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Volume 39, Issue 2
On the Kernel of the Borel’s Characteristic Map of Lie Groups

Haibao Duan & Xuezhi Zhao

DOI: 10.4208/cmr.2022-0041

Commun. Math. Res., 39 (2023), pp. 173-189.

Published online: 2023-04

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  • Abstract

For compact and connected Lie group $G$ with a maximal torus $T$ the quotient space $G/T$ is canonically a smooth projective manifold, known as the complete flag manifold of the group $G.$ The cohomology ring map $c^∗: H^∗ (B_T) → H^∗ (G/T)$ induced by the inclusion $c:G/T→B_T$ is called the Borel’s characteristic map of the group $G [7, 8],$ where $B_T$ denotes the classifying space of $T.$ Let $G$ be simply-connected and simple. Based on the Schubert presentation of the cohomology $H^∗ (G/T)$ of the flag manifold $G/T$ obtained in $[10, 11],$ we develop a method to find a basic set of explicit generators for the kernel ker$c^∗ ⊂ H^∗ (B_T)$ of the characteristic map $c.$

  • Keywords

Lie group, flag manifold, Schubert calculus.

  • AMS Subject Headings

14M15, 55T10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-39-173, author = {Duan , Haibao and Zhao , Xuezhi}, title = {On the Kernel of the Borel’s Characteristic Map of Lie Groups}, journal = {Communications in Mathematical Research }, year = {2023}, volume = {39}, number = {2}, pages = {173--189}, abstract = {

For compact and connected Lie group $G$ with a maximal torus $T$ the quotient space $G/T$ is canonically a smooth projective manifold, known as the complete flag manifold of the group $G.$ The cohomology ring map $c^∗: H^∗ (B_T) → H^∗ (G/T)$ induced by the inclusion $c:G/T→B_T$ is called the Borel’s characteristic map of the group $G [7, 8],$ where $B_T$ denotes the classifying space of $T.$ Let $G$ be simply-connected and simple. Based on the Schubert presentation of the cohomology $H^∗ (G/T)$ of the flag manifold $G/T$ obtained in $[10, 11],$ we develop a method to find a basic set of explicit generators for the kernel ker$c^∗ ⊂ H^∗ (B_T)$ of the characteristic map $c.$

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2022-0041}, url = {http://global-sci.org/intro/article_detail/cmr/21543.html} }
TY - JOUR T1 - On the Kernel of the Borel’s Characteristic Map of Lie Groups AU - Duan , Haibao AU - Zhao , Xuezhi JO - Communications in Mathematical Research VL - 2 SP - 173 EP - 189 PY - 2023 DA - 2023/04 SN - 39 DO - http://doi.org/10.4208/cmr.2022-0041 UR - https://global-sci.org/intro/article_detail/cmr/21543.html KW - Lie group, flag manifold, Schubert calculus. AB -

For compact and connected Lie group $G$ with a maximal torus $T$ the quotient space $G/T$ is canonically a smooth projective manifold, known as the complete flag manifold of the group $G.$ The cohomology ring map $c^∗: H^∗ (B_T) → H^∗ (G/T)$ induced by the inclusion $c:G/T→B_T$ is called the Borel’s characteristic map of the group $G [7, 8],$ where $B_T$ denotes the classifying space of $T.$ Let $G$ be simply-connected and simple. Based on the Schubert presentation of the cohomology $H^∗ (G/T)$ of the flag manifold $G/T$ obtained in $[10, 11],$ we develop a method to find a basic set of explicit generators for the kernel ker$c^∗ ⊂ H^∗ (B_T)$ of the characteristic map $c.$

Duan , Haibao and Zhao , Xuezhi. (2023). On the Kernel of the Borel’s Characteristic Map of Lie Groups. Communications in Mathematical Research . 39 (2). 173-189. doi:10.4208/cmr.2022-0041
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