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DUAL CANONICAL BASES FOR THE QUANTUM GENERAL LINEAR SUPERGROUP
, 2005
"... Abstract. Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the quantum special linear supergroup Oq(SL mn). We apply ..."
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Abstract. Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the quantum special linear supergroup Oq(SL mn). We apply the canonical bases to study invariant subalgebras of the quantum supergroups under left and right translations. In the case n = 1, it is shown that each invariant subalgebra is spanned by a part of the dual canonical bases. This in turn leads to dual canonical bases for any Kac module constructed by using an analogue of BorelWeil theorem. 1.
TWO EXTERIOR ALGEBRAS FOR ORTHOGONAL AND SYMPLECTIC QUANTUM GROUPS
, 1999
"... Let Γ be one of the N 2dimensional bicovariant first order differential calculi on the quantum groups Oq(N) or Sp q(N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra sΓ ∧ is the quotient of the universal exterior algebra uΓ ∧ by the principal ideal genera ..."
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Let Γ be one of the N 2dimensional bicovariant first order differential calculi on the quantum groups Oq(N) or Sp q(N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra sΓ ∧ is the quotient of the universal exterior algebra uΓ ∧ by the principal ideal generated by θ∧θ. Here θ denotes the unique up to scalars biinvariant 1form. Moreover θ∧θ is central in uΓ ∧ and uΓ ∧ is an inner differential calculus.
(l, q)Deformed Grassmann Field and the Twodimensional Ising Model
, 1995
"... In this paper we construct the exact representation of the Ising partition function in the form of the SLq(2,R)invariant functional integral for the lattice free (l,q)fermion field theory (l = q = −1). It is shown that the (l,q)fermionization allows one to reexpress the partition function of the ..."
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In this paper we construct the exact representation of the Ising partition function in the form of the SLq(2,R)invariant functional integral for the lattice free (l,q)fermion field theory (l = q = −1). It is shown that the (l,q)fermionization allows one to reexpress the partition function of the eightvertex model in external field through functional integral with fourfermion interaction. To construct these representations, we define a lattice (l,q,s)deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At l = q = −1,s = 1 we obtain the lattice (l,q)fermion field which allows us to fermionize the twodimensional Ising model. We show that the Gaussian integral over (q,s)Grassmann variables is expressed through the (q,s)deformed Pfaffian which is equal to square root of the determinant of some matrix at q = ±1,s = ±1.
DAMTP/9147/Revised NIKHEF 95023 Nonstandard Quantum Groups and Superization
, 1995
"... We obtain the universal Rmatrix of the nonstandard quantum group associated to the AlexanderConway knot polynomial. We show further that this nonstandard quantum group is related to the superquantum group Uqgl(11) by a general process of superization, which we describe. We also study a twisted ..."
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We obtain the universal Rmatrix of the nonstandard quantum group associated to the AlexanderConway knot polynomial. We show further that this nonstandard quantum group is related to the superquantum group Uqgl(11) by a general process of superization, which we describe. We also study a twisted variant of this nonstandard quantum group and obtain, as a result, a twisted version of Uqgl(11) as a qsupersymmetry of the exterior differential calculus of any quantum plane of Hecke type, acting by mixing the bosonic xi coordinates and the forms dxi. 1
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, 2005
"... The GL(mn) type quantum matrix algebras II: the structure of the characteristic subalgebra and its spectral parameterization ..."
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The GL(mn) type quantum matrix algebras II: the structure of the characteristic subalgebra and its spectral parameterization
The quantum supergroup SPOq(2n2m) and an SPOq(2n2m)–covariant Quantum Weyl Superalgebra
, 2000
"... Recently, the R–matrix of the symplecto–orthogonal quantum superalgebra Uq(spo(2n2m)) in the vector representation has been calculated. In the present work, this R–matrix is used to introduce the corresponding quantum supergroup SPOq(2n2m) and to construct an SPOq(2n2m)–covariant quantum Weyl sup ..."
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Recently, the R–matrix of the symplecto–orthogonal quantum superalgebra Uq(spo(2n2m)) in the vector representation has been calculated. In the present work, this R–matrix is used to introduce the corresponding quantum supergroup SPOq(2n2m) and to construct an SPOq(2n2m)–covariant quantum Weyl superalgebra.