Abstract
We explore the Fock spaces of the parafermionic algebra introduced by H.S. Green. Each parafermionic Fock space allows for a free minimal resolution by graded modules of the graded two-step nilpotent subalgebra of the parafermionic creation operators. Such a free resolution is constructed with the help of a classical Kostant’s theorem computing Lie algebra cohomologies of the nilpotent subalgebra with values in the parafermionic Fock space. The Euler-Poincaré characteristic of the parafermionic Fock space free resolution yields some interesting identities between Schur polynomials. Finally we briefly comment on parabosonic and general parastatistics Fock spaces.
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Notes
- 1.
The self-conjugacy \(\mathcal{V}(p)\mathop{\cong}\mathcal{V}(p)^{{\ast}}\) allows to switch between x i : = exp(±e i ) without a conflict.
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Popov, T. (2014). Parafermionic Algebras, Their Modules and Cohomologies. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 111. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55285-7_39
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DOI: https://doi.org/10.1007/978-4-431-55285-7_39
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