Homogeneous Hermitian holomorphic vector bundles and operators in the Cowen-Douglas class over the ...

@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.