Quiz About Affine Hecke Algebra

1) Who is Ivan Cherednik?

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

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

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

Explanation

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

2) 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

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.

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.

3) What's the other term for Quantum KZ equation?

Ppq

Kzq

Lkz

Qkz

The other term for Quantum KZ equation is qkz.

Explanation

The other term for Quantum KZ equation is qkz.

4) What's Weyl group?

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 isometry group of the root system.

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

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.

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.

5) How is an affine Hecke algebra depicted?

L (Σ, Q)

H (Σ, q)

Q(Σ, h)

H (L,H)

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.

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.

6) What's an associative algebra?

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, multiplication, and a scalar 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.

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.

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.

7) What's an affine root system?

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 affine-linear function on a Euclidean space.

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

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.

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.

8) When were Macdonald polynomials introduced?

In 1986

In 1987

In 1988

In 1990

In 1987, Macdonald polynomials were introduced.

Explanation

In 1987, Macdonald polynomials were introduced.

9) When does the affine Hecke reduce to C [W]?

Q-1

Q=0

Q=1

Q=-3

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Explanation

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10) 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

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.

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.