What Do You Know About Semi-simplicity?

10 Questions | Total Attempts: 106

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What Do You Know About Semi-simplicity? - Quiz

Semi-simplicity is a concept widely used in many branches of mathematics, which include representation theory, linear and abstract algebra, algebraic geometry, and category theory. An object that has semi-simplicity can be deteriorated into a total of simple objects. To know more about semi-simplicity, take this short, intelligent quiz.


Questions and Answers
  • 1. 
    Simple objects are those that do not contain which of the following?
    • A. 

      Trivial objects

    • B. 

      Trivial sub-objects

    • C. 

      Non-trivial objects

    • D. 

      Non-trivial sub-objects

  • 2. 
    A semi-simple object is one that can be decomposed into a sum of what?
    • A. 

      Trivial objects

    • B. 

      Simple objects

    • C. 

      Non-simple objects

    • D. 

      Non-trivial objects

  • 3. 
    What is a nonzero representation that has no proper subrepresentation? 
    • A. 

      A semi-simplicity of sets

    • B. 

      