What Do You Know About Satake Isomorphism?
Algebraic braic groups mathematician Ichirō Satake introduced Satake isomorphism (along with Satake diagrams) in the early 1960s.
It recognizes the Hecke algebra or polynomial math of a reductive group over a local field with a ring of constants of the Weyl group.
To find out more, take this short, intelligent quiz.
- 1.
When was Satake isomorphism introduced?
- A.
1963
- B.
1964
- C.
1965
- D.
1966
Correct Answer
A. 1963Explanation
The Satake isomorphism was introduced in 1963.Rate this question:
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- 2.
What does G represent?
- A.
Trojan group
- B.
Chevalley group
- C.
Chivallas group
- D.
Martins theorem
Correct Answer
B. Chevalley groupExplanation
G represents the Chevalley group. Chevalley groups are a class of algebraic groups introduced by Claude Chevalley. They are defined over a field and have a specific structure that makes them important in the study of algebraic groups and Lie algebras. The other options, Trojan group, Chivallas group, and Martins theorem, are unrelated to the concept of Chevalley groups.Rate this question:
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- 3.
What does K represent?
- A.
Non-archimedean local field
- B.
Archimedean local field
- C.
Local field
- D.
Global field
Correct Answer
A. Non-archimedean local fieldExplanation
K represents a non-archimedean local field. This type of field is a complete field with a non-archimedean absolute value, meaning that it does not satisfy the archimedean property. In a non-archimedean local field, the absolute value of a nonzero element is a positive real number, and the field is equipped with a topology induced by this absolute value. This is in contrast to an archimedean local field, where the absolute value satisfies the archimedean property. A local field is a field that is complete with respect to a non-trivial absolute value, and a global field is a field that is finite-dimensional over its prime field.Rate this question:
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- 4.
What does O denote?
- A.
Group of integers
- B.
Ordinate value
- C.
Orifices
- D.
Ring of integers
Correct Answer
D. Ring of integersExplanation
The letter "O" is commonly used to denote the set or ring of integers. In mathematics, a ring is a set equipped with two operations, addition and multiplication, that satisfy certain properties. The set of integers, denoted by the symbol "Z", is a well-known example of a ring. Therefore, the correct answer is "Ring of integers".Rate this question:
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- 5.
Over a local field, which group does it identifies?
- A.
Reductive group
- B.
Recursive group
- C.
Finite group
- D.
Infinite group
Correct Answer
A. Reductive groupExplanation
Over a local field, the group that is identified is a reductive group. A reductive group is a type of algebraic group that is defined over a field and has no nontrivial connected normal unipotent subgroup. In the context of a local field, which is a field that is complete with respect to a non-Archimedean absolute value, a reductive group is a natural choice as it allows for the study of various properties and structures of the field.Rate this question:
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- 6.
When was the geometric version of the Satake isomorphism introduced?
- A.
2006
- B.
2007
- C.
2008
- D.
2009
Correct Answer
B. 2007Explanation
The geometric version of the Satake isomorphism was introduced in 2007.Rate this question:
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- 7.
The Satake isomorphism identifies the Grothendieck group of complex representations of which of these?
- A.
Langlands single
- B.
Langlands dual
- C.
Langlands triple
- D.
Free space
Correct Answer
B. Langlands dualExplanation
The Satake isomorphism identifies the Grothendieck group of complex representations of the Langlands dual. The Langlands dual is a mathematical concept that relates the representation theory of a group to the representation theory of its dual group. The Satake isomorphism is a result in algebraic geometry that establishes an isomorphism between the cohomology ring of a compact connected Lie group and the ring of symmetric polynomials. Therefore, the Satake isomorphism is used to study the representation theory of the Langlands dual group.Rate this question:
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- 8.
For a Satake isomorphism in characteristic p, what is K?
- A.
Special maximal compact subgroup of G (F)
- B.
Hyperspecial maximal compact subgroup of G (F)
- C.
Krystal group
- D.
Subgroup of G (F)
Correct Answer
B. Hyperspecial maximal compact subgroup of G (F)Explanation
The correct answer is the hyperspecial maximal compact subgroup of G (F). In the context of a Satake isomorphism in characteristic p, K refers to the hyperspecial maximal compact subgroup of G (F). This subgroup plays a crucial role in the theory of reductive algebraic groups and is used to study the structure and representation theory of these groups.Rate this question:
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- 9.
For a Satake isomorphism in characteristic p, what is V?
- A.
Irreducible representation of K
- B.
Reducible representation of K
- C.
Reductive representation of K
- D.
Isomorphic representation of K
Correct Answer
A. Irreducible representation of KExplanation
The correct answer is "Irreducible representation of K". In the context of a Satake isomorphism in characteristic p, V refers to the irreducible representation of K. This means that the representation cannot be further decomposed into smaller subrepresentations.Rate this question:
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- 10.
Who introduced the geometric version of the Satake isomorphism?
- A.
Mirković and Vilonen
- B.
Mirković and Markus
- C.
Vivian and Vilonen
- D.
Charles and Armstrong
Correct Answer
A. Mirković and VilonenExplanation
Mirković and Vilonen introduced the geometric version of the Satake isomorphism.Rate this question:
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Mar 18, 2023Quiz Edited by
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Jun 17, 2018Quiz Created by
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