A parabolic subgroup of a connected reductive group is a closed subgroup containing a Borel subgroup. Its quotient in the group is projective.
For a connected reductive algebraic group and a Borel subgroup , the quotient is its complete flag variety. Quotients by parabolic subgroups are partial flag varieties.
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In the context of algebraic groups and group theory, a **Borel subgroup** is a specific type of subgroup that is particularly important in the study of linear algebraic groups. Here are the key points regarding Borel subgroups: 1. **Definition**: A Borel subgroup of an algebraic group \( G \) is a maximal connected solvable subgroup of \( G \). This means that it cannot be contained in any larger connected solvable subgroup of \( G \).