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Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. (English) Zbl 0694.16008
The “Physics for algebraists” is in the context of quantum mechanics combined with gravity, describing Yang-Baxter equations, position observables, momentum space, momentum and position quantization, etc. This physics leads to the search for self-dual algebraic structures, and finally to a bicrossproduct construction of non-commutative and non- cocommutative Hopf algebras. There are numerous examples, including a modification of the Weyl algebra. In the appropriate context, the compatibility conditions on the structure maps reduce to the classical Yang-Baxter equations. An example is given related to the double Hopf algebra D(H) of Drinfeld.
Reviewer: E.J.Taft

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81S05 Commutation relations and statistics as related to quantum mechanics (general)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text: DOI
[1] Takeuchi, M, Matched pairs of groups and bismash products of Hopf algebras, Comm. algebra, 9, No. 8, (1981) · Zbl 0456.16011
[2] \scV. G. Drinfel’d, Quantum groups, in “ICM Proceedings, Berkeley, 1986.”
[3] Molnar, R, Semidirect products of Hopf algebras, J. algebra, 47, 29-51, (1977) · Zbl 0353.16004
[4] Sweedler, M.E, Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901
[5] Abe, E, Hopf algebras, (1977), Cambridge Univ. Press London/New York
[6] Sakai, S, C∗ algebras and W∗ algebras, (1971), Springer-Verlag New York/Berlin · Zbl 0219.46042
[7] Kobayashi, S; Nomizu, K, Foundations of differential geometry, (1963), Wiley New York · Zbl 0119.37502
[8] Sniatycki, J, Geometric quantization and quantum mechanics, (), No. 30 · Zbl 0429.58007
[9] Majid, S, Hopf algebras for physics at the Planck scale, J. classical quantum gravity, 5, 1587-1606, (1988) · Zbl 0672.16009
[10] Bratteli, O; Robinson, D.W, ()
[11] McConnell, J.C; Sweedler, M.E, Simplicity of smash products, (), 251-266 · Zbl 0221.16009
[12] Einstein, A, The meaning of relativity, (1967), Chapman & Hall London, (1922) · Zbl 0063.01229
[13] Mach, E, The science of mechanics, (1960), Open Court Publ.,, Chap. 2, 1893, transl. T. J. McCormack · JFM 45.0956.03
[14] Mackey, G, Unitary group representations, (1978), Benjamin New York
[15] Segal, I, Mathematical problems of relativistic physics, (1963), Amer. Math. Soc., Providence, RI · Zbl 0112.45307
[16] Moore, G; Seiberg, N, Polynomial equations for rational conformal field theories, Princeton IAS physics preprint IASSNS-HEP-88/18, (1988)
[17] Jones, V.F.R, Subfactors and related topics, (1988), Berkeley mathematics preprint · Zbl 0692.46049
[18] Enock, M; Schwartz, J-M, Une dualité dans LES algèbres de von Neumann, Bull. soc. math. France mem., 44, (1975) · Zbl 0343.46044
[19] Connes, A, C∗ algèbres et géométrie différentielle, C.R. acad. sci. Paris, 290, 599-604, (1980) · Zbl 0433.46057
[20] Rieffel, M.A, Projective modules over higher dimensional non-commutative tori, Canad. J. math., 40, No. 2, 257-338, (1988) · Zbl 0663.46073
[21] Michaelis, W, Lie coalgebras, Adv. in math., 38, 1-54, (1980) · Zbl 0451.16006
[22] Hilton, P.J; Stammbach, U.S, A course in homological algebra, (1970), Springer-Verlag New York/Berlin · Zbl 0863.18001
[23] Drinfel’d, V.G, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Soviet math. dokl., 27, 68, (1983) · Zbl 0526.58017
[24] Babelon, O, Jimbo’s q-analogues and current algebras, Lett. math. phys., 15, 111-117, (1987) · Zbl 0648.16006
[25] Radford, D, Hopf algebras with projection, J. algebra, 92, 322-347, (1985) · Zbl 0549.16003
[26] \scS. Majid, Matched pairs of Lie groups associated to solutions of the classical Yang-Baxter equations, Pacific J. Math., in press. · Zbl 0735.17017
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