Quantised $\mathfrak{sl}_2$-differential algebras

Krutov, Andrey ; Pandžić, Pavle

Archivum mathematicum, Tome 60 (2024) no. 5, pp. 351-364

Abstract (VO)

We propose a definition of a quantised ${\mathfrak{sl}}_2$-differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of ${\mathfrak{sl}}_2$ are natural examples of such algebras.

Détail

DOI : 10.5817/AM2024-5-351

Classification : 16T20, 17B37, 81R50
Keywords: quantum group; Clifford algebra; quantised exterior algebra; $\mathfrak{g}$-differential algebra

Bibliographie

[1] Alekseev, A., Krutov, A.: Group-valued moment maps on even and odd simple $\mathfrak{g}$-modules. In preparations.

[2] Alekseev, A., Meinrenken, E.: The non-commutative Weil algebra. Invent. Math. 139 (1) (2000), 135–172, arXiv:math/9903052 [math.DG]. | DOI | MR

[3] Alekseev, A., Meinrenken, E.: Lie theory and the Chern-Weil homomorphism. Ann. Sci. Éc. Norm. Supér. (4) 38 (2) (2005), 303–338, arXiv:math/0308135 [math.RT]. | MR

[4] Aschieri, P., Castellani, L.: An introduction to noncommutative differential geometry on quantum groups. Internat. J. Modern Phys. A 8 (10) (1993), 1667–1706. | DOI | MR

[5] Aschieri, P., Schupp, P.: Vector fields on quantum groups. Internat. J. Modern Phys. A 11 (6) (1996), 1077–1100. | DOI | MR

[6] Berenstein, A., Zwicknagl, S.: Braided symmetric and exterior algebras. Trans. Amer. Math. Soc. 360 (71) (2008), 3429–3472, arXiv:math/0504155 [math.QA]. | DOI | MR

[7] Bouarroudj, S., Krutov, A., Leites, D., Shchepochkina, I.: Non-degenerate invariant (super) symmetric bilinear forms on simple Lie (super)algebras. Algebr. Represent. Theory 21 (5) (2018), 897–941, arXiv:1806.05505 [math.RT]. | DOI | MR

[8] Cartan, H.: La transgression dans un groupe de Lie et dans un espace fibré principal. Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 57–71. | MR

[9] Cartan, H.: Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie. Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson $\&$ Cie, Paris, 1951, pp. 15–27. | MR

[10] Cheng, S.-J.: Differentiably simple Lie superalgebras and representations of semisimple Lie superalgebras. J. Algebra 173 (1) (1995), 1–43. | DOI | MR

[11] Drinfeld, V. G.: Quasi-Hopf algebras. Algebra i Analiz 1 (6) (1989), 114–148. | MR

[12] Drinfeld, V.G.: Quantum groups. Proceedings of the International Congress of Mathematicians, Amer. Math. Soc., Providence, RI, Berkeley, Calif., 1986, 1987, pp. 798–820. | MR

[13] Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2015. | MR

[14] Guillemin, V.W., Sternberg, S.: Supersymmetry and equivariant de Rham theory. Mathematics Past and Present, Springer-Verlag, Berlin, 1999, With an appendix containing two reprints by Henri Cartan [MR0042426 (13,107e); MR0042427 (13,107f)]. | MR

[15] Henriques, A., Kamnitzer, J.: Crystals and coboundary categories. Duke Math. J. 132 (2) (2006), 191–216. | DOI | MR

[16] Huang, J.-S., Pandžić, P.: Dirac cohomology, unitary representations and a proof of a conjecture of Vogan. J. Amer. Math. Soc. 15 (1) (2002), 185–202. | DOI | MR

[17] Huang, J.-S., Pandžić, P.: Dirac operators in representation theory. Mathematics: Theory $\&$ Applications, Birkhäuser Boston, Inc., Boston, MA,, 2006. | MR | Zbl

[18] Jurčo, B.: Differential calculus on quantized simple Lie groups. Lett. Math. Phys. 22 (3) (1991), 177–186. | DOI | MR

[19] Klimyk, A., Schmüdgen, K.: Quantum groups and their representations. Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. | MR

[20] Kostant, B.: Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the $\rho $-decomposition $C(\mathfrak{g}) =$ End$V_\rho \otimes C(P)$, and the $\mathfrak{g}$-module structure of $\bigwedge \mathfrak{g}$. Adv. Math. 125 (2) (1997), 275–350. | DOI | MR

[21] Krutov, A., Pandžić, P.: Cubic Dirac operator for $U_q({\mathfrak{sl}}_2)$. arXiv:2209.09591 [math.RT].

[22] Krutov, A.O., Ó Buachalla, R., Strung, K.R.: Nichols algebras and quantum principal bundles. Int. Math. Res. Not. 2023 (23) (2023), 20076–20117. | DOI | MR

[23] Lu, J.-H.: Moment maps at the quantum level. Comm. Math. Phys. 157 (2) (1993), 389–404. | DOI | MR

[24] Meinrenken, E.: Clifford algebras and Lie theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 58, Springer, Heidelberg, 2013. | MR

[25] Schupp, P., Watts, P., Zumino, B.: Cartan calculus on quantum Lie algebras. Differential geometric methods in theoretical physics (Ixtapa-Zihuatanejo, 1993), 1994, pp. 125–134. | MR

[26] Woronowicz, S.L.: Twisted SU(2) group. An example of a noncommutative differential calculus. (2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1) (1987), 117–181. | DOI | MR