensions. For integer \ell we give an explicit formula of Kac determinant. It results immediately that the Verma modules are irreducible for nonvanishing highest weights. It is also shown that the Verma modules are reducible for vanishing highest weights. For half-integer \ell it is shown that all the Verma module is reducible. These results are independent of the central charges.

ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

We construct, for any given l=12+N0, the second-order, linear partial differential equations (PDEs) which are invariant under the centrally extended conformal Galilei algebra. At the given l, two invariant equations in one time and l+12 space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrodinger equation (recovered for l=12) in 1 + 1 dimension. The second equation (the “l-oscillator”) possesses a discrete, positive spectrum. It generalizes the 1 + 1-dimensional harmonic oscillator (recovered for l=12). The spectrum of the l-oscillator, derived from a specific osp(1|2l + 1) h.w.r., is explicitly presented. The two sets of invariant PDEs are determined by imposing (representation-dependent) on-shell invariant conditions both for degree 1 operators (those with continuum spectrum) and for degree 0 operators (those with discrete spectrum). The on-shell condition is better understood by enlarging the conformal Galilei algebras with the addition of certa...

Some Properties of Planar Galilean Conformal Algebras

An infinite dimensional extension of the spin 1 Galilean conformal algebra in the plane is investigated. We present the coadjoint representation and a classification of all possible central extensions. Furthermore, we study representations of the algebra with central extensions. Kac determinant for the highest weight Verma modules is given explicitly which shows that the Verma modules are irreducible for nonvanishing highest weights. A boson realization corresponding to unit central charge is also presented.