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- Beidar, K.I., et al.
(författare)
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On antipodes and integrals in Hopf algebras over rings and the quantum Yang-Baxter equation
- 1997
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Ingår i: Journal of Algebra. - : Elsevier BV. - 1090-266X .- 0021-8693. ; 194:1, s. 36-52
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Tidskriftsartikel (refereegranskat)abstract
- The authors showed previously (on Frobenius algebras and quantum Yang-Baxter equation, II, preprint, TRITA-MAT-1995, February 1995) that every Frobenius algebra over a commutative ring defines a solution of the quantum Yang-Baxter equation. Applying this result to Hopf algebras over commutative rings which are finitely generated and projective as modules, we obtain an explicit formula for this solution. It turns out that this solution can be expressed in terms of the integral and antipode. We use this solution to characterize separable Hopf algebras over rings. Some results on the order of the antipode are also obtained.
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| 2. |
- Beidar, K.I., et al.
(författare)
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On Frobenius algebras and the quantum Yang-Baxter equation
- 1997
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Ingår i: Transactions of the American Mathematical Society. - 0002-9947 .- 1088-6850. ; 349:9, s. 3823-3836
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that every Frobenius algebra over a commutative ring determines a class of solutions of the quantum Yang-Baxter equation, which forms a subbimodule of its tensor square. Moreover, this subbimodule is free of rank one as a left (right) submodule. An explicit form of a generator is given in terms of the Frobenius homomorphism. It turns out that the generator is invertible in the tensor square if and only if the algebra is Azumaya.
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| 8. |
- Khoroshkin, S., et al.
(författare)
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Classical limit of the scaled elliptic algebra A(h,eta)((sl)over-cap(2))
- 1999
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Ingår i: Compositio Mathematica. - 0010-437X .- 1570-5846. ; 115:2, s. 205-230
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Tidskriftsartikel (refereegranskat)abstract
- The classical limit of the scaled elliptic algebra A((h) over bar eta)( (2)) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl(2) valued functions on a strip and as an extended algebra of decreasing automorphic sl(2) valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra A(q,p)( (2)) is also briefly presented.
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| 9. |
- Khoroshkin, S.M., et al.
(författare)
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Deformation of Yangian Y(sl(2))
- 1998
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Ingår i: Communications in Algebra. - : Informa UK Limited. - 0092-7872 .- 1532-4125. ; 26:4, s. 1041-1055
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Tidskriftsartikel (refereegranskat)abstract
- A quantization of a non-standard rational solution of CYBE for sit is given explicitly. We obtain the quantization with the help of a twisting of the usual Yangian Y (sl(2)). This quantum object (deformed Yangian Y-eta,Y-xi(sl(2))) is a two-parametric deformation of the universal enveloping algebra U(sl(2)[u]) of the polynomial current algebra sl(2)[u]. We consider the pseudotriangular structure on Y-eta,Y-xi(sl(2)), the quantum double DYeta,xi(sl(2)), its the universal R-matrix and also the RTT-realization of Y-eta,Y-xi(sl(2)).
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| 10. |
- Khoroshkin, S. M., et al.
(författare)
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Generalized Gauss decomposition of trigonometric R-matrices
- 1995
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Ingår i: Modern Physics Letters A. - 0217-7323. ; 10:19, s. 1375-1392
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Tidskriftsartikel (refereegranskat)abstract
- The general formula for the universal R-matrix for quantified nontwisted affine algebras, obtained by the first and third authors, is applied to zero central charge, highest weight modules of the quantized affine algebras. It is shown how the universal R-matrix produces the Gauss decomposition of trigonometric R-matrix in tenser product of these modules. In particular, A(1)((1)) current realization of the universal R-matrix is presented. It gives a new universal presentation for the trigonometric R-matrix with a parameter in tenser product of U-q(sl(2))-Verma modules. Detailed analysis of a scalar factor arising in finite-dimensional representations of the universal R-matrix for any U-q(g) is given. We interpret this scalar factor as a multiplicative bilinear form on highest weight polynomials of irreducible representations and express this form in terms of infinite q-shifted factorials.
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