Lesson plan
Classical group
In mathematics, the classical groups are the matrix groups arising from finite-dimensional vector spaces and from nondegenerate bilinear, sesquilinear, quadratic, and Hermitian forms. In the traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues. In the language of linear algebraic groups, the connected classical groups are the connected reductive groups of Dynkin types A_n , B_n , C_n , and D_n , together with their forms over arbitrary fields. Over \mathbb R and \mathbb C this recovers the familiar classical Lie groups, while over finite fields one obtains the finite classical groups. The term goes back to Hermann Weyl's book The Classical Groups. Among the simple Lie groups, the classical groups are in contrast to the exceptional Lie groups, G 2 , F 4 , E 6 , E 7 , E 8 , which share their abstract properties, but not their familiarity. This article begins with the classical Lie groups over \mathbb R , \mathbb C , and \mathbb H , and later discusses the more general formulation over arbitrary fields.
Article summary
In mathematics, the classical groups are the matrix groups arising from finite-dimensional vector spaces and from nondegenerate bilinear, sesquilinear, quadratic, and Hermitian forms. In the traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues. In the language of linear algebraic groups, the connected classical groups are the connected reductive groups of Dynkin types A_n , B_n , C_n , and D_n , together with their forms over arbitrary fields. Over \mathbb R and \mathbb C this recovers the familiar classical Lie groups, while over finite fields one obtains the finite classical groups. The term goes back to Hermann Weyl's book The Classical Groups. Among the simple Lie groups, the classical groups are in contrast to the exceptional Lie groups, G 2 , F 4 , E 6 , E 7 , E 8 , which share their abstract properties, but not their familiarity. This article begins with the classical Lie groups over \mathbb R , \mathbb C , and \mathbb H , and later discusses the more general formulation over arbitrary fields.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.