Abstract
Until now, the cubic Dirac operator 
was viewed in a purely algebraic setting, as an element of the relative quantum Weil algebra. In this short chapter, we show that if \(\mathfrak{g}\) and \(\mathfrak{k}\) are the Lie algebras of a Lie group G with a closed subgroup K, then 
is realized as a geometric Dirac operator over the homogeneous space G/K, for a left-invariant connection with nonzero torsion. Such Dirac operators had been studied by Slebarski in the late 1980s.
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Notes
- 1.
The definition of the Dirac operator given below works more generally for possibly degenerate metrics on T ∗ M.
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Meinrenken, E. (2013).
as a geometric Dirac operator.
In: Clifford Algebras and Lie Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36216-3_9
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DOI: https://doi.org/10.1007/978-3-642-36216-3_9
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