# Noncubic Dirac Operators for finite-dimensional modules

Abstract : This thesis focuses on the study of noncubic Dirac operators within the framework of representation theory of Lie groups. After recalling basic notions of Lie theory and Clifford algebras, we present the main properties of cubic Dirac operators D introduced by Kostant in 1999. These results quickly aroused great interest. In particular, in the late 1990’s, Vogan introduced a cohomology defined by Kostant operator D and suggested a cohomological classification of representations. Dirac cohomology was computed for various families of representations, such as the discrete series, Aq(>) modules or finite dimensional representations. It turns out that for finite dimensional modules, Dirac cohomology coincides with the kernel of D. It appears that Kostant’s Dirac operator is an algebraic version of a specific member of a continuous family of geometric Dirac operators introduced by Slebarski in the mid 1980’s in the context of bundles over homogeneous spaces G/H of compact groups. What distinguishes the cubic Dirac operator is that it is the only member of this family whose square, generalizing Parthasarathy’s formula, differs from the Casimir operator up to a scalar. This property has important applications in representation theory of Lie groups. The square of the noncubic Dirac operators, i.e. of the other members of Slebarski’s family, was calculated by Agricola who also established precise links between these noncubic operators and string theory in physics. Actually, noncubic Dirac operators are invariant differential operators, and therefore their kernels define (finite-dimensional) representations of compact groups. In this thesis we study the kernel of noncubic Dirac operators, and we show that, under certain conditions on the homogeneous spaces G/H, the kernel contains the kernel of the cubic Dirac operator. We obtain an explicit formula for the kernel which we apply to the case of classical Lie algebras and of exceptional Lie algebras. We remark that some properties of noncubic operators are analogous to those of Kostant cubic Dirac operator, such as the index. We also deduce some observations on noncubic geometric Dirac operators.
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Theses
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https://docnum.univ-lorraine.fr/ulprive/DDOC_T_2021_0035_AFENTOULIDIS_ALMPANIS.pdf

https://hal.univ-lorraine.fr/tel-03229094
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Submitted on : Tuesday, May 18, 2021 - 4:25:56 PM
Last modification on : Saturday, October 16, 2021 - 11:18:03 AM

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• HAL Id : tel-03229094, version 1

### Citation

Spyridon Afentoulidis Almpanis. Noncubic Dirac Operators for finite-dimensional modules. Mathematics [math]. Université de Lorraine, 2021. English. ⟨NNT : 2021LORR0035⟩. ⟨tel-03229094⟩

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