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  • 1
    Electronic Resource
    Electronic Resource
    Classification of all star and grade star irreps of gl(n||1) (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 1524-1534 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A method for inducing nondegenerate forms on irreducible gl(m||n) modules that implies some general results on star and grade star representations is investigated. These results are applied to obtain a complete classification, in terms of highest weights, of the irreducible star and grade star representations of gl(n||1). It is demonstrated that while gl(n||1) admits a large class of star representations the irreducible grade star representations are comparatively rare. Moreover, for n≠2 all grade star irreducible representations are also star representations and, for n〉2, are atypical. The superalgebra gl(2||1) proves to be a special case and admits a two-parameter family of four-dimensional typical grade star irreducible representations that are not star representations. In particular, typical grade star irreducible representations of gl(n||1) exist only for n=1,2.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528695
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  • 2
    Electronic Resource
    Electronic Resource
    Classification of star and grade-star representations of C(n+1) (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 1889-1897 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The two types of * and grade-* representations of the Lie superalgebra C(n+1) are classified. A type (1) irreducible *-representation is characterized by the single condition (Λ,αs)≥0, Λ and αs being the highest weight and the odd simple root, respectively, while the type (2) *-representations are duals of type (1) *-representations. For n〉1, only the identity and vector representations of C(n+1) are shown to be grade -*. The superalgebra C(2) proves to be an exception and admits several classes of nontrivial irreducible grade -* representations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528687
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  • 3
    Electronic Resource
    Electronic Resource
    Universal R matrices and invariants of quantum supergroups (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 3261-3267 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A general method is developed for constructing invariants of quantum supergroups using universal R matrices. Applied to Uq(gl(2/1)), this method yields the invariants of this quantum supergroup in explicit form. The eigenvalues of these invariants in arbitrary irreducible highest weight representations are computed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529487
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  • 4
    Electronic Resource
    Electronic Resource
    Braid group representations arising from quantum supergroups with arbitrary q and link polynomials (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 3918-3930 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The theory of quantum supergroups is applied to the construction of braid group representations and link invariants. A method developed in an earlier publication is reviewed, which produces link polynomials from braid group representations generated by quantum supergroup invariant R matrices. When q is a generic complex parameter, this method, applied to the universal R matrices, yields hierarchies of link polynomials associated with different irreps of quantum supergroups. When q is a root of unity, suitable constraints are imposed on quantum supergroups to turn them into finite-dimensional quasitriangular Z2-graded Hopf algebras. The corresponding universal R matrices are obtained, which, in turn, lead to series of link polynomials upon using the above mentioned method. As concrete examples, the link polynomials related to the vector irreps of the quantum supergroups Uq(osp(M/2n)), all the irreps of Uq(osp(1/2)) with generic q, and a class of doubly atypical irreps of Uq(gl(2/1)) with q a root of unity are studied in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529840
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  • 5
    Electronic Resource
    Electronic Resource
    Universal L operator and invariants of the quantum supergroup Uq(gl(m/n)) (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 1970-1979 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A spectral parameter-dependent solution of the graded Yang–Baxter equation is obtained, which is universal in the sense that it lives in Uq(gl(m/n))⊗End(V) with V the vector module of Uq(gl(m/n)). The invariants of this quantum supergroup are constructed using this solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529672
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  • 6
    Electronic Resource
    Electronic Resource
    Graded representations of the Temperley–Lieb algebra, quantum supergroups, and the Jones polynomial (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 2605-2613 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown that from each self-dual representation of a quantum supergroup with nonvanishing q-superdimension, a graded representation of the Temperley–Lieb algebra can be constructed, which, in turn, gives rise to a solution of the graded Yang–Baxter equation. As examples, the vector irrep of Uq (osp(M/2n)) and all the finite-dimensional irreps of Uq (osp(1/2)) are studied, and the corresponding graded R-matrices are obtained explicitly. A general method is also given to obtain link polynomials from graded R-matrices. Applied to the ones associated with graded representations of the Temperley–Lieb algebra, the method yields the Jones polynomial.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529105
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  • 7
    Electronic Resource
    Electronic Resource
    Classification of all star irreps of gl(m||n) (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 2552-2559 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: All the finite-dimensional star irreps of gl(m||n) are classified in terms of their highest weights, thereby completing the classification of all finite-dimensional star irreps of the basic classical Lie superalgebras. The lowest weights of such irreps are determined explicitly and it is shown that the contravariant and covariant tensor irreps of gl(m||n) are star irreps of type (1) and (2), respectively. This explains the applicability of the Young diagram method to these two classes of representations. However, it is shown that one-parameter families of typical star irreps occur that are intrinsically different from the tensorial irreps.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529001
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  • 8
    Electronic Resource
    Electronic Resource
    Generalized Gel'fand invariants and characteristic identities for quantum groups (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 2298-2303 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The generalized Gel'fand invariants for an arbitrary quantum group are explicitly constructed, and their eigenvalues in any irreducible representation are computed. These invariants enable one to develop characteristic identities for the quantum group, and as a natural application, these identities are used to construct projection operators for tensor products of representations. To illustrate the general theory, the quantum group Uq(gl(m)) is studied in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529152
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  • 9
    Electronic Resource
    Electronic Resource
    Structure and representations on the quantum supergroup OSPq(2|2n) (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 6639-6656 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The structure and representations of the quantum supergroup OSPq(2|2n) are studied systematically. The algebra of functions on the quantum supergroup, which specifies the quantum supergroup itself, is taken to be the superalgebra generated by the matrix elements of the vector representation of the quantized universal superalgebra Uq(osp(2|2n)). It is shown that the algebra of functions is dense in the full dual Uq(osp(2|2n))* of Uq(osp(2|2n)) and possesses a Hopf superalgebra structure. The left integral and right integral on the quantum supergroup are discussed. Induced representations are developed using the noncommutative geometry of quantum homogeneous supervector bundles, and a geometric realization of irreducible representations is obtained. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1286881
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  • 10
    Electronic Resource
    Electronic Resource
    Symmetrizable quantum affine superalgebras and their representations (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 535-543 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable Cartan matrices are studied. The irreducible integrable highest weight representations are classified, and shown to be deformations of their classical counterparts. It is also shown that Jimbo-type quantum affine superalgebras can be obtained by deforming universal enveloping algebras of ordinary (i.e., nongraded) affine algebras supplemented by certain parity operators. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531833
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