Abstract
We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ1 over a Hopf algebra A which are quotients of the augmentation ideal A
+ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation provides a bicovariant differential graded algebra on A. We introduce Λ
m
a
x
providing the maximal prolongation, while the canonical braided-exterior algebra Λ
min
= B
−(Λ1) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ♯ by super-braided Fourier transform on B
−(Λ1) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S
3] with its 3D calculus and obeying the q-Hecke relation ♯2 = 1 + (q − q
−1)♯ in middle degree on k
q
[S
L
2] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra \(\mathcal {L}\subseteq A\) defines a sub-braided Lie algebra and \({\Lambda }^{1}\subseteq \mathcal {L}^{*}\) provides the required data A
+ → Λ1.
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References
Bazlov, Y.: Nichols-Woronowicz algebra model for Schubert calculus on Coxeter groups. J. Algebra 297, 372–399 (2006)
Bespalov, Y., Kerler, T., Lyubashenko, V., Turaev, V.: Integrals for braided Hopf algebras. J. Pure Appl. Algebra 148, 113–164 (2000)
Brzezinski, T.: Remarks on bicovariant differential calculi and exterior Hopf algebras. Lett. Math. Phys. 27, 287–300 (1993)
Castellani, L., Catenacci, R., Grassi, P.A.: The geometry of supermanifolds and new supersymmetric actions. Nucl. Phys. B 899, 112–148 (2015)
Drinfeld, V.G.: Quantum groups. In: Proceedings of the ICM. AMS (1987)
Fomin, S., Kirillov, A.N.: Quadratic algebras, Dunkl elements, and Schubert calculus. Adv. Geom, Progr. Math. 172, 147–182 (1989)
Gomez, X., Majid, S.: Noncommutative cohomology and electromagnetism on \(\mathbb {C}_{q}[SL2]\) at roots of unity. Lett. Math. Phys. 60, 221–237 (2002)
Gomez, X., Majid, S.: Braided Lie algebras and bicovariant differential calculi over coquasitriangular Hopf algebras. J. Algebra 261, 334–388 (2003)
Heckenberger, I.: Hodge and Laplace-Beltrami operators for bicovariant differential calculi on quantum groups. Compos. Math. 123, 329–354 (2000)
Jurco, B.: Differential calculus on quantized simple Lie groups. Lett. Math. Phys. 22, 177–186 (1991)
Kempf, A., Majid, S.: Algebraic q-integration and Fourier theory on quantum and braided spaces. J. Math. Phys. 35, 6802–6837 (1994)
Kustermans, J., Murphy, G.J., Tuset, L.: Quantum groups, differential calculi and the eigenvalues of the Laplacian. Trans. Amer. Math. Soc. 357, 4681–4717 (2005)
Lopez Pena, J., Majid, S., Rietsch, K.: Lie theory of finite simple groups and the Roth property, in press Math. Proc. Camb. Phil. Soc. 39pp
Lusztig, G.: Introduction to Quantum Groups. Birkhauser (1993)
Lyubashenko, V., Majid, S.: Braided groups and quantum Fourier transform. J. Algebra 166, 506–528 (1994)
Lyubashenko, V.: Modular transformations for tensor categories. J. Pure Appl. Algebra 98, 279–327 (1995)
Majid, S.: The self-representing Universe. In: Eckstein, M., Heller, M., Szybka, S. (eds.) Mathematical Structures of he Universe, pp. 357–387. Copernicus Center Press (2014)
Majid, S.: Foundations of Quantum Group Theory, p. 640. C.U.P (2000)
Majid, S.: Algebras and Hopf algebras in braided categories. In: Lecture Notes Pure and Applied Maths, vol. 158, pp. 55–105. Marcel Dekker (1994)
Majid, S.: Examples of braided groups and braided matrices. J. Math. Phys. 32, 3246–3253 (1991)
Majid, S.: Braided groups. J. Pure Appl. Algebra 86, 187–221 (1993)
Majid, S.: Cross products by braided groups and bosonization. J. Algebra 163, 165–190 (1994)
Majid, S.: Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group. Comm. Math. Phys. 156, 607–638 (1993)
Majid, S.: Free braided differential calculus, braided binomial theorem and the braided exponential map. J. Math. Phys. 34, 4843–4856 (1993)
Majid, S.: Quantum and braided linear algebra. J. Math. Phys. 34, 1176–1196 (1993)
Majid, S.: Quantum and braided Lie algebras. J. Geom. Phys. 13, 307–356 (1994)
Majid, S.: Double bosonisation of braided groups and the construction of U q (g). Math. Proc. Camb. Phil. Soc. 125, 151–192 (1999)
Majid, S.: Classification of differentials on quantum doubles and finite noncommutative geometry. In: Lecture Notes Pure Applied Mathematics, vol. 239, pp. 167–188. Marcel Dekker (2004)
Majid, S.: Noncommutative differentials and Yang-Mills on permutation groups S N . In: Lecture Notes Pure Applied Mathematics, vol. 239, pp. 189–214. Marcel Dekker (2004)
Majid, S.: Noncommutative Ricci curvature and Dirac operator on \(\mathbb {C}_{q}[SL2]\) at roots of unity. Lett. Math. Phys. 63, 39–54 (2003)
Majid, S.: Algebraic approach to quantum gravity III: Noncommutative Riemannian geometry. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds.) Mathematical and Physical Aspects of Quantum Gravity, pp. 77–100. Birkhauser (2006)
Majid, S.: Reconstruction and quantisation of Riemannian structures, 40p. arXiv:1307.2778 (math.QA)
Majid, S., Raineri, E.: Electromagnetism and gauge theory on the permutation group S 3. J. Geom. Phys. 44, 129–155 (2002)
Majid, S., Rietsch, K.: Lie theory and coverings of finite groups. J. Algebra 389, 137–150 (2013)
Majid, S., Tao, W.-Q.: Duality for generalised differentials on quantum groups. J. Algebra 439, 67–109 (2015)
Manin, Yu.: Quantum Groups and Noncommutative Geometry. Centre De Recherches Mathematiques (1988)
Nichols, W.: Bialgebras of type I. Comm. Algebra 15, 1521–1552 (1978)
Radford, D.: The structure of Hopf algebras with a projection. J. Algebra 92, 322–347 (1985)
Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Comm. Math. Phys. 122, 125–170 (1989)
Whitehead, J.H.C.: Combinatorial homotopy, II. Bull. Amer. Math. Soc. 55, 453–496 (1949)
Yetter, D.N.: Quantum groups and representations of monoidal categories. Math. Proc. Camb. Phil. Soc. 108, 261–290 (1990)
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Majid, S. Hodge Star as Braided Fourier Transform. Algebr Represent Theor 20, 695–733 (2017). https://doi.org/10.1007/s10468-016-9661-0
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DOI: https://doi.org/10.1007/s10468-016-9661-0
Keywords
- Noncommutative geometry
- Differential calculus
- Finite groups
- Hopf algebra
- Bicovariant
- Quantum group
- Fourier duality
- q-Hecke algebra
- Maxwell equations