Ruibin Zhang

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Hung Vuong University

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Papers by Ruibin Zhang

Topological Invariants for Lens Spaces and Exceptional Quantum Groups

Letters in Mathematical Physics, 1996

The Reshetikhin–Turaev invariants arising from the quantum groups associated with the exceptional... more

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Quantum group actions on rings and equivariant K-theory

Algebraic Groups and Quantum Groups, 2012

Let U q (g) be a quantum group. Regarding a (noncommutative) space with U q (g)-symmetry as a U q... more Let U q (g) be a quantum group. Regarding a (noncommutative) space with U q (g)-symmetry as a U q (g)-module algebra A, we may think of equivariant vector bundles on A as projective A-modules with compatible U q (g)-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. Contents 1. Introduction 1 2. Equivariant K-theory of quantum group actions 4 2.1. Module algebras and equivariant modules 4 2.2. Equivariant K-theory of quantum group actions 6 3. Categories of equivariant modules 7 4. Equivariant K-theory of filtered module algebras 10 4.1. Graded module algebras 11 4.2. Filtered module algebras 12 5. Quantum symmetric algebras 16 5.1. Equivariant K-theory of quantum symmetric algebras 16 5.2. Quadratic algebras and Koszul complexes 16 5.3. Proof of Theorem 5.3 20 5.4. Examples 20 6. Quantum homogeneous spaces 24 6.1. Quantum homogenous spaces 24 6.2. Equivariant K-theory of quantum homogeneous spaces 25 6.3. Proof of Theorem 6.4 27 Appendix A. Comodules and smash products 29 References 30

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The first fundamental theorem of invariant theory for the orthosymplectic super group

Advances in Mathematics, 2017

We give an elementary and explicit proof of the first fundamental theorem of invariant theory for... more We give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of Atiyah, Bott and Patodi to the supergroup context. We use methods from super-algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space. In this way, super Schur-Weyl-Brauer duality is established between the orthosymplectic supergroup of superdimension (m|2n) and the Brauer algebra with parameter m − 2n. The result may be interpreted either in terms of the group scheme OSp(V) over C, where V is a finite dimensional super space, or as a statement about the orthosymplectic Lie supergroup over the infinite dimensional Grassmann algebra. We take the latter point of view here, and also state a corresponding theorem for the orthosymplectic Lie superalgebra, which involves an extra invariant generator, the super-Pfaffian.

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Invariants of the orthosymplectic Lie superalgebra and super Pfaffians

Mathematische Zeitschrift, 2016

Given a complex orthosymplectic superspace V , the orthosymplectic Lie superalgebra osp(V) and ge... more Given a complex orthosymplectic superspace V , the orthosymplectic Lie superalgebra osp(V) and general linear algebra gl N both act naturally on the coordinate super-ring S(N) of the dual space of V ⊗ C N , and their actions commute. Hence the subalgebra S(N) osp(V) of osp(V)-invariants in S(N) has a gl N-module structure. We introduce the space of super Pfaffians as a simple gl Nsubmodule of S(N) osp(V) , give an explicit formula for its highest weight vector, and show that the super Pfaffians and the elementary (or 'Brauer') OSp(V)-invariants together generate S(N) osp(V) as an algebra. The decomposition of S(N) osp(V) as a direct sum of simple gl N-submodules is obtained and shown to be multiplicity free. Using Howe's (gl(V), gl N)-duality on S(N), we deduce from the decomposition that the subspace of osp(V)-invariants in any simple gl(V)-tensor module is either 0 or 1-dimensional. These results also enable us to determine the osp(V)-invariants in the tensor powers V ⊗r for all r.

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Integrable representations of the quantum affine special linear superalgebra

Advances in Theoretical and Mathematical Physics, 2016

The simple integrable modules with finite dimensional weight spaces are classified for the quantu... more The simple integrable modules with finite dimensional weight spaces are classified for the quantum affine special linear superalgebra U q (sl(M|N)) at generic q. Any such module is shown to be a highest weight or lowest weight module with respect to one of the two natural triangular decompositions of the quantum affine superalgebra depending on whether the level of the module is zero or not. Furthermore, integrable U q (sl(M|N))-modules at nonzero levels exist only if M or N is 1.

