Extensions of the conformal representations for orthogonal Lie algebras

JOURNAL OF ALGEBRA

卷:377

Article

Xu, Xiaoping¹  Zhao, Yufeng² 

[1] Chinese Acad Sci, Acad Math & Syst Sci, Inst Math, Hua Loo Keng Key Math Lab, Beijing 100190, Peoples R China

[2] Peking Univ, Sch Math Sci, LMAM, Beijing 100871, Peoples R China

关键词: Conformal transformation;    Polynomial algebra;    Tensor;    Pieri's formula;    Generalizations of the conformal representation;    Orthogonal Lie algebra;    Irreducibility;   

DOI  :  10.1016/j.jalgebra.2012.11.040

来源: Elsevier

【 摘 要 】

The conformal transformations with respect to the metric defining o(n, C) give rise to an inhomogeneous polynomial representation of o(n + 2, C). Using Shen's technique of mixed product, we generalize the above representation to an inhomogeneous representation of o(n + 2, C) on the tensor space of any finite-dimensional irreducible o(n, C)-module with the polynomial space, where a hidden central transformation is involved. Moreover, we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible. In our approach, Pieri's formulas, invariant operators and the idea of Kostant's characteristic identities play key roles. The result could be useful in understanding higher-dimensional conformal field theory with the constant value taken by the central transformation as the central charge. Our representations virtually provide natural extensions of the conformal transformations on a Riemannian manifold to its vector bundles. (c) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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