1.
Simple objects are those that do not contain which of the following?
Correct Answer
D. Non-trivial sub-objects
Explanation
Simple objects are those that do not contain non-trivial sub-objects. This means that simple objects do not have any sub-objects that are considered complex or non-trivial. Trivial objects and trivial sub-objects may still be present in simple objects, but non-trivial sub-objects are not.
2.
A semi-simple object is one that can be decomposed into a sum of what?
Correct Answer
B. Simple objects
Explanation
A semi-simple object is one that can be decomposed into a sum of simple objects. Simple objects are the building blocks or irreducible components of a semi-simple object. They cannot be further decomposed into smaller objects. Therefore, the correct answer is "Simple objects."
3.
What is a nonzero representation that has no proper subrepresentation?
Correct Answer
C. An irreducible representation of an algebraic structure
Explanation
An irreducible representation of an algebraic structure is a nonzero representation that has no proper subrepresentation. This means that it cannot be broken down or decomposed into smaller parts that still retain the same structure. It is a fundamental building block of the algebraic structure and cannot be further simplified.
4.
"Any finite-dimensional representation is a direct sum of simple representations" is a postulate of which of the following?
Correct Answer
C. Maschke's theorem
Explanation
Maschke's theorem states that any finite-dimensional representation of a finite group is completely reducible, meaning it can be decomposed into a direct sum of simple representations. This theorem is a fundamental result in the theory of group representations and has important applications in various areas of mathematics and physics. It provides a powerful tool for studying the structure and properties of representations of finite groups.
5.
What is semi-simplicity also called?
Correct Answer
B. Complete reducibility
Explanation
Semi-simplicity is also known as complete reducibility. This term refers to the property of a representation of a group or algebra being completely decomposable into irreducible subrepresentations. In other words, a representation is semi-simple if it can be broken down into simpler, irreducible components. This concept is important in the study of group theory and linear algebra, as it allows for a deeper understanding of the structure and behavior of representations.
6.
Which of these describes every representation of a finite group?
Correct Answer
A. Semi-simple
Explanation
Every representation of a finite group can be described as semi-simple. This means that the representation can be decomposed into a direct sum of irreducible representations, where irreducible representations cannot be further decomposed. In other words, every representation can be broken down into simpler, irreducible components. This property is true for all finite groups, making the statement "semi-simple" applicable to every representation.
7.
What is a finite-dimensional representation of a semisimple compact Lie group?
Correct Answer
B. Semisimple
Explanation
A finite-dimensional representation of a semisimple compact Lie group is a representation that decomposes into a direct sum of irreducible representations. Semisimple groups have no non-trivial, proper, closed, connected normal subgroups, and their representations can be fully understood by studying irreducible representations. Therefore, the correct answer is "Semisimple".
8.
Which is a feature of fusion category?
Correct Answer
C. Monoidal
Explanation
A fusion category is a mathematical structure that combines the properties of a category and a tensor product. The concept of "monoidal" refers to the ability to define a tensor product operation on objects in the category. This means that objects can be combined in a way that is associative and has an identity element. Therefore, "monoidal" is a feature of a fusion category, indicating its ability to support tensor product operations.
9.
What is a collection of objects and maps between such objects?
Correct Answer
A. Category
Explanation
A category is a collection of objects and maps between such objects. In category theory, objects are the basic elements, and maps (also called morphisms) are the relationships between these objects. A category consists of a set of objects and a set of maps, along with operations that define how these maps can be composed and associated. It provides a framework for studying and analyzing mathematical structures and their relationships.
10.
Which of these defines the following statement?
While the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities.
Correct Answer
B. Jordan–Hölder theorem
Explanation
The Jordan-Hölder theorem defines the statement given in the question. This theorem states that any two composition series for a module have the same length and the same isomorphism factors, up to permutation and isomorphism. In other words, while the decomposition into irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. Therefore, the Jordan-Hölder theorem is the correct answer for this question.