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Seiberg--Witten Monopoles in Three Dimensions

Letters in Mathematical Physics

Dimensional reduction of the Seiberg--Witten equations leads to the equations of motion of a U(1)... more Dimensional reduction of the Seiberg--Witten equations leads to the equations of motion of a U(1) Chern--Simons theory coupled to a massless spinorial field. A topological quantum field theory is constructed for the moduli space of gauge equivalence classes of solutions of these equations. The Euler characteristic of the moduli space is obtained as the partition function which yields an analogue of Casson's invariant.A mathematically rigorous definition of the invariant isdeveloped for homology spheres using the theory of spectral flow ofself-adjoint Fredholm operators.

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Comments on the Noncommutative Differential Geometry of Quantum Homogeneous Vector Bundles

Differential calculi are obtained for quantum homogeneous spaces by extend- ing Woronowicz' ... more Differential calculi are obtained for quantum homogeneous spaces by extend- ing Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum homogeneous vector bundles are classified and explicitly constructed by using the theory of projective modules.

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En">The Second Cohomology of sl ( m | 1 ) with Coefficients in its Enveloping Algebra is Trivial

Letters in Mathematical Physics - LETT MATH PHYS, 1999

Using techniques developed in a recent article by the authors, it is proved that the 2-cohomology... more

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Projective Module Description of Embedded Noncommutative Spaces

Reviews in Mathematical Physics, 2010

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra d... more An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. 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 [8]. 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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Cohomology of Lie superalgebras 𝔰𝔩m|nand 𝔬𝔰𝔭2|2n

Proceedings of the London Mathematical Society, 2006

We explicitly compute the first and second cohomology groups of the classical Lie superalgebras s... more We explicitly compute the first and second cohomology groups of the classical Lie superalgebras sl m|n and osp 2|2n with coefficients in the finite dimensional irreducible modules and the Kac modules. We also show that the second cohomology groups of these Lie superalgebras with coefficients in the respective universal enveloping algebras (under the adjoint action) vanish. The latter result in particular implies that the universal enveloping algebras U(sl m|n) and U(osp 2|2n) do not admit any non-trivial formal deformations of Gerstenhaber type. µ and some ϕ(f α) = 0. 12 5.2. The case with all ϕ(e α) ∈ L bottom µ and some ϕ(e α) = 0 13 5.3. The remaining case 13 5.4. Main result on first cohomology groups 15 6. Second cohomology groups with coefficients in irreducible modules 19 6.1. Primitive weight graphs 19 6.2. Technical lemmas 21 6.3. Computation of second cohomology groups 25 7. Cohomology groups with coefficients in enveloping algebra 33 8. Cohomology groups of the Lie superalgebra C(n) 37 Appendix A. Atypicality 38 References 41

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Non-linear sigma models on arbitrary genus Riemann surfaces

Physics Letters B, 1987

A Ward-Takahashi type identity is obtained for two insertions of the energy-momentum tensor of th... more A Ward-Takahashi type identity is obtained for two insertions of the energy-momentum tensor of the non-linear sigma model on an arbitrary Riemann surface. The identity shows explicitly how the Virasoro algebra is violated by spurious terms generated by the trace anomaly. Requiring these terms to vanish leads to a set of constraints on the graviton and dilaton background fields, which are necessary for the algebra to be restored. Although the modular parameters play an important role in the computation, the background field equations turn out to be genus independent up to order a'.

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Grand Unification and Grassmannian Kaluza-Klein Theory

Modern Physics Letters A, 1988

It is possible to construct a grand unified model, including gravity, by adjoining further antico... more It is possible to construct a grand unified model, including gravity, by adjoining further anticommuting coordinates to space-time. We carry out this program for SU(5) and SO(10) unified models; curiously, the former is much more economical than the latter and simply requires five complex Grassmann variables ξm in addition to the four xμ.

