Lowest Weight Vectors Research Articles

Bases for infinite dimensional simple osp(1|2n)-modules respecting the branching osp(1|2n)⊃gl(n)

We study the effects of the branching osp(1|2n)⊃gl(n) on a particular class of simple infinite-dimensional osp(1|2n)-modules L(p) characterized by a positive integer p. In the first part (Sec. III), we use combinatorial methods, such as Young tableaux and Young subgroups, to construct a new basis for L(p) that respects this branching, and we express the basis elements explicitly in two distinct ways: first, as monomials of negative root vectors of gl(n) acting on certain gl(n)-highest weight vectors in L(p) and then as polynomials in the generators of osp(1|2n) acting on a osp(1|2n)-lowest weight vector in L(p). In the second part (Sec. IV), we use extremal projectors and the theory of Mickelsson–Zhelobenko algebras to give new explicit constructions of raising and lowering operators related to the branching osp(1|2n)⊃gl(n). We use the raising operators to give new expressions for the elements of the Gel’fand–Zetlin basis for L(p) as monomials of operators from U(osp(1|2n)) acting on a osp(1|2n)-lowest weight vector in L(p). We observe that the Gel’fand–Zetlin basis for L(p) is related to the basis constructed earlier in this paper by a triangular transition matrix. We end this paper (Sec. V) with a detailed example treating the case n = 3.

$\mathfrak{q}$-Crystal Structure on Primed Tableaux and on Signed Unimodal Factorizations of Reduced Words of Type $B$

Crystal basis theory for the queer Lie superalgebra was developed in Grantcharov et al. (Trans. Amer. Math. Soc. 366, 2013, 457–489) and Grantcharov et al. (J. Europ. Math. Soc. 17, 2015, 1593–1627), where it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply \mathfrak{q} -crystal structure. In this paper, we explore the \mathfrak{q} -crystal structure of primed tableaux (Hawkes et al., Electron. J. Combin. 24, 2017, P3.51) (semistandard marked shifted tableaux; Cho, Trans. Amer. Math. Soc. 365, 2013, 939–972) and that of signed unimodal factorizations of reduced words of type B (Hawkes et al., Electron. J. Combin. 24 (2017) P3.51). We give explicit odd Kashiwara operators on primed tableaux and the forms of the highest and lowest weight vectors. We clarify the relation between signed unimodal factorizations and the type- B Coxeter-Knuth relation of reduced words. We also give explicit algorithms for odd Kashiwara operators on signed unimodal factorizations of reduced words of type B .

Symmetry of asymmetric quantum Rabi models

The aim of this paper is a better understanding for the eigenstates of the asymmetric quantum Rabi model by Lie algebra representations of . We define a second order element of the universal enveloping algebra of , which, through the action of a certain infinite dimensional representation of , provides a picture of the asymmetric quantum Rabi model equivalent to the one drawn by confluent Heun ordinary differential equations. Using this description, we prove the existence of level crossings in the spectral graph of the asymmetric quantum Rabi model when the symmetry-breaking parameter ϵ is equal to , and conjecture a formula that ensures likewise the presence of level crossings for general . This result on level crossings was demonstrated numerically by Li and Batchelor in 2015, investigating an earlier empirical observation by Braak (2011). The first analysis of the degenerate spectrum was given for the symmetric quantum Rabi model by Kuś in 1985. In our picture, we find a certain reciprocity (or -symmetry) for if the spectrum is described by representations of . We further discuss briefly the non-degenerate part of the exceptional spectrum from the viewpoint of infinite dimensional representations of having lowest weight vectors.

${\mathbbm {C}}$ C -Graded vertex algebras and conformal flow

We consider $\mathbbm {C}$C-graded vertex algebras, which are vertex algebras V with a $\mathbbm {C}$C-grading such that V is an admissible V-module generated by “lowest weight vectors.” We show that such vertex algebras have a “good” representation theory in the sense that there is a Zhu algebra A(V) and a bijection between simple admissible V-modules and simple A(V)-modules. We also consider pseudo vertex operator algebras (PVOAs), which are $\mathbbm {C}$C-graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L(0); essentially, these are VOAs that lack any semisimplicity or integrality assumptions on L(0). As a motivating example, we show that deformation of the conformal structure (conformal flow) of a strongly regular VOA (e.g., a lattice theory, or Wess-Zumino-Witten model) is a path in a space whose points are PVOAs.

