What's an Affine algebraic variety?

It's the zero-locus in the affine n-space $$K^n$$

It's the affine n-space $$K^n$$

What's a reductive group?

It's a type of linear algebraic group over a field.

It's a type of line of group over a field.

It's a type of linear algebraic group over a curve.

It's a type of continuous algebraic group over a field.

What's a symmetric variety?

It's an algebraic analogue of a symmetric space in differential geometry.

It's an algebraic analogue of a symmetric space in general.

It's an algebraic analogue of space in differential geometry.

It's the symmetrical space in differential geometry.

What's a spherical variety?

It's a G-variety with an open dense B-orbit

It's an A-variety with an open dense B-orbit

It's a B-variety with an open dense B-orbit

It's a B-variety with an open dense A-orbit

What's a matrix coefficient?

It's a function on a group of a special form, which depends on a linear representation of the group and additional data.

It's a function on a group of a special form, which depends on a representation of some additional data.

It's a function on a group, which depends on a representation of the group and additional data.

It's a group of a special form, which depends on a linear representation of the group and additional data.

What's the Peter-Weyl theorem?

It's a basic result in the theory of harmonic analysis applying to topological groups that are compact but are not necessarily abelian.

It's a basic result in the theory of harmonic analysis applying to topological groups that are compact but are not necessarily real numbers.

It's a basic result in the theory of harmonic analysis applying to topological groups that are liquid but are not necessarily abelian.

It's a basic result in the theory applying to topological groups that are solid but are not necessarily abelian.

What's a complex number?

It's a number that can be expressed in the form a+bi

It's a number that can be expressed in the form a-bi

It's a number that can be expressed in the form b+ai

It's a number that can be expressed in the form c+bi

What's a real number?

It's a valve of a continuous quantity that can represent a distance along a line.

It's a valve of a continuous quantity that can represent a distance along a curve.

It's a valve of a continuous quantity that can represent a distance along a circle.

It's a valve of a constant quantity that can represent a distance along a line.

Take Our Quiz About The Representation On Coordinate Rings

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Take Our Quiz About The Representation On Coordinate Rings - Quiz

For those who love mathematics, representation on coordinate rings is defined as the representation of a group on coordinate rings of affine varieties. Now, the question is do you know all the symbols and terms involved with this discipline? Do you use them in your every day life? Take our quiz and see if you really know what a representation on coordinate rings is.

Questions and Answers

1.

What's an Affine algebraic variety?
- A.
  
  It's the affine n-space K^n
- B.
  
  It's the equivalent of K^n
- C.
  
  It's simply the n-space
- D.
  
  It's the zero-locus in the affine n-space K^n
Correct Answer
D. It's the zero-locus in the affine n-space K^n

Explanation
An affine algebraic variety is defined as the zero-locus in the affine n-space K^n. This means that it is the set of points in the n-dimensional space where a given polynomial equation becomes zero. This definition helps to distinguish an affine algebraic variety from the entire affine n-space and emphasizes its connection to algebraic geometry.

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2.

What's a reductive group?
- A.
  
  It's a type of linear algebraic group over a field.
- B.
  
  It's a type of line of group over a field.
- C.
  
  It's a type of linear algebraic group over a curve.
- D.
  
  It's a type of continuous algebraic group over a field.
Correct Answer
A. It's a type of linear algebraic group over a field.
3.

What's a symmetric variety?
- A.
  
  It's an algebraic analogue of a symmetric space in general.
- B.
  
  It's an algebraic analogue of space in differential geometry.
- C.
  
  It's the symmetrical space in differential geometry.
- D.
  
  It's an algebraic analogue of a symmetric space in differential geometry.
Correct Answer
D. It's an algebraic analogue of a symmetric space in differential geometry.

Explanation
A symmetric variety is an algebraic analogue of a symmetric space in differential geometry. In differential geometry, a symmetric space is a manifold that is equipped with a symmetric bilinear form. Similarly, in algebraic geometry, a symmetric variety is a variety that possesses a similar symmetry structure. This concept allows for the study of symmetry properties in algebraic settings, providing a bridge between algebraic and differential geometry.

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4.

What's a spherical variety?
- A.
  
  It's an A-variety with an open dense B-orbit
- B.
  
