Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum Lie algebra”. (English) Zbl 0751.16014
For a bialgebra \(A\), the category of left \(A\)-modules is a monoidal category. If \(A\) is quasitriangular, then \(_ A\text{Mod}\) is a braided monoidal category. In this case, the matrix \(R=\sum_ iR_ i\otimes R_ i'\) in \(A\otimes A\) involved in the definition is used to construct a map \(\sigma(E,F)\) of \(E\otimes F\) to \(F\otimes E\), for \(E\), \(F\) in \(_ A\text{Mod}\), namely \(\sigma(E,F)\) \((e\otimes f)=\sum_ i(R_ i,f)\otimes (R_ i',e)\), thus replacing the usual twist map which is not a map of left \(A\)-modules. If \(A\) is not finite-dimensional, some applications require \(R\) to be taken in a suitable topological completion of \(A\otimes A\). Majid has tried to circumvent this by considering pairings of bialgebras to the base field. In the paper under review, the authors propose a different approach.
They first consider the category of left \(A\)-comodules, which is a monoidal category in an obviuos way, and ask when it is a braided monoidal category. The answer involves a pairing of \(A\otimes A\) to the base field, and in this case, \(A\) is called a braided bialgebra, and suitable equivalent sets of axioms are developed. If \(R: V\otimes V\to V\otimes V\) for a finite dimensional vector space \(V\), then a somewhat universal construction of a bialgebra \(A(R)\) is given, such that \(V\) is a left \(A(R)\)-comodule and \(R\) is an \(A(R)\)-comodule map. If \(R\) is Yang-Baxter, then \(A(R)\) is a braided bialgebra. The authors feel that working with comodules, rather than modules, avoids topological questions involving completions. The final section of the paper extends the concept of a quasitriangular bialgebra to what the authors call a “completed-triangular” bialgebra. Here the bialgebras are cofinitary topological modules over the base ring \(K\) (a Dedekind domain), the \(R\) involved is in the completed tensor product, and the category of continuous right modules is a braided monoidal category. Also there is a continuous dual which is a braided bialgebra, and over a field there is a suitable converse (i.e. every braided bialgebra gives rise to a completed-triangular bialgebra).
They first consider the category of left \(A\)-comodules, which is a monoidal category in an obviuos way, and ask when it is a braided monoidal category. The answer involves a pairing of \(A\otimes A\) to the base field, and in this case, \(A\) is called a braided bialgebra, and suitable equivalent sets of axioms are developed. If \(R: V\otimes V\to V\otimes V\) for a finite dimensional vector space \(V\), then a somewhat universal construction of a bialgebra \(A(R)\) is given, such that \(V\) is a left \(A(R)\)-comodule and \(R\) is an \(A(R)\)-comodule map. If \(R\) is Yang-Baxter, then \(A(R)\) is a braided bialgebra. The authors feel that working with comodules, rather than modules, avoids topological questions involving completions. The final section of the paper extends the concept of a quasitriangular bialgebra to what the authors call a “completed-triangular” bialgebra. Here the bialgebras are cofinitary topological modules over the base ring \(K\) (a Dedekind domain), the \(R\) involved is in the completed tensor product, and the category of continuous right modules is a braided monoidal category. Also there is a continuous dual which is a braided bialgebra, and over a field there is a suitable converse (i.e. every braided bialgebra gives rise to a completed-triangular bialgebra).
