Quaternifications and extensions of current algebras on S3

Tosiaki Kori, Yuto Imai

    Research output: Contribution to journalArticle

    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 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 ⊗ X11a+μ1d, φ2 ⊗ X22a+μ2d]g^ = (φ1φ2) ⊗ (X1 X2) - (φ2φ1) ⊗ (X2X1)+μ1d0φ2 ⊗ X22d0φ1 ⊗ 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).

    Original languageEnglish
    Pages (from-to)2150-2180
    Number of pages31
    JournalSymmetry
    Volume7
    Issue number4
    DOIs
    Publication statusPublished - 2015

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    Current Algebra
    current algebra
    Algebra
    Subalgebra
    algebra
    Lie Algebra
    Spinors
    Central Extension
    Brackets
    Vector Field
    Cartan Subalgebra
    Noncommutative Algebra
    Quaternion Algebra
    Laurent Polynomials
    Enveloping Algebra
    Simple Lie Algebra
    Dirac Operator
    Cocycle
    brackets
    Vector space

    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

    Quaternifications and extensions of current algebras on S3 . / Kori, Tosiaki; Imai, Yuto.

    In: Symmetry, Vol. 7, No. 4, 2015, p. 2150-2180.

    Research output: Contribution to journalArticle

    Kori, Tosiaki ; Imai, Yuto. / Quaternifications and extensions of current algebras on S3 In: Symmetry. 2015 ; Vol. 7, No. 4. pp. 2150-2180.
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