Gelfand-Kirillov dimension of the quantized algebra of regular functions on homogeneous spaces

Chakraborty, Partha; Saurabh, Bipul

doi:10.1090/proc/14522

Gelfand-Kirillov dimension of the quantized algebra of regular functions on homogeneous spaces
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Proc. Amer. Math. Soc. 147 (2019), 3289-3302 Request permission

Abstract:

In this article, we prove that the Gelfand-Kirillov dimension of the quantized algebra of regular functions on certain homogeneous spaces of types $A$, $C$, and $D$ is equal to the dimension of the homogeneous space as a real differentiable manifold.

References

Teodor Banica and Roland Vergnioux, Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), no. 2, 321–340. MR 2541400, DOI 10.1142/S0219025709003677
Teodor Banica, Fusion rules for representations of compact quantum groups, Exposition. Math. 17 (1999), no. 4, 313–337. MR 1734250
Julien Bichon, An De Rijdt, and Stefaan Vaes, Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (2006), no. 3, 703–728. MR 2202309, DOI 10.1007/s00220-005-1442-2
José L. Bueso, J. Gómez Torrecillas, F. J. Lobillo, and F. J. Castro, An introduction to effective calculus in quantum groups, Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996) Lecture Notes in Pure and Appl. Math., vol. 197, Dekker, New York, 1998, pp. 55–83. MR 1615821
Partha Sarathi Chakraborty and Bipul Saurabh, Gelfand-Kirillov dimension of some simple unitarizable modules, J. Algebra 514 (2018), 199–218. MR 3853063, DOI 10.1016/j.jalgebra.2018.08.007
Alexandru Chirvasitu, Chelsea Walton, and Xingting Wang, Gelfand-Kirillov dimension of cosemisimple Hopf algebras, arXiv:1807.01894v2, (2018).
Alessandro D’Andrea, Claudia Pinzari, and Stefano Rossi, Polynomial growth of discrete quantum groups, topological dimension of the dual and $*$-regularity of the Fourier algebra, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 5, 2003–2027 (English, with English and French summaries). MR 3732682, DOI 10.5802/aif.3127
I. M. Gelfand and A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 5–19 (French). MR 207918
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Anatoli Klimyk and Konrad Schmüdgen, Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1492989, DOI 10.1007/978-3-642-60896-4
Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups. Part I, Mathematical Surveys and Monographs, vol. 56, American Mathematical Society, Providence, RI, 1998. MR 1614943, DOI 10.1090/surv/056
Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
Sergey Neshveyev and Lars Tuset, Quantized algebras of functions on homogeneous spaces with Poisson stabilizers, Comm. Math. Phys. 312 (2012), no. 1, 223–250. MR 2914062, DOI 10.1007/s00220-012-1455-6
Piotr Podleś, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $\textrm {SU}(2)$ and $\textrm {SO}(3)$ groups, Comm. Math. Phys. 170 (1995), no. 1, 1–20. MR 1331688, DOI 10.1007/BF02099436
Louis H. Rowen, Ring theory, Student edition, Academic Press, Inc., Boston, MA, 1991. MR 1095047
Jasper V. Stokman and Mathijs S. Dijkhuizen, Quantized flag manifolds and irreducible $*$-representations, Comm. Math. Phys. 203 (1999), no. 2, 297–324. MR 1697598, DOI 10.1007/s002200050613