1.
What does s(α) represent?
Correct Answer
A. The starting vertices of α
Explanation
The correct answer is "The starting vertices of α". This means that s(α) represents the vertices from which the path or edge α begins. It does not represent the ending vertices or the vertices in the middle of the path. It specifically refers to the initial vertices of α.
2.
What does t(α) represent?
Correct Answer
B. The ending vertices of α
Explanation
The correct answer is "The ending vertices of α." This means that t(α) represents the vertices at the end of α, indicating the final position or destination of α.
3.
What do we call the quivers of finite representation-type?
Correct Answer
A. Dynkin quivers
Explanation
Dynkin quivers are the correct answer because they are the quivers of finite representation-type. Dynkin quivers are a type of directed graph that represent the relations between simple modules in a finite-dimensional algebra. They are named after Eugene Dynkin, a mathematician who made significant contributions to the theory of Lie groups and Lie algebras. Dynkin quivers are widely used in the study of representation theory and have important applications in various areas of mathematics and physics.
4.
What does K represent?
Correct Answer
D. Vector space
Explanation
K represents a vector space. In mathematics, a vector space is a collection of objects called vectors, which can be added together and multiplied by scalars (numbers), satisfying certain axioms. K is commonly used to denote a vector space, and it is a standard notation in linear algebra. Therefore, the correct answer is vector space.
5.
Which of these is used to represent a quiver?
Correct Answer
A. Matrices
Explanation
Matrices are used to represent a quiver. A quiver is a directed graph where the nodes represent objects and the arrows represent morphisms between the objects. Matrices can be used to represent the adjacency matrix of a quiver, where each entry in the matrix represents the number of arrows between two objects. Therefore, matrices are the correct answer for representing a quiver.
6.
What does R(α) represent?
Correct Answer
C. Representation space
Explanation
The correct answer is Representation space. R(α) represents the representation space, which is a mathematical concept used in the field of representation theory. In representation theory, the focus is on studying abstract algebraic structures by representing them as linear transformations on vector spaces. The representation space refers to the vector space on which these transformations act. It is an important concept in understanding the behavior and properties of abstract algebraic structures.
7.
Why do we refer them as semi-invariants?
Correct Answer
B. Invariants are up to the character of a group
Explanation
The term "semi-invariants" suggests that the invariants are limited or restricted in some way, but still encompass the character of a group. The phrase "up to the character of a group" implies that the invariants are within the boundaries or constraints set by the character of a group. Therefore, referring to them as semi-invariants highlights that they possess certain characteristics of invariants, but may not fully meet all the requirements or conditions.
8.
Which form of shape do they represent?
Correct Answer
B. Ring shapes
Explanation
The given answer, "Ring shapes," refers to a form of shape that has a circular structure with a hole in the center. These shapes resemble a ring or a circular band. They can be seen in various objects such as jewelry, wheels, or even certain architectural designs.
9.
In case Q has neither loops nor cycles, what happens to the invariant function?
Correct Answer
B. It remains constant
Explanation
If the given question states that there are neither loops nor cycles, it means that the process is linear and does not repeat. In such a case, the invariant function, which is a function that remains unchanged throughout the process, will also remain constant. Since there are no loops or cycles to affect the values of the variables involved, the invariant function will not change and will remain constant throughout the process.
10.
What do we call a function belonging to SI (Q, d) σ?
Correct Answer
A. Semi-invariant of weight σ
Explanation
A function belonging to SI (Q, d) σ is called a semi-invariant of weight σ.