SU(2) COHERENT STATE PATH INTEGRALS LABELED BY A FULL SET OF EULER ANGLES: BASIC FORMULATION

We develop a basic formulation of the spin (SU(2)) coherent state path integrals based not on the conventional highest or lowest weight vectors but on arbitrary fiducial vectors. The coherent states, being defined on a 3-sphere, are specified by a full set of Euler angles. They are generally considered as states without classical analogues. The overcompleteness relation holds for the states, by which we obtain the time evolution of general systems in terms of the path integral representation; the resultant Lagrangian in the action has a monopole-type term à la Balachandran et al. as well as some additional terms, both of which depend on fiducial vectors in a simple way. The process of the discrete path integrals to the continuous ones is clarified. Complex variable forms of the states and path integrals are also obtained. During the course of all steps, we emphasize the analogies and correspondences to the general canonical coherent states and path integrals that we proposed some time ago. In this paper we concentrate on the basic formulation. The physical applications as well as criteria in choosing fiducial vectors for real Lagrangians, in relation to fictitious monopoles and geometric phases, will be treated in subsequent papers separately.

Young wall realization of crystal bases for classical Lie algebras

In this paper, we give a new realization of crystal bases for finite-dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.

Spectra of PP-Wave Limits of M-/Superstring Theory on AdSp× SqSpaces

In this paper we show how one can obtain very simply the spectra of the PP-wave limits of M-theory over AdS_7(4) x S^4(7) spaces and IIB superstring theory over AdS_5 x S^5 from the oscillator construction of the Kaluza-Klein spectra of these theories over the corresponding spaces. The PP-wave symmetry superalgebras are obtained by taking the number P of ``colors'' of oscillators to be large (infinite). In this large P limit, the symmetry superalgebra osp(8*|4) of AdS_7 x S^4 and the symmetry superalgebra osp(8|4,R) of AdS_4 x S^7 lead to isomorphic PP-wave algebras, which is the semi-direct sum of su(4|2) with H^(18,16), while the symmetry superalgebra su(2,2|4) of AdS_5 x S^5 leads to the semi-direct sum of [psu(2|2) + psu(2|2) + u(1)] with H^(16,16) as its PP-wave algebra [H^(m,n) denoting a super-Heisenberg algebra with m bosonic and n fermionic generators]. The zero mode spectra of M-theory or IIB superstring theory in the PP-wave limit corresponds simply to the unitary positive energy representations of these algebras whose lowest weight vector is the Fock vacuum of all the oscillators. General positive energy supermultiplets including those corresponding to higher modes can similarly be constructed by the oscillator method.

Finite dimensional irreducible representations of the quantum supergroup Uq (gl(m/n))

Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n)) at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction. The representation theory is rather similar to that of gl(m/n) at generic q, but drastically different when q is a root of unity. In the latter case, atypicality conditions of highest weight irreducible representations (irreps) are substantially altered, and such finite-dimensional irreps arise that do not have highest weight and/or lowest weight vectors. As concrete examples, the irreps of Uq(gl(2/1)) are classified.

A projection based approach to the Clebsch-Gordan multiplicity problem for compact semisimple Lie groups: I. General formalism

The authors describe a method, analogous to the Elliott method (1958) for O(3) contained in/implied by U(3), for resolution of the multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a compact semisimple Lie group G. The method is based on a decomposition of highest weight vectors in the product representation into direct product states, focusing on components of the form e(X)e+mu with e a weight vector and e+mu a highest weight vector. The weight of the vector e corresponds to the 'shift' weight in a tensor operator formulation of the problem. A result from an earlier paper (Gould et al. 1984) based on the fact that a tensor product representation can be generated cyclically from the product of a highest and a lowest weight vector is used to give an explicit characterisation of the space of shift weight vectors e that can appear in the decomposition of a highest weight state. This characterisation is in terms of lowering operators in the complexified Lie algebra L of G, and closely parallels Verma's well known enveloping algebra characterisation (1968) of the highest weight states of finite-dimensional irreducible representations of complex semisimple Lie algebra.