Lie monads and dualities. (English) Zbl 1312.17016
Summary: We study dualities between Lie algebras and Lie coalgebras, and their respective (co)representations. To allow a study of dualities in an infinite-dimensional setting, we introduce the notions of Lie monads and Lie comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive monoidal categories. We show that (strong) dualities between Lie algebras and Lie coalgebras are closely related to (iso)morphisms between associated Lie monads and Lie comonads. In the case of a duality between two Hopf algebras – in the sense of Takeuchi – we recover a duality between a Lie algebra and a Lie coalgebra – in the sense defined in this note – by computing the primitive and the indecomposable elements, respectively.
MSC:
| 17B62 | Lie bialgebras; Lie coalgebras |
| 16T05 | Hopf algebras and their applications |
| 18C15 | Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads |
References:
| [1] | Bruguières, A.; Lack, S.; Virelizier, A., Hopf monads on monoidal categories, Adv. Math., 227, 745-800 (2011) · Zbl 1233.18002 |
| [2] | Böhm, G.; Brzeziński, T.; Wisbauer, R., Monads and comonads on module categories, J. Algebra, 322, 1719-1747 (2009) · Zbl 1208.18003 |
| [3] | Bulacu, D.; Caenepeel, S.; Torrecillas, B., The braided monoidal structures on the category of vector spaces graded by the Klein group, Proc. Edinb. Math. Soc., 54, 613-641 (2011) · Zbl 1262.16027 |
| [4] | Borceux, F., Handbook of Categorical Algebra I, Encyclopedia Math. Appl., vol. 50 (1994), Cambridge University Press · Zbl 0803.18001 |
| [5] | Caenepeel, S.; Goyvaerts, I., Monoidal Hom-Hopf algebras, Comm. Algebra, 39, 2216-2240 (2011) · Zbl 1255.16032 |
| [6] | Eilenberg, S.; Moore, J. C., Adjoint functors and triples, Illinois J. Math., 9, 381-398 (1965) · Zbl 0135.02103 |
| [7] | Goyvaerts, I.; Vercruysse, J., A note on the categorification of Lie algebras, (Lie Theory and Its Applications in Physics. Lie Theory and Its Applications in Physics, Springer Proc. Math. Statist., vol. 36 (2013)) · Zbl 1280.17027 |
| [8] | Goyvaerts, I.; Vercruysse, J., On the duality of generalized Lie and Hopf algebras, Adv. Math., 258, 154-190 (2014) · Zbl 1295.16021 |
| [9] | Kassel, C., Quantum Groups, Grad. Texts in Math., vol. 155 (1995), Springer Verlag: Springer Verlag Berlin · Zbl 0808.17003 |
| [10] | Kelly, G. M., Basic concepts of enriched category theory, Repr. Theory Appl. Categ., 10, 1-136 (2005) · Zbl 1086.18001 |
| [11] | Kharchenko, V. K.; Shestakov, I. P., Generalizations of Lie algebras, Adv. Appl. Clifford Algebr., 22, 721-743 (2012) · Zbl 1263.17006 |
| [12] | Majid, S., Quantum and braided-Lie algebras, J. Geom. Phys., 13, 307-356 (1994) · Zbl 0821.17007 |
| [13] | Michaelis, W., Lie coalgebras, Adv. Math., 38, 1-54 (1980) · Zbl 0451.16006 |
| [14] | Michaelis, W., The primitives of the continuous linear dual of a Hopf algebra as the dual Lie algebra of a Lie coalgebra, Contemp. Math., 110, 125-176 (1990) · Zbl 0712.16022 |
| [15] | Nichita, F., Self-inverse Yang-Baxter operators from (co)algebra structures, J. Algebra, 218, 738-759 (1999) · Zbl 0944.16033 |
| [16] | Pareigis, B., On Lie algebras in the category of Yetter-Drinfeld modules, Appl. Categ. Structures, 6, 151-175 (1998) · Zbl 0907.16022 |
| [17] | Porst, H. E., On corings and comodules, Arch. Math., 42, 419-425 (2006) · Zbl 1152.16030 |
| [18] | Takeuchi, M., Survey of braided Hopf algebras, Contemp. Math., 267, 301-323 (2000) · Zbl 0978.16035 |
| [19] | Vercruysse, J., Hopf algebras - variant notions and reconstruction theorems, (Heunen, C.; Sadrzadeh, M.; Grefenstette, E., Compositional Methods in Physics and Linguistics (2013), Oxford University Press) · Zbl 1321.16022 |
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