Article
Full entry |
PDF
(0.6 MB)
Feedback
Keywords:
symplectic Lie groups; quasi-Frobenius Lie algebras; Lie bialgebras; Drinfeld double; group actions
symplectic Lie groups; quasi-Frobenius Lie algebras; Lie bialgebras; Drinfeld double; group actions
Summary:
A Lie version of Turaev’s $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $\mathfrak{g}$-quasi-Frobenius Lie algebra for $\mathfrak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\mathfrak{q},\beta )$ together with a left $\mathfrak{g}$-module structure which acts on $\mathfrak{q}$ via derivations and for which $\beta $ is $\mathfrak{g}$-invariant. Geometrically, $\mathfrak{g}$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group $G$ which acts via symplectic Lie group automorphisms. In addition to geometry, $\mathfrak{g}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\mathfrak{g}$-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in $\mathbf{Rep}(\mathfrak{g})$. If $\mathfrak{g}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline{G}$-Frobenius algebras given in [16] suggests that the Lie version of a $\overline{G}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf{Rep}(D(\mathfrak{g}))$, where $D(\mathfrak{g})$ is the associated (semiclassical) Drinfeld double. We show that if $\mathfrak{g}$ is a quasitriangular Lie bialgebra, then every $\mathfrak{g}$-quasi-Frobenius Lie algebra has an induced $D(\mathfrak{g})$-action which gives it the structure of a $D(\mathfrak{g})$-quasi-Frobenius Lie algebra.
References:
[1] Abrams, L.: Two dimensional topological quantum field theories and Frobenius algebras. J. Knot Theory Ramifications 5 (1996), 569–587. DOI 10.1142/S0218216596000333 | MR 1414088 | Zbl 0897.57015
[2] Agaoka, Y.: Uniqueness of left invariant symplectic structures on the affine Lie group. Proc. Amer. Math. Soc. 129 (9) (2001), 2753–2762, (electronic). DOI 10.1090/S0002-9939-01-05828-2 | MR 1838799 | Zbl 1021.53052
[3] Atiyah, M.F.: Topological quantum field theory. Publ. Math. Inst. Hautes Études Sci. 68 (1988), 175–186. DOI 10.1007/BF02698547 | MR 1001453
[4] Baues, O., Cortés, V.: Symplectic Lie Groups I–III. arXiv:1307.1629.
[5] Boyom, N.: Models for solvable symplectic Lie groups. Indiana Univ. Math. J. 42 (4) (1993), 1149–1168. DOI 10.1512/iumj.1993.42.42053 | MR 1266088 | Zbl 0846.58023
[6] Burde, D.: Characteristically nilpotent Lie algebras and symplectic structures. Forum Math. 18 (5) (2006), 769–787. DOI 10.1515/FORUM.2006.038 | MR 2265899 | Zbl 1206.17009
[7] Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, 1994. MR 1300632 | Zbl 0839.17009
[8] Chu, B.: Symplectic homogenous spaces. Trans. Amer. Math. Soc. 197 (1974), 145–159. DOI 10.1090/S0002-9947-1974-0342642-7 | MR 0342642
[9] Drinfeld, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the classical Yang-Baxter equations. Dokl. Akad. Nauk SSSR 268 (2) (1983), 285–287, (Russian). MR 0688240 | Zbl 0526.58017
[10] Etingof, P., Schiffman, O.: Lectures on Quantum Groups. Lect. Math. Phys., Int. Press, 1998. MR 1698405
[11] Golubitsky, M., Guillemin, V.: Stable mappings and their singularities. Grad. Texts in Math., vol. 14, Springer, Berlin, 1973. DOI 10.1007/978-1-4615-7904-5_3 | MR 0341518 | Zbl 0294.58004
[12] Goyvaerts, I., Vercruysse, J.: A Note on the Categorification of Lie Algebras. Lie Theory and Its Applications in Physics, Springer Proceedings in Math. $\&$ Stat., 2013, pp. 541–550. MR 3070680 | Zbl 1280.17027
[13] Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall, 1974. MR 0348781 | Zbl 0361.57001
[14] Helgason, S.: Differential Geometry, Lie groups, and Symmetric Spaces. Pure Appl. Math., 1978. MR 1834454 | Zbl 0451.53038
[15] Kaufmann, R.: Orbifolding Frobenius algebras. Internat. J. Math. 14 (6) (2003), 573–617. DOI 10.1142/S0129167X03001831 | MR 1997832 | Zbl 1083.57037
[16] Kaufmann, R., Pham, D.: The Drinfel’d double and twisting in stringy orbifold theory. Internat. J. Math. 20 (5) (2009), 623–657. DOI 10.1142/S0129167X09005431 | MR 2526310 | Zbl 1174.14048
[17] Kosmann-Schwarzbach, Y.: Poisson-Drinfel’d groups. Publ. Inst. Rech. Math. Av. 5 (12) (1987).
[18] Kosmann-Schwarzbach, Y.: Lie Bialgebras, Poisson Lie groups and dressing transformations. Integrability of nonlinear systems (Pondicherry, 1996), vol. 495, Lecture Notes in Phys., 1997, pp. 104–170. MR 1636293 | Zbl 1078.37517
[19] Lee, J.: Introduction to Smooth Manifolds. Springer-Verlag, New York Inc., 2003. MR 1930091
[20] Lu, J., Weinstein, A.: Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Differential Geom. 31 (2) (1990), 501–526. DOI 10.4310/jdg/1214444324 | MR 1037412 | Zbl 0673.58018
[21] Mikami, K.: Symplectic and Poisson structures on some loop groups. Contemp. Math. 179 (1994), 173–192. DOI 10.1090/conm/179/01940 | MR 1319608 | Zbl 0820.58021
[22] Ooms, A.: On Lie algebras having a primitive universal enveloping algebra. J. Algebra 32 (1974), 488–500. DOI 10.1016/0021-8693(74)90154-9 | MR 0387365 | Zbl 0355.17014
[23] Ooms, A.: On Frobenius Lie algebras. Comm. Algebra 8 (1) (1980), 13–52. DOI 10.1080/00927878008822445 | MR 0556091 | Zbl 0421.17004
[24] Semenov-Tian-Shansky, M.A.: What is a classical $r$-matrix?. Funct. Anal. Appl. 17 (1983), 259–272. DOI 10.1007/BF01076717
[25] Turaev, V.: Homotopy field theory in dimension 2 and group-algebras. arXiv.org:math/9910010, (1999).
[26] Vinberg, E.B.: A course in algebra. Graduate Studies in Math., vol. 56, AMS, 2003. MR 1974508 | Zbl 1016.00003
[27] Warner, F.: Foundations of Differentiable Manifolds and Lie Groups. Springer, 1983. MR 0722297 | Zbl 0516.58001
[28] Witten, E.: Topological quantum field theory. Comm. Math. Phys. 117 (1988), 353–386. DOI 10.1007/BF01223371 | MR 0953828 | Zbl 0656.53078
Search
Partner of