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**1 - 6**of**6**### The quantum supergroup SPOq(2n|2m) and an SPOq(2n|2m)–covariant

, 2000

"... Recently, the R–matrix of the symplecto–orthogonal quantum superalgebra Uq(spo(2n|2m)) in the vector representation has been calculated. In the present work, this R–matrix is used to introduce the corresponding quantum supergroup SPOq(2n|2m) and to construct an SPOq(2n|2m)–covariant quantum Weyl sup ..."

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Recently, the R–matrix of the symplecto–orthogonal quantum superalgebra Uq(spo(2n|2m)) in the vector representation has been calculated. In the present work, this R–matrix is used to introduce the corresponding quantum supergroup SPOq(2n|2m) and to construct an SPOq(2n|2m)–covariant quantum Weyl superalgebra. math.QA/0004033

### unknown title

, 2005

"... The GL(m|n) type quantum matrix algebras II: the structure of the characteristic subalgebra and its spectral parameterization ..."

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The GL(m|n) type quantum matrix algebras II: the structure of the characteristic subalgebra and its spectral parameterization

### TWO EXTERIOR ALGEBRAS FOR ORTHOGONAL AND SYMPLECTIC QUANTUM GROUPS

, 1999

"... Abstract. Let Γ be one of the N 2-dimensional 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 ide ..."

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Abstract. Let Γ be one of the N 2-dimensional 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 bi-invariant 1-form. Moreover θ∧θ is central in uΓ ∧ and uΓ ∧ is an inner differential calculus. AMS subject classification: 58B30, 81R50 1.

### 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 m|n). 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 m|n). 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 Borel-Weil theorem. 1.

### hep-th/9501116 (l, q)-Deformed Grassmann Field and the Two-dimensional 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 re-express 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 re-express the partition function of the eight-vertex model in external field through functional integral with four-fermion 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 two-dimensional 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. 1 During the last decade, a considerable progress in understanding the meaning of the infinitedimensional

### DAMTP/91-47/Revised NIKHEF 95-023 Non-standard Quantum Groups and Superization

, 1995

"... We obtain the universal R-matrix of the non-standard quantum group associated to the Alexander-Conway knot polynomial. We show further that this nonstandard quantum group is related to the super-quantum group Uqgl(1|1) by a general process of superization, which we describe. We also study a twisted ..."

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We obtain the universal R-matrix of the non-standard quantum group associated to the Alexander-Conway knot polynomial. We show further that this nonstandard quantum group is related to the super-quantum group Uqgl(1|1) by a general process of superization, which we describe. We also study a twisted variant of this non-standard quantum group and obtain, as a result, a twisted version of Uqgl(1|1) as a q-supersymmetry of the exterior differential calculus of any quantum plane of Hecke type, acting by mixing the bosonic xi co-ordinates and the forms dxi. 1