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LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART

Part of: Algebraic combinatorics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  02 August 2021

VOLODYMYR MAZORCHUK Open the ORCID record for VOLODYMYR MAZORCHUK [Opens in a new window]  and
RAFAEL MRÐEN Open the ORCID record for RAFAEL MRÐEN [Opens in a new window]
VOLODYMYR MAZORCHUK
Affiliation:
Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, [email protected]
RAFAEL MRÐEN
Affiliation:
Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, Sweden (On leave from: Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, Zagreb 10000, Croatia) [email protected]
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Abstract

For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

MSC classification

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 17B30: Solvable, nilpotent (super)algebras 05E10: Combinatorial aspects of representation theory
Type
Article
Information
Nagoya Mathematical Journal , Volume 246 , June 2022 , pp. 430 - 470
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Copyright
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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Footnotes

This research was partially supported by the Swedish Research Council, Göran Gustafsson Stiftelse and Vergstiftelsen. Rafael Mrđen was also partially supported by the QuantiXLie Center of Excellence grant no. KK.01.1.1.01.0004 funded by the European Regional Development Fund.

References

Alshammari, F., Isaac, P. S., and Marquette, I., On Casimir operators of conformal Galilei algebras , J. Math. Phys. 60 (2019), 013509.CrossRefGoogle Scholar
Andersen, H. H. and Stroppel, C., Twisting functors on $\;\mathcal {O}$ , Represent. Theory 7 (2003), 681–699.CrossRefGoogle Scholar
Arkhipov, S., “Algebraic construction of contragradient quasi-Verma modules in positive characteristic ”, in Representation Theory of Algebraic Groups and Quantum Groups , Adv. Stud. Pure Math. 40, Math. Soc. Japan, Tokyo, 2004, 27–68.10.2969/aspm/04010027CrossRefGoogle Scholar
Bagchi, A. and Gopakumar, R., Galilean conformal algebras and AdS/CFT , J. High Energy Phys. 2009 (2009), 37.10.1088/1126-6708/2009/07/037CrossRefGoogle Scholar
Batra, P. and Mazorchuk, V., Blocks and modules for Whittaker pairs , J. Pure Appl. Algebra 215 (2011), 1552–1568.CrossRefGoogle Scholar
Bavula, V. V. and Lu, T., Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules , J. Math. Phys. 58 (2017), 011701.CrossRefGoogle Scholar
Bavula, V. V. and Lu, T., Classification of simple weight modules over the Schrödinger algebra , Canad. Math. Bull. 61 (2018), 16–39.CrossRefGoogle Scholar
Block, R. E., The irreducible representations of the Lie algebra $\mathrm{sl}(2)$ and of the Weyl algebra , Adv. Math. 39 (1981), 69–110.10.1016/0001-8708(81)90058-XCrossRefGoogle Scholar
Bourbaki, N., Lie Groups and Lie Algebras. Chapters 1–3, Elements of Mathematics, Springer, Berlin, 1989, Translated from the French, Reprint of the 1975 edition.Google Scholar
Cai, Y. and Chen, Q., Quasi-Whittaker modules over the conformal Galilei algebras , Linear Multilinear Algebra 65 (2017), 313–324.CrossRefGoogle Scholar
Cai, Y., Cheng, Y., and Shen, R., Quasi-Whittaker modules for the Schrödinger algebra , Linear Algebra Appl. 463 (2014), 16–32.CrossRefGoogle Scholar
Cai, Y., Shen, R., and Zhang, J., Whittaker modules and quasi-Whittaker modules for the Euclidean Lie algebra e(3) , J. Pure Appl. Algebra 220 (2016), 1419–1433.10.1016/j.jpaa.2015.09.009CrossRefGoogle Scholar
Chen, C.-W., Coulembier, K., and Mazorchuk, V., Translated simple modules for Lie algebras and simple supermodules for Lie superalgebras , Math. Z. 297 (2021), 255–281.CrossRefGoogle Scholar
Chen, C.-W. and Mazorchuk, V., Simple supermodules over Lie superalgebras , Trans. Amer. Math. Soc. 374 (2021), no. 2, 899–921.CrossRefGoogle Scholar
Deodhar, V. V., On a construction of representations and a problem of Enright , Invent. Math. 57 (1980), 101–118.CrossRefGoogle Scholar
Dimitrov, I., Mathieu, O., and Penkov, I., On the structure of weight modules , Trans. Amer. Math. Soc. 352 (2000), 2857–2869.10.1090/S0002-9947-00-02390-4CrossRefGoogle Scholar
Dobrev, V. K., Doebner, H.-D., and Mrugalla, C., Lowest weight representations of the Schrödinger algebra and generalized heat Schrödinger equations , Rep. Math. Phys. 39 (1997), 201–218.CrossRefGoogle Scholar
Drozd, Yu. A., Ovsienko, S. A., and Futorny, V. M.. “On Gelfand–Zetlin modules ”, in Proceedings of the Winter School on Geometry and Physics, Vol. 26 (Srní, 1990) (ed. Bureš, , , J. and Souček, , , V.), Circolo Matematico di Palermo, Palermo, Italy, 1991, 143–147.Google Scholar
Dubsky, B., Classification of simple weight modules with finite-dimensional weight spaces over the Schrödinger algebra , Linear Algebra Appl. 443 (2014), 204–214.CrossRefGoogle Scholar
Dubsky, B., Lü, R., Mazorchuk, V., and Zhao, K., Category $\;\mathbf{\mathcal{O}}$ for the Schrödinger algebra , Linear Algebra Appl. 460 (2014), 17–50.CrossRefGoogle Scholar
Early, N., Mazorchuk, V., and Vishnyakova, E., Canonical Gelfand–Zeitlin modules over orthogonal Gelfand–Zeitlin algebras , Int. Math. Res. Not. IMRN 2020 (2020), no. 20, 6947–6966.CrossRefGoogle Scholar
Enright, T. J., On the fundamental series of a real semisimple Lie algebra: Their irreducibility, resolutions and multiplicity formulae , Ann. of Math. (2) 110 (1979), 1–82.CrossRefGoogle Scholar
Futorny, V. and Ovsienko, S., Kostant’s theorem for special filtered algebras , Bull. Lond. Math. Soc. 37 (2005), 187–199.CrossRefGoogle Scholar
Geoffriau, F., Sur le centre de l’algèbre enveloppante d’une algèbre de Takiff , Ann. Math. Blaise Pascal 1 (1994), 15–31 (1995).CrossRefGoogle Scholar
Geoffriau, F., Homomorphisme de Harish–Chandra pour les algèbres de Takiff généralisées , J. Algebra 171 (1995), 444–456.10.1006/jabr.1995.1021CrossRefGoogle Scholar
Gomis, J. and Kamimura, K., Schrödinger equations for higher order nonrelativistic particles and $N$ -Galilean conformal symmetry , Phys. Rev. D 85 (2012), 045023.CrossRefGoogle Scholar
Hahn, H., Huh, J., Lim, E., and Sohn, J., From partition identities to a combinatorial approach to explicit Satake inversion , Ann. Comb. 22 (2018), 543–562.CrossRefGoogle Scholar
Han, B., Higher spin algebras and universal enveloping algebras, Bachelor’s thesis, Australian National University, 2019.Google Scholar
Hilbert, D., Theory of Algebraic Invariants , Cambridge University Press, Cambridge, 1993, Translated from the German and with a preface by Reinhard C. Laubenbacher, Edited and with an introduction by Bernd Sturmfels.Google Scholar
Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Grad. Texts Math. 9, Springer, New York and Berlin, 1972.CrossRefGoogle Scholar
Humphreys, J. E., Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$ , Grad. Stud. Math. 94, Amer. Math. Soc., Providence, RI, 2008.Google Scholar
Jantzen, J. C., Einhüllende Algebren halbeinfacher Lie-Algebren [Enveloping algebras of semisimple Lie algebras], Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 3, Springer, Berlin, 1983.CrossRefGoogle Scholar
Kac, V. G., Infinite-Dimensional Lie Algebras , 3rd ed., Cambridge University Press, Cambridge, 1990.CrossRefGoogle Scholar
Khomenko, O. and Mazorchuk, V., On Arkhipov’s and Enright’s functors , Math. Z. 249 (2005), 357–386.CrossRefGoogle Scholar
Knapp, A. W., Lie Groups, Lie Algebras, and Cohomology, Math. Notes 34, Princeton University Press, Princeton, NJ, 1988.Google Scholar
König, S. and Mazorchuk, V., Enright’s completions and injectively copresented modules , Trans. Amer. Math. Soc. 354 (2002), 2725–2743.CrossRefGoogle Scholar
Kostant, B., On Whittaker vectors and representation theory , Invent. Math. 48 (1978), 101–184.CrossRefGoogle Scholar
Krause, G. R. and Lenagan, T. H., Growth of algebras and Gelfand–Kirillov dimension, revised ed., Grad. Stud. Math. 22, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
Lau, M., Classification of Harish–Chandra modules for current algebras , Proc. Amer. Math. Soc. 146 (2018), 1015–1029.CrossRefGoogle Scholar
Lü, R., Mazorchuk, V., and Zhao, K., On simple modules over conformal Galilei algebras , J. Pure Appl. Algebra 218 (2014), 1885–1899.CrossRefGoogle Scholar
Mathieu, O., Classification of irreducible weight modules , Ann. Inst. Fourier (Grenoble) 50 (2000), 537–592.CrossRefGoogle Scholar
Mazorchuk, V., Lectures on $\;{sl}_2\left(\mathbb{C}\right)$ -Modules, Imperial College Press, London, 2010.Google Scholar
Mazorchuk, V. and Söderberg, C., Category $\;\mathbf{\mathcal{O}}$ for Takiff sl2 , J. Math. Phys. 60 (2019), 111702.CrossRefGoogle Scholar
Mazorchuk, V. and Stroppel, C., On functors associated to a simple root , J. Algebra 314 (2007), 97–128.CrossRefGoogle Scholar
Molev, A., “Casimir elements for certain polynomial current lie algebras”, in Group 21: Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras (ed. Doebner, H.-D., Nattermann, P., and Scherer, W.), World Scientific, Singapore, 1997, 172–176.Google Scholar
Perroud, M., Projective representations of the Schrödinger group , Helv. Phys. Acta 50 (1977), 233–252.Google Scholar
Pope, C. N., Romans, L. J., and Shen, X., W∞ and the Racah–Wigner algebra , Nuclear Phys. B 339 (1990), 191–221.10.1016/0550-3213(90)90539-PCrossRefGoogle Scholar
Takiff, S. J., Rings of invariant polynomials for a class of Lie algebras , Trans. Amer. Math. Soc. 160 (1971), 249–262.CrossRefGoogle Scholar
Verma, D.-N., Structure of certain induced representations of complex semisimple Lie algebras , Bull. Amer. Math. Soc. 74 (1968), 160–166.CrossRefGoogle Scholar
Vogan, D. A., Representations of Real Reductive Lie Groups, Progr. Math. 15, Birkhäuser, Boston, MA, 1981.Google Scholar
Webster, B., Gelfand–Zeitlin modules in the Coulomb context, preprint, 2020, arXiv:1904.05415.Google Scholar
Wilson, B. J., Highest-weight theory for truncated current Lie algebras , J. Algebra 336 (2011), 1–27.CrossRefGoogle Scholar