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@article{65304, author = {Deb, Prahllad and Hazra, Somnath}, article_location = {San Diego (USA)}, article_number = {2}, doi = {http://dx.doi.org/10.1016/j.jmaa.2022.126031}, keywords = {Cowen-Douglas class; Homogeneous operators; Hermitian holomorphic homogeneous vector bundles; Curvature; Representation; Lie algebra}, language = {eng}, issn = {0022-247X}, journal = {Journal of Mathematical Analysis and Applications}, title = {Homogeneous Hermitian holomorphic vector bundles and operators in the Cowen-Douglas class over the poly-disc}, url = {https://www.sciencedirect.com/science/article/pii/S0022247X22000452}, volume = {510}, year = {2022} }
TY - JOUR ID - 65304 AU - Deb, Prahllad - Hazra, Somnath PY - 2022 TI - Homogeneous Hermitian holomorphic vector bundles and operators in the Cowen-Douglas class over the poly-disc JF - Journal of Mathematical Analysis and Applications VL - 510 IS - 2 SP - "126031-1"-"126031-32" EP - "126031-1"-"126031-32" PB - Academic Press Inc. Elsevier Science SN - 0022247X KW - Cowen-Douglas class KW - Homogeneous operators KW - Hermitian holomorphic homogeneous vector bundles KW - Curvature KW - Representation KW - Lie algebra UR - https://www.sciencedirect.com/science/article/pii/S0022247X22000452 N2 - In this article, we obtain two sets of results. The first set of results are for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc. A classification of irreducible hermitian holomorphic vector bundles over D-2, homogeneous with respect to Mob x Mob, is obtained assuming that the associated representations are multiplicity-free. Among these the ones that give rise to an operator in the Cowen-Douglas class of D-2 of rank 1, 2 or 3 are determined. Any hermitian holomorphic vector bundle of rank 2 over D-n, homogeneous with respect to the n-fold direct product of the group Mob is shown to be a tensor product of n hermitian holomorphic vector bundles over D. Among them, n - 1 are shown to be the line bundles and one is shown to be a rank 2 bundle. Also, each of the bundles are homogeneous with respect to Mob. The classification of irreducible homogeneous hermitian holomorphic vector bundles over D-2 of rank 3 (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of D-n, n > 2. It is shown that there is no irreducible n - tuple of operators in the Cowen-Douglas class B-2 (D-n) that is homogeneous with respect to Aut(D-n), n > 1. Also, pairs of operators in B-3(D-2) homogeneous with respect to Aut(D-2) are produced, while it is shown that no n - tuple of operators in B-3(D-n) is homogeneous with respect to Aut(D-n), n > 2. ER -
DEB, Prahllad and Somnath HAZRA. Homogeneous Hermitian holomorphic vector bundles and operators in the Cowen-Douglas class over the poly-disc. \textit{Journal of Mathematical Analysis and Applications}. San Diego (USA): Academic Press Inc. Elsevier Science, 2022, vol.~510, No~2, p.~''126031-1''-''126031-32'', 32 pp. ISSN~0022-247X. Available from: https://dx.doi.org/10.1016/j.jmaa.2022.126031.
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