Positive Energy Unitary Irreducible Representations of the Superalgebras osp(1|2n,R)
We give the classification of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R).
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arXiv subjects
R. B. Zhang
Publications and source records attributed to R. B. Zhang.
The Brauer Category and Invariant Theory
A category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, and a presentation of the category is given; specifically, we prove that seven relations among its four generating homomorphisms suffice to deduce all equations among the morphisms. Full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group $O(V)$ or the symplectic group $Sp(V)$ over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain new presentations for the endomorphism algebras of the module $V^{\otimes r}$. These are obtained by appending to the standard presentation of the Brauer algebra of degree $r$ one additional relation. This relation stipulates the vanishing of an element of the Brauer algebra which is quasi-idempotent, and which we describe explicitly both in terms of diagrams and algebraically. In the symplectic case, if $\dim V = 2n$, the element is precisely the central idempotent in the Brauer subalgebra of degree $n + 1$, which corresponds to its trivial representation. Since this is the Brauer algebra of highest degree which is semisimple, our generator is an exact analogue for the Brauer algebra of the Jones idempotent of the Temperley-Lieb algebra. In the orthogonal case the additional relation is also a quasi-idempotent in the integral Brauer algebra. Both integral and quantum analogues of these results are given, the latter of which involve the BMW algebras.
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Extended Poincaré supersymmetry in three dimensions and supersymmetric anyons
We classify the unitary representations of the extended Poincaré supergroups in three dimensions. Irreducible unitary representations of any spin can appear, which correspond to supersymmetric anyons. Our results also show that all irreducible unitary representations necessarily have physical momenta. This is in sharp contrast to the ordinary Poincaré group in three dimensions, that admits in addition irreducible unitary representations with non-physical momenta, which are discarded on physical grounds.
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A noncommutative geometric approach to the quantum structure of spacetime
Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to construct a noncommutative version of general relativity, which is expected to capture some essential structural features of spacetime at the Planck scale. Examples of noncommutative spacetimes were investigated in detail. These include quantisations of plane-fronted gravitational waves, quantum Schwarzschild spacetime and Schwarzschild-de Sitter spacetime, and a quantun Tolman spacetime which is relevant to gravitational collapse. Here we briefly review the theory and its application in the study of quantum structure of spacetime.
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Serre presentations of Lie superalgebras
An analogue of Serre's theorem is established for finite dimensional simple Lie superalgebras, which describes presentations in terms of Chevalley generators and Serre type relations relative to all possible choices of Borel subalgebras. The proof of the theorem is conceptually transparent; it also provides an alternative approach to Serre's theorem for ordinary Lie algebras.
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Generalised Jantzen filtration of Lie superalgebras I
A Jantzen type filtration for generalised Varma modules of Lie superalgebras is introduced. In the case of type I Lie superalgebras, it is shown that the generalised Jantzen filtration for any Kac module is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan-Lusztig polynomials. Furthermore, the length of the Jantzen filtration for any Kac module is determined explicitly in terms of the degree of atypicality of the highest weight. These results are applied to obtain a detailed description of the submodule lattices of Kac modules.
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Projective module description of embedded noncommutative spaces
An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in \cite{CTZZ}. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi-Civita connection and related curvatures, which were introduced geometrically in \cite{CTZZ}. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.
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Gravitational collapse of spherically symmetric stars in noncommutative general relativity
Gravitational collapse of a class of spherically symmetric stars are investigated. We quantise the geometries describing the gravitational collapse by a deformation quantisation procedure. This gives rise to noncommutative spacetimes with gravitational collapse.
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Generalised Verma modules for the orthosymplectic Lie superalgebra osp(k|2)
The composition factors and their multiplicities are determined for generalised Verma modules over the orthosymplectic Lie superalgebra osp(k|2). The results enable us to obtain explicit formulae for the formal characters and dimensions of the finite-dimensional irreducible modules. Applying these results, we also compute the first and second cohomology groups of the Lie superalgebra with coefficients in finite-dimensional Kac modules and irreducible modules.
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Quantum group actions on rings and equivariant K-theory
Let $\Uq$ be a quantum group. Regarding a (noncommutative) space with $\Uq$-symmetry as a $\Uq$-module algebra $A$, we may think of equivariant vector bundles on $A$ as projective $A$-modules with compatible $\Uq$-action. We construct an equivariant K-theory of such quantum vector bundles using Quillen's exact categories, and provide means for its compution. The equivariant K-groups of quantum homogeneous spaces and quantum symmetric algebras of classical type are computed.
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Exact solutions of noncommutative vacuum Einstein field equations and plane-fronted gravitational waves
We construct a class of exact solutions of the noncommutative vacuum Einstein field equations, which are noncommutative analogues of the plane-fronted gravitational waves in classical gravity.
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A quantum analogue of the first fundamental theorem of invariant theory
We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Thus by taking the limit as $q\to 1$, our results imply the first fundamental theorem of classical invariant theory, and therefore generalise them to the noncommutative case.
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Quantum deformations of Schwarzschild and Schwarzschild-de Sitter spacetimes
A quantum Schwarzschild spacetime and a quantum Schwarzschild-de Sitter spacetime with cosmological constant $Λ$ are constructed within the framework of a noncommutative Riemannian geometry developed in an earlier publication. The metrics and curvatures of the quantum Schwarzschild spacetime and the quantum Schwarzschild-de Sitter spacetime are computed. It is shown that up to the second order in the deformation parameter, the quantum spacetimes are solutions of a noncommutative Einstein equation which is proposed in this paper.
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Riemannian Geometry of Noncommutative Surfaces
A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the underlying structure of Einstein's theory of general relativity and led to further developments of the latter. The notions of metric and connections on such noncommutative surfaces are introduced and it is shown that the connections are metric-compatible, giving rise to the corresponding Riemann curvature. The latter also satisfies the noncommutative analogue of the first and second Bianchi identities. As examples, noncommutative analogues of the sphere, torus and hyperboloid are studied in detail. The problem of covariance under appropriately defined general coordinate transformations is also discussed and commented on as compared with other treatments.
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A Temperley-Lieb analogue for the BMW algebra
The Temperley-Lieb algebra may be thought of as a quotient of the Hecke algebra of type A, acting on tensor space as the commutant of the usual action of quantum sl(2) on the n-th tensor power of the 2-dimensional irreducible module. We define and study a quotient of the Birman-Wenzl-Murakami algebra, which plays an analogous role for the 3-dimensional representation of quantum sl(2). In the course of the discussion we prove some general results about the radical of a cellular algebra, which may be of independent interest.
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Noncommutative fields and actions of twisted Poincare algebra
Within the context of the twisted Poincaré algebra, there exists no noncommutative analogue of the Minkowski space interpreted as the homogeneous space of the Poincaré group quotiented by the Lorentz group. The usual definition of commutative classical fields as sections of associated vector bundles on the homogeneous space does not generalise to the noncommutative setting, and the twisted Poincaré algebra does not act on noncommutative fields in a canonical way. We make a tentative proposal for the definition of noncommutative classical fields of any spin over the Moyal space, which has the desired representation theoretical properties. We also suggest a way to search for noncommutative Minkowski spaces suitable for studying noncommutative field theory with deformed Poincaré symmetries.
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Orthosymplectic Lie superalgebras in superspace analogues of quantum Kepler problems
A Schroedinger type equation on the superspace R^{D|2n} is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant "distance" away from the origin. An osp(2,D+1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well as their respective dimensions.
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Super duality and Kazhdan-Lusztig polynomials
We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type A. In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental representations of type A.
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