The gl (M|N) super Yangian and its finite-dimensional representations

Letters in Mathematical Physics, 1996

Methods are developed for systematically constructing the finite dimensional irreducible represen... more Methods are developed for systematically constructing the finite dimensional irreducible representations of the super Yangian Y (gl(M |N)) associated with the Lie superalgebra gl(M |N). It is also shown that every finite dimensional irreducible representation of Y (gl(M |N)) is of highest weight type, and is uniquely characterized by a highest weight. The necessary and sufficient conditions for an irrep to be finite dimensional are given.

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Unitary representations of basic classical Lie superalgebras

Letters in Mathematical Physics, 1990

We have obtained all the finite-dimensional umtary irreps of gl(m I n) and C(n), which also exhau... more We have obtained all the finite-dimensional umtary irreps of gl(m I n) and C(n), which also exhaust such irreps of all the basic classical Lie superalgebras. The lowest weights of such irreps are worked out explimtly. It is also shown that the contravariant and covariant tensor irreps of gl(m I n) are unitary irreps of type (I) and type (2) respectively, explaining the applicability of the Young diagram method to these two types of tensor irreps.

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Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

Letters in Mathematical Physics, 1992

It is shown that the quantum supergroup Uq(osp(1/2n)) is essentially isomorphic to the quantum gr... more It is shown that the quantum supergroup Uq(osp(1/2n)) is essentially isomorphic to the quantum group U q(SO(2n + 1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of Uq(osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.

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On Endomorphisms of Quantum Tensor Space

Letters in Mathematical Physics, 2008

We give a presentation of the endomorphism algebra End Uq(sl2) (V ⊗r ), where V is the 3-dimensio... more We give a presentation of the endomorphism algebra End Uq(sl2) (V ⊗r ), where V is the 3-dimensional irreducible module for quantum sl 2 over the function field C(q 1 2 ). This will be as a quotient of the Birman-Wenzl-Murakami algebra BM W r (q) := BM W r (q -4 , q 2q -2 ) by an ideal generated by a single idempotent Φ q . Our presentation is in analogy with the case where V is replaced by the 2-dimensional irreducible U q (sl 2 )-module, the BMW algebra is replaced by the Hecke algebra H r (q) of type A r-1 , Φ q is replaced by the quantum alternator in H 3 (q), and the endomorphism algebra is the classical realisation of the Temperley-Lieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the R-matrices on V ⊗r are consequences of relations among the three R-matrices acting on V ⊗4 . The proof makes extensive use of the theory of cellular algebras. Potential applications include the decomposition of tensor powers when q is a root of unity.

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Dual Canonical Bases for the Quantum Special Linear Group and Invariant Subalgebras

Letters in Mathematical Physics, 2005

A string basis is constructed for each subalgebra of invariants of the function algebra on the qu... more A string basis is constructed for each subalgebra of invariants of the function algebra on the quantum special linear group. By analyzing the string basis for a particular subalgebra of invariants, we obtain a "canonical basis" for every finite dimensional irreducible U q (sl(n))-module. It is also shown that the algebra of functions on any quantum homogeneous space is generated by quantum minors.

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The quantum super-Yangian and Casimir operators of Uq(gl(M |N))

Letters in Mathematical Physics, 1995

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BRST cohomology for U q (sl(2)) representations

Journal of Physics A: Mathematical and General, 1992

Page 1. BRST cohomology for Uq(sl(2)) representations This article has been downloaded from IOPsc... more

Multiparameter dependent solutions of the Yang-Baxter equation

Journal of Physics A: Mathematical and General, 1991

A general method is developed for constructing representations of the Temperley-Lieb algebra, whi... more A general method is developed for constructing representations of the Temperley-Lieb algebra, which in turn yield solutions of the Yang-Baxter equation. Several classes of multiparameter dependent R-matrices are obtained using this method, and the connection of the method with quantum groups is discussed.

Topological Invariants for Lens Spaces and Exceptional Quantum Groups

Letters in Mathematical Physics, 1996

The Reshetikhin–Turaev invariants arising from the quantum groups associated with the exceptional... more

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Quantum group actions on rings and equivariant K-theory

Algebraic Groups and Quantum Groups, 2012

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The first fundamental theorem of invariant theory for the orthosymplectic super group

Advances in Mathematics, 2017

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Invariants of the orthosymplectic Lie superalgebra and super Pfaffians

Mathematische Zeitschrift, 2016