Reviewer: E.J.Taft (New Brunswick)
MSC:
16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
57T05 | Hopf algebras (aspects of homology and homotopy of topological groups) |
18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
16D90 | Module categories in associative algebras |
18E30 | Derived categories, triangulated categories (MSC2010) |
Keywords:
category of left modules; braided monoidal category; twist map; pairings of bialgebras; category of left comodules; braided bialgebra; quasitriangular bialgebra; cofinitary topological modulesReferences:
[1] | Abe E., Hopf (1980) |
[2] | Artin E., Hamb. Abh. pp 47– (1925) · JFM 51.0450.01 · doi:10.1007/BF02950718 |
[3] | DOI: 10.1016/0001-8708(87)90034-X · Zbl 0633.16001 · doi:10.1016/0001-8708(87)90034-X |
[4] | Baxter R. J., Exactly Solvable Models in Statistical Mechanics (1953) |
[5] | Belavin A. A., Func. Anal. and Appl. 16 pp 1– (1982) · Zbl 0498.32010 · doi:10.1007/BF01081801 |
[6] | Cartan H., Homological Algebra (1956) |
[7] | Deligne P., Lecture Notes in Math. 900 |
[8] | Drinfel’d V. G., Sov. Math. Dokl. 32 pp 254– (1985) |
[9] | Drinfel’d V. G., Zap. Nauch. Semin. Leningr. Otd. Mat. Inst. 155 pp 18– (1986) |
[10] | Drinfel’d V. G., Proceedings of the International Congress of Mathematicians pp 798– (1987) |
[11] | Drinfel’d V. G., Alg. and Anali. 1 pp 30– (1989) |
[12] | DOI: 10.1090/S0273-0979-1985-15361-3 · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3 |
[13] | Jacobson N., Amer. Math. Soc. (1956) |
[14] | DOI: 10.1007/BF00704588 · Zbl 0587.17004 · doi:10.1007/BF00704588 |
[15] | DOI: 10.1007/BF00400222 · Zbl 0602.17005 · doi:10.1007/BF00400222 |
[16] | Jimbo M., Comm. Math. Physics 102 pp 537– (1985) · Zbl 0604.58013 · doi:10.1007/BF01221646 |
[17] | DOI: 10.2307/1971403 · Zbl 0631.57005 · doi:10.2307/1971403 |
[18] | DOI: 10.1090/S0273-0979-1985-15304-2 · Zbl 0564.57006 · doi:10.1090/S0273-0979-1985-15304-2 |
[19] | Joyal A., Macquarie Math. Reports (1986) |
[20] | Kaufman L. H., Conference 163 pp 137– |
[21] | Kelley J.L., General Topology (1955) · Zbl 0066.16604 |
[22] | Kirillov A. N., Quantum Clebsch-Gordan coefficients 168 pp 67– (1988) · Zbl 0684.17003 |
[23] | Kirillov A. N., q-orthogonal polynomials and invariants of links LOMI (1988) |
[24] | Kohno T., Int. Conf. in Honour of J. K. Koszul (1987) |
[25] | Kulish P. P., A two-parameter quantum group and a guage transform 180 pp 89– (1990) |
[26] | Kulish P. P., On the solution of the Yang-Baxter equations 95 pp 129– (1980) |
[27] | DOI: 10.1016/0021-8693(71)90018-4 · Zbl 0217.33801 · doi:10.1016/0021-8693(71)90018-4 |
[28] | Lusztig G., Adv. in Math. 70 pp 237– (1988) · Zbl 0651.17007 · doi:10.1016/0001-8708(88)90056-4 |
[29] | Lusztig G., Contemp. Math. 82 pp 59– (1989) · doi:10.1090/conm/082/982278 |
[30] | Lyubashenko V. V., Usp. Mat. Nauk 41 pp 185– (1986) |
[31] | MacLane S., Rice University studies 49 pp 28– (1963) |
[32] | Majid S., Int. J. Math. Physics A (1989) |
[33] | Majid S., Supl. Rend. Circ. Mat. Palermo |
[34] | Majid S., Israel J. Math. |
[35] | Manin Yu., U. of Montreal Lectures (1988) |
[36] | Radford D. E., J. Algebra |
[37] | Reshitikhin N. Yu., LOMI reprints E–4–87 pp 17– |
[38] | Reshetikhin N. Yu., Invariants of 3-manifolds via link polynomials and quantum groups · Zbl 0725.57007 · doi:10.1007/BF01239527 |
[39] | DOI: 10.1007/BF01218386 · Zbl 0651.17008 · doi:10.1007/BF01218386 |
[40] | Rosso M., 1 305 pp 587– (1987) |
[41] | Sweedler M., Hopf Algebras (1968) |
[42] | Taft E., Journal of Algebra · Zbl 0169.05204 |
[43] | Takeuchi M., Proc. Japan Acad. 65 (1989) |
[44] | DOI: 10.3792/pjaa.66.112 · Zbl 0723.17012 · doi:10.3792/pjaa.66.112 |
[45] | Takhtajan L. A., Quantum inverse scattering method and the algebrized matrix Bethe ansatz 101 pp 158– (1981) |
[46] | Tsuchiya A., LMP 13 pp 303– (1987) |
[47] | DOI: 10.1007/BF01393746 · Zbl 0648.57003 · doi:10.1007/BF01393746 |
[48] | Ulbrich K.-H., Israel J. Math. |
[49] | Verlinde E., Nuc. Phys. 300 (1988) · Zbl 1180.81120 · doi:10.1016/0550-3213(88)90603-7 |
[50] | Yetter D. N., Math. Proc. Cambridge Phil. Soc. |
[51] | DOI: 10.1007/BF01217730 · Zbl 0667.57005 · doi:10.1007/BF01217730 |
[52] | Woronowicz , S. L. 1987.Twisted SU(2) group. An example of a non–commutative differential calculus, Vol. 23, 117–181. Publ. RIMS, Kyoto Univ. · Zbl 0676.46050 |
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