• Corpus ID: 231662409

Current algebras on S^3 of complex Lie algebras

@inproceedings{Kori2021CurrentAO,
  title={Current algebras on S^3 of complex Lie algebras},
  author={Tosiaki Kori},
  year={2021}
}

• Tosiaki Kori
• Published 21 January 2021
• Mathematics

Let L be the space of spinors on S3 that are the restrictions to S3 of the Laurent polynomial type harmonic spinors on C2. L becomes an associative algebra. For a simple Lie algebra g the real Lie algebra Lg generated by L⊗C g is called g-current algebra. The real part K of L becomes a commutative subalgebra of L. For the Cartan subalgebra h of g , Kh = K ⊗R h becomes a Cartan subalgebra of Lg. We investigate the adjoint representation of Kh and find that the set of non-zero weights corresponds bijectively to the root space of g. Let g = h + e + f be the standard triangular decomposition of g, and let L⊗Ch, L⊗Ce and L⊗Cf generate respectively the Lie subalgebras Lh, Le and Lf of Lg. Then we have the triangular decomposition Lg = Lh+ Le+ Lf , that is also associated with the weight space decomposition of Lg. With the aid of the basic vector fields on S3 that arise from the infinitesimal representation of SO(3) we introduce a triple of 2-cocycles {ck; k = 0, 1, 2 } on the Lie algebra Lg. Then we have the central extenstion Lg⊕ ⊕k=0Cak associated to the 2-cocycles {ck}k=0,1,2. Adjoining a derivation coming from the radial vector field n on S3 we obtain the second central extension ĝ = Lg⊕⊕k=0Cak ⊕Cn. The root space decomposition and the Chevalley generators of ĝwill be given. Mathematics Subject Classification. 17B65, 17B67, 22E67, 81R10, 81R25, 15B33.