### Abstract

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification g^{H} = (H ⊗ U(g), [,]_{g}H) of g is defined by the bracket [z ⊗ X, w ⊗ Y]_{g}H = (z · w) ⊗ (XY) - (w · z) ⊗ (YX), for z, w e H and the basis vectors X and Y of U(g). Let S^{3}H be the (non-commutative) algebra of H-valued smooth mappings over S^{3} and let S^{3}g^{H} = S^{3}H ⊗ U(g). The Lie algebra structure on S^{3}g^{H} is induced naturally from that of g^{H}. We introduce a 2-cocycle on S^{3}g^{H} by the aid of a tangential vector field on S^{3} ⊂ C^{2} and have the corresponding central extension S^{3}g^{H}⊕(Ca). As a subalgebra of S^{3}H we have the algebra of Laurent polynomial spinors C[φ^{±}] spanned by a complete orthogonal system of eigen spinors (φ^{±}
^{(m, l,k)})_{m, l,k} of the tangential Dirac operator on S^{3}. Then C[φ^{±}] ⊗ U(g) is a Lie subalgebra of S^{3}g^{H}. We have the central extension g^(a) = (C[φ^{±}] ⊗ U(g)) ⊕ (Ca) as a Lie-subalgebra of S^{3}g^{H} ⊕ (Ca). Finally we have a Lie algebra g^ which is obtained by adding to g^(a) a derivation d which acts on g^(a) by the Euler vector field d_{0}. That is the C-vector space g^ = (C[φ^{±}] ⊗ U(g)) ⊕ (Ca)⊕(Cd) endowed with the bracket [φ_{1} ⊗ X_{1} +λ_{1}a+μ_{1}d, φ_{2} ⊗ X_{2} +λ_{2}a+μ_{2}d]_{g^} = (φ_{1}φ_{2}) ⊗ (X_{1} X_{2}) - (φ_{2}φ_{1}) ⊗ (X_{2}X_{1})+μ_{1}d_{0}φ_{2} ⊗ X_{2} -μ_{2}d_{0}φ_{1} ⊗ X_{1}+(X_{1}|X_{2})c(φ_{1}, φ_{2})a: When g is a simple Lie algebra with its Cartan subalgebra h we shall investigate the weight space decomposition of g^ with respect to the subalgebra h^ = (φ+^{(0,0,1)} ⊗ h)⊕(Ca)⊕(Cd).

Original language | English |
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Pages (from-to) | 2150-2180 |

Number of pages | 31 |

Journal | Symmetry |

Volume | 7 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2015 |

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### Keywords

- Current algebra
- Infinite dimensional lie algebras
- Lie algebra extensions
- Quaternion analysis

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science (miscellaneous)
- Chemistry (miscellaneous)
- Physics and Astronomy (miscellaneous)

### Cite this

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*Symmetry*,

*7*(4), 2150-2180. https://doi.org/10.3390/sym7042025