What's an affine root system?

It's a root system of affine-linear function on a Euclidean space.

It's a system of affine-linear function on a Euclidean space.

It's a root system of affine-linear function on a space.

It's a root system of a linear function on a Euclidean space.

What's an associative algebra?

It's an algebraic structure with compatible operation of addition, multiplication, and a scalar multiplication by elements in some field.

It's an algebraic structure with compatible operation of addition by elements in some field.

It's an algebraic structure with compatible operation of multiplication by elements in some field.

It's an algebraic structure with compatible operation of addition, and a scalar multiplication by elements in some field.

It's a subgroup of isometry group of the root system.

It's a subgroup of isometry group of the system.

It's a subgroup of metric group of the root system.

It's a subgroup of geometry group of the root system.

Who is Ivan Cherednik?

He is a Russian mathematician who introduced the double affine Hecke algebras

He is a French mathematician who introduced the double affine Hecke algebras

He is a Russian mathematician who introduced the triple affine Hecke algebras

He is a Russian technician who introduced the double affine Hecke algebras

What's a quantum KZ equation?

It's an analogue for quantum affine algebras of Knizhnik

It's an analogue for affine algebras of Knizhnik

It's an analogue for quantity affine algebras of Knizhnik

It's an analogue for quality affine algebras of Knizhnik

What's a multiplicity function?

It's a number of spin states such as n of the N spins point in the Z direction

It's a number of states such as n of the A spins point in the Z direction

It's a number of spin states such as n of the N spins point in the M direction

It's a number of spin states such as n of the Z spins point in the N direction

Quiz About Affine Hecke Algebra

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Questions: 10 | Attempts: 127 | Updated: Mar 22, 2023

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The name might sound a bit strange but the Hecke Algebra is a type of affine Weyle group. And as strange as it may also sound, people actually use it everyday. So, are you a math genius? What do you know about this theory? Take our quiz and find out.

Questions and Answers

1.

What's an affine root system?
- A.
  
  It's a system of affine-linear function on a Euclidean space.
- B.
  
  It's a root system of affine-linear function on a space.
- C.
  
  It's a root system of affine-linear function on a Euclidean space.
- D.
  
  It's a root system of a linear function on a Euclidean space.
Correct Answer
C. It's a root system of affine-linear function on a Euclidean space.

Explanation
An affine root system refers to a root system of affine-linear functions on a Euclidean space. This means that the root system consists of vectors that satisfy certain properties under affine-linear transformations, such as translations and reflections, in a Euclidean space. The term "affine-linear" emphasizes the combination of affine transformations and linear functions in defining the root system.

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2.

What's an associative algebra?
- A.
  
  It's an algebraic structure with compatible operation of addition by elements in some field.
- B.
  
  It's an algebraic structure with compatible operation of multiplication by elements in some field.
- C.
  
  It's an algebraic structure with compatible operation of addition, multiplication, and a scalar multiplication by elements in some field.
- D.
  
  It's an algebraic structure with compatible operation of addition, and a scalar multiplication by elements in some field.
Correct Answer
C. It's an algebraic structure with compatible operation of addition, multiplication, and a scalar multiplication by elements in some field.

Explanation
The correct answer explains that an associative algebra is an algebraic structure that has compatible operations of addition, multiplication, and scalar multiplication by elements in some field. This means that the algebra satisfies the associative property for addition and multiplication, and also allows for scalar multiplication by elements from a field. This definition encompasses all the necessary operations for an associative algebra.

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3.

What's Weyl group?
- A.
  
  It's a subgroup of isometry group of the system.
- B.
  
  It's a subgroup of metric group of the root system.
- C.
  
  It's a subgroup of isometry group of the root system.
- D.
  
  It's a subgroup of geometry group of the root system.
Correct Answer
C. It's a subgroup of isometry group of the root system.

Explanation
The Weyl group is a subgroup of the isometry group of the root system. The root system is a collection of vectors that encode the geometric properties of a Lie algebra. The Weyl group acts on the root system by permuting and reflecting the roots while preserving the angles between them. It plays a fundamental role in the study of Lie algebras and Lie groups, as it captures the symmetries of the root system and provides important information about the structure and representation theory of these mathematical objects.

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4.

How is an affine Hecke algebra depicted?
- A.
  
  L (Σ, Q)
- B.
  
  H (Σ, q)
- C.
  
  Q(Σ, h)
- D.
  
  H (L,H)
Correct Answer
B. H (Σ, q)

Explanation
An affine Hecke algebra is depicted by the notation H (Σ, q). This notation represents the algebraic structure used to study certain types of symmetries in mathematics. The Σ represents a set of generators, which are elements that generate the algebra. The q represents a parameter that is used in defining the algebraic operations within the Hecke algebra. Therefore, H (Σ, q) is the correct depiction of an affine Hecke algebra.

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5.

Who is Ivan Cherednik?
- A.
  
  He is a French mathematician who introduced the double affine Hecke algebras
- B.
  
  He is a Russian mathematician who introduced the triple affine Hecke algebras
- C.
  
  He is a Russian technician who introduced the double affine Hecke algebras
- D.
  
  He is a Russian mathematician who introduced the double affine Hecke algebras
Correct Answer
D. He is a Russian mathematician who introduced the double affine Hecke algebras

Explanation
Ivan Cherednik is a Russian mathematician who is known for introducing the double affine Hecke algebras.

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6.

What's a quantum KZ equation?
- A.
  
  It's an analogue for quantum affine algebras of Knizhnik
- B.
  
  It's an analogue for affine algebras of Knizhnik
- C.
  
  It's an analogue for quantity affine algebras of Knizhnik
- D.
  
  It's an analogue for quality affine algebras of Knizhnik
Correct Answer
A. It's an analogue for quantum affine algebras of Knizhnik

Explanation
A quantum KZ equation is an analogue for quantum affine algebras of Knizhnik. This means that it is a mathematical equation or relationship that is similar to or corresponds to the concept of quantum affine algebras in the context of Knizhnik.

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7.

What's the other term for Quantum KZ equation?
- A.
  
  Ppq
- B.
  
  Kzq
- C.
  
  Lkz
- D.
  
  Qkz
Correct Answer
D. Qkz

Explanation
The other term for Quantum KZ equation is qkz.

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8.

When were Macdonald polynomials introduced?
- A.
  
  In 1986
- B.
  
  In 1987
- C.
  
  In 1988
- D.
  
  In 1990
Correct Answer
B. In 1987

Explanation
In 1987, Macdonald polynomials were introduced.

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9.

When does the affine Hecke reduce to C [W]?
- A.
  
  Q-1
- B.
  
  Q=0
- C.
  
  Q=1
- D.
  
  Q=-3
Correct Answer
C. Q=1
10.

What's a multiplicity function?
- A.
  
  It's a number of spin states such as n of the N spins point in the Z direction
- B.
  
  It's a number of states such as n of the A spins point in the Z direction
- C.
  
  It's a number of spin states such as n of the N spins point in the M direction
- D.
  
  It's a number of spin states such as n of the Z spins point in the N direction
Correct Answer
A. It's a number of spin states such as n of the N spins point in the Z direction

Explanation
The multiplicity function refers to the number of spin states in which n of the N spins point in the Z direction. In other words, it represents the number of possible arrangements or configurations of spins where a specific number of spins align in the Z direction out of the total N spins.

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