### 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 |
---|---|

Pages (from-to) | 2150-2180 |

Number of pages | 31 |

Journal | Symmetry |

Volume | 7 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### 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

^{3}

*Symmetry*,

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

**Quaternifications and extensions of current algebras on S ^{3}
.** / Kori, Tosiaki; Imai, Yuto.

Research output: Contribution to journal › Article

^{3}',

*Symmetry*, vol. 7, no. 4, pp. 2150-2180. https://doi.org/10.3390/sym7042025

^{3}Symmetry. 2015;7(4):2150-2180. https://doi.org/10.3390/sym7042025

}

TY - JOUR

T1 - Quaternifications and extensions of current algebras on S3

AU - Kori, Tosiaki

AU - Imai, Yuto

PY - 2015

Y1 - 2015

N2 - 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 gH = (H ⊗ U(g), [,]gH) of g is defined by the bracket [z ⊗ X, w ⊗ Y]gH = (z · w) ⊗ (XY) - (w · z) ⊗ (YX), for z, w e H and the basis vectors X and Y of U(g). Let S3H be the (non-commutative) algebra of H-valued smooth mappings over S3 and let S3gH = S3H ⊗ U(g). The Lie algebra structure on S3gH is induced naturally from that of gH. We introduce a 2-cocycle on S3gH by the aid of a tangential vector field on S3 ⊂ C2 and have the corresponding central extension S3gH⊕(Ca). As a subalgebra of S3H 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 S3. Then C[φ±] ⊗ U(g) is a Lie subalgebra of S3gH. We have the central extension g^(a) = (C[φ±] ⊗ U(g)) ⊕ (Ca) as a Lie-subalgebra of S3gH ⊕ (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 d0. That is the C-vector space g^ = (C[φ±] ⊗ U(g)) ⊕ (Ca)⊕(Cd) endowed with the bracket [φ1 ⊗ X1 +λ1a+μ1d, φ2 ⊗ X2 +λ2a+μ2d]g^ = (φ1φ2) ⊗ (X1 X2) - (φ2φ1) ⊗ (X2X1)+μ1d0φ2 ⊗ X2 -μ2d0φ1 ⊗ X1+(X1|X2)c(φ1, φ2)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).

AB - 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 gH = (H ⊗ U(g), [,]gH) of g is defined by the bracket [z ⊗ X, w ⊗ Y]gH = (z · w) ⊗ (XY) - (w · z) ⊗ (YX), for z, w e H and the basis vectors X and Y of U(g). Let S3H be the (non-commutative) algebra of H-valued smooth mappings over S3 and let S3gH = S3H ⊗ U(g). The Lie algebra structure on S3gH is induced naturally from that of gH. We introduce a 2-cocycle on S3gH by the aid of a tangential vector field on S3 ⊂ C2 and have the corresponding central extension S3gH⊕(Ca). As a subalgebra of S3H 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 S3. Then C[φ±] ⊗ U(g) is a Lie subalgebra of S3gH. We have the central extension g^(a) = (C[φ±] ⊗ U(g)) ⊕ (Ca) as a Lie-subalgebra of S3gH ⊕ (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 d0. That is the C-vector space g^ = (C[φ±] ⊗ U(g)) ⊕ (Ca)⊕(Cd) endowed with the bracket [φ1 ⊗ X1 +λ1a+μ1d, φ2 ⊗ X2 +λ2a+μ2d]g^ = (φ1φ2) ⊗ (X1 X2) - (φ2φ1) ⊗ (X2X1)+μ1d0φ2 ⊗ X2 -μ2d0φ1 ⊗ X1+(X1|X2)c(φ1, φ2)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).

KW - Current algebra

KW - Infinite dimensional lie algebras

KW - Lie algebra extensions

KW - Quaternion analysis

UR - http://www.scopus.com/inward/record.url?scp=84952837718&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84952837718&partnerID=8YFLogxK

U2 - 10.3390/sym7042025

DO - 10.3390/sym7042025

M3 - Article

VL - 7

SP - 2150

EP - 2180

JO - Symmetry

JF - Symmetry

SN - 2073-8994

IS - 4

ER -