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تسجيل مستخدم جديدQuaternifiations and extensions of current algebras on S^3
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Let $H$ be the quaternion algebra. Let $g$ be a complex Lie algebra and let $U(g)$ be the enveloping algebra of $g$. We define a Lie algebra structure on the tensor product space of $H$ and $U(g)$, and obtain the quaternification $g^H$ of $g$. Let $S^3g^H$ be the set of $g^H$-valued smooth mappings over $S^3$. The Lie algebra structure on $S^3g^H$ is induced naturally from that of $g^H$. On $S^3$ exists the space of Laurent polynomial spinors spanned by a complete orthogonal system of eigen spinors of the tangential Dirac operator on $S^3$. Tensoring $U(g)$ we have the space of $U(g)$-valued Laurent polynomial spinors, which is a Lie subalgebra of $S^3g^H$. We introduce a 2-cocycle on the space of $U(g)$-valued Laurent polynomial spinors by the aid of a tangential vector field on $S^3$. Then we have the corresponding central extension $hat g(a)$ of the Lie algebra of $U(g)$-valued Laurent polynomial spinors. Finally we have the a Lie algebra $hat g=hat g(a)+Cd$ which is obtained by adding to $hat g(a)$ a derivation $d$ which acts on $hat g(a)$ as the radial derivation. When $g$ is a simple Lie algebra with its Cartan subalgebra $h$, We shall investigate the weight space decomposition of $(hat g, ad(hat h))$, where $hat h=h+Ca+Cd$ . The previo
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Let L be the space of spinors on the 3-sphere that are the restrictions of the Laurent polynomial type harmonic spinors on C^2. L becomes an associative algebra. For a simple Lie algebra g, the real Lie algebra Lg generated by the tensor product of L
and g is called the g-current algebra. The real part K of L becomes a commutative subalgebra of L. For a Cartan subalgebra h of g, h tensored by K becomes a Cartan subalgebra Kh of Lg. The set of non-zero weights of the adjoint representation of Kh corresponds bijectively to the root space of g. Let g=h+e+ f be the standard triangular decomposition of g, and let Lh, Le and Lf respectively be the Lie subalgebras of Lg generated by the tensor products of L with h, e and f respectively . 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 the 3-shpere that arise from the infinitesimal representation of SO(3) we introduce a triple of 2-cocycles {c_k; k=0,1,2} on Lg. Then we have the central extension: Lg+ sum Ca_k associated to the 2-cocycles {c_k; k=0,1,2}. Adjoining a derivation coming from the radial vector field on S^3 we obtain the second central extension g^=Lg+ sum Ca_k + Cn. The root space decomposition of g^ as welll as the Chevalley generators of g^ will be given.
Let Omega^3(SU(n)) be the Lie group of based mappings from S^3 to SU(n). We construct a Lie group extension of Omega^3(SU(n)) for n>2 by the abelian group of the affine dual space of SU(n)-connections on S^3. In this article we give several improveme
nt of J. Mickelssons results in 1987, especially we give a precise description of the extension of those components that are not the identity component,. We also correct several argument about the extension of Omega^3(SU(2)) which seems not to be exact in Mickelssons work, though his observation about the fact that the extension of Omega^3(SU(2)) reduces to the extension by Z_2 is correct. Then we shall investigate the adjoint representation of the Lie group extension of Omega^3(SU(n)) for n>2.
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