  It's a B-variety with an open dense B-orbit
- C.
  
  It's a G-variety with an open dense B-orbit
- D.
  
  It's a B-variety with an open dense A-orbit
Correct Answer
C. It's a G-variety with an open dense B-orbit
5.

What's a matrix coefficient?
- A.
  
  It's a function on a group of a special form, which depends on a representation of some additional data.
- B.
  
  It's a function on a group of a special form, which depends on a linear representation of the group and additional data.
- C.
  
  It's a function on a group, which depends on a representation of the group and additional data.
- D.
  
  It's a group of a special form, which depends on a linear representation of the group and additional data.
Correct Answer
B. It's a function on a group of a special form, which depends on a linear representation of the group and additional data.

Explanation
A matrix coefficient is a function on a group of a special form, which depends on a linear representation of the group and additional data. This means that the function is defined on a specific kind of group and is related to a linear representation of that group. The additional data could refer to any extra information that is required to define the function.

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6.

What's the Peter-Weyl theorem?
- A.
  
  It's a basic result in the theory of harmonic analysis applying to topological groups that are compact but are not necessarily real numbers.
- B.
  
  It's a basic result in the theory of harmonic analysis applying to topological groups that are liquid but are not necessarily abelian.
- C.
  
  It's a basic result in the theory applying to topological groups that are solid but are not necessarily abelian.
- D.
  
  It's a basic result in the theory of harmonic analysis applying to topological groups that are compact but are not necessarily abelian.
Correct Answer
D. It's a basic result in the theory of harmonic analysis applying to topological groups that are compact but are not necessarily abelian.

Explanation
The Peter-Weyl theorem is a fundamental result in the theory of harmonic analysis that applies to topological groups. It states that for a topological group that is compact but not necessarily abelian, any square-integrable function on the group can be decomposed into a sum of irreducible representations. This theorem is important in understanding the structure and properties of these groups and has applications in various areas of mathematics and physics.

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7.

What's a complex number?
- A.
  
  It's a number that can be expressed in the form a-bi
- B.
  
  It's a number that can be expressed in the form a+bi
- C.
  
  It's a number that can be expressed in the form b+ai
- D.
  
  It's a number that can be expressed in the form c+bi
Correct Answer
B. It's a number that can be expressed in the form a+bi

Explanation
A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit. The imaginary unit i is defined as the square root of -1. The real part of the complex number is represented by a, while the imaginary part is represented by bi. The complex number system extends the real number system by introducing the concept of imaginary numbers, allowing for the representation of numbers that cannot be expressed as real numbers.

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8.

What's a real number?
- A.
  
  It's a valve of a continuous quantity that can represent a distance along a line.
- B.
  
  It's a valve of a continuous quantity that can represent a distance along a curve.
- C.
  
  It's a valve of a continuous quantity that can represent a distance along a circle.
- D.
  
  It's a valve of a constant quantity that can represent a distance along a line.
Correct Answer
A. It's a valve of a continuous quantity that can represent a distance along a line.
9.

What's the other term for a polynomial ring?
- A.
  
  Polynomial mathematics
- B.
  
  Polynomial algebra
- C.
  
  Polynomial bond
- D.
  
  Polynomials
Correct Answer
B. Polynomial algebra

Explanation
A polynomial ring is another term for polynomial algebra. This refers to the mathematical structure that deals with polynomials, which are expressions with variables and coefficients. Polynomial algebra involves operations such as addition, subtraction, multiplication, and division of polynomials, as well as the study of their properties and relationships. Therefore, "Polynomial algebra" is the correct answer as it accurately represents the alternative term for a polynomial ring.

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10.

What's the other term for an eigenvector?
- A.
  
  Eightvector
- B.
  
  Vectors
- C.
  
  Characteristic vector
- D.
  
  Sevenvector
Correct Answer
C. Characteristic vector

Explanation
The other term for an eigenvector is a characteristic vector. Eigenvectors are vectors that do not change their direction when a linear transformation is applied to them, only their magnitude is scaled. These vectors are associated with eigenvalues, which represent the scaling factor for the eigenvectors. The term "characteristic vector" is often used in the context of linear algebra to describe these special vectors.

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