Abstract
The aim of this paper is to define cubic Dirac operators for colour Lie algebras. We give a necessary and sufficient condition to construct a colour Lie algebra from an ϵ-orthogonal representation of an ϵ-quadratic colour Lie algebra. This is used to prove a strange Freudenthal–de Vries formula for basic colour Lie algebras as well as a Parthasarathy formula for cubic Dirac operators of colour Lie algebras. We calculate the cohomology induced by this Dirac operator, analogously to the algebraic Vogan conjecture proved by Huang and Pandžić.
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Atiyah, M., Schmid, W.: A geometric construction of the discrete series for semisimple Lie groups. Invent. Math. 42, 1–62 (1977)
Barbasch, D., Ciubotaru, D., Trapa, P.E.: Dirac cohomology for graded afine Hecke algebras. Acta Math. 209(2), 197–227 (2012)
Z. Chen, Y. Kang, An analogue of the Kostant criterion for quadratic Lie superalgebras, preprint.
Chen, Z., Kang, Y.: Generalized Clifford theory for graded spaces. J. Pure Appl. Algebra. 220(2), 647–665 (2016)
D. Ciubotaru, Dirac cohomology for symplectic reection algebras, Selecta Math. (N.S.) 22 (2016), no. 1, 111–144.
Ciubotaru, D., De Martino, M.: Dirac induction for rational Cherednik algebras. IMRN. 2020(17), 5155–5214 (2018)
Fegan, H.D., Steer, B.: On the “strange formula” of Freudenthal and de Vries. Math. Proc. Cambridge Philos. Soc. 105(2), 249–252 (1989)
H. Freudenthal, H. de Vries, Linear Lie Groups, Pure and Applied Mathematics, Vol. 35, Academic Press, New York, 1969.
Huang, J.S., Pandžić, P.: Dirac cohomology, unitary representations and a proof of a conjecture of Vogan. J. Amer. Math. Soc. 15(1), 185–202 (2002)
Huang, J.S., Pandžić, P.: Dirac cohomology for Lie superalgebras. Transform. Groups. 10(2), 201–209 (2005)
Huang, J.S., Pandžić, P., Renard, D.: Dirac operators and Lie algebra cohomology. Represent. Theory. 10(2), 299–313 (2006)
V. G. Kac, P. Möseneder Frajria, P. Papi, Dirac operators and the very strange formula for Lie superalgebras, Advances in Lie Superalgebras, Springer INdAM Ser. 7 (2014), 121–147.
Kostant, B.: A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups. Duke Math. J. 100(3), 447–501 (1999)
Kostant, B.: The Weyl algebra and the structure of all Lie superalgebras of Riemannian type. Transform. Groups. 6(3), 215–226 (2001)
B. Kostant, Dirac cohomology for the cubic Dirac operator, in: Studies in Memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math., Vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 69–93.
Mehdi, S., Zierau, R.: The Dirac cohomology of a finite dimensional representation. Proc. Amer. Math. Soc. 142(5), 1507–1512 (2014)
P. Meyer, Representations associated to gradations of Lie algebras and colour Lie algebras, Thesis IRMA Université de Strasbourg (2019).
Meyer, P.: The Kostant invariant and special ϵ-orthogonal representations for ϵ- quadratic colour Lie algebras. J. Algebra. 572, 337–380 (2021)
Nishiyama, K.: Oscillator representations for orthosymplectic algebras. J. Algebra. 129(1), 231–262 (1990)
R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30.
Pengpan, T.: Kostant’s cubic Dirac operator of Lie superalgebras. J. Math. Phys. 40(12), 6577–6588 (1999)
Ree, R.: Generalized Lie elements. Canad. J. Math. 12, 493–502 (1960)
Rittenberg, V., Wyler, D.: Generalized superalgebras. Nuclear Phys. B. 139(3), 189–202 (1978)
Rittenberg, V., Wyler, D.: Sequences of Z2 ⊕ Z2 graded Lie algebras and superalgebras. J. Math. Phys. 19(10), 2193–2200 (1978)
Scheunert, M.: Generalized Lie algebras. J. Math. Phys. 20(4), 712–720 (1979)
Scheunert, M.: Graded tensor calculus. J. Math. Phys. 24(11), 2658–2670 (1983)
Scheunert, M.: Casimir elements of 𝜀 Lie algebras. J. Math. Phys. 24(11), 2671–2680 (1983)
Xiao, W.: Dirac operators and cohomology for Lie superalgebra of type I. J. Lie Theory. 27(1), 111–121 (2017)
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Philippe Meyer is supported by the EPSRC grant [EP/N033922/1].
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MEYER, P. CUBIC DIRAC OPERATORS AND THE STRANGE FREUDENTHAL–DE VRIES FORMULA FOR COLOUR LIE ALGEBRAS. Transformation Groups 27, 1307–1336 (2022). https://doi.org/10.1007/s00031-021-09680-x
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DOI: https://doi.org/10.1007/s00031-021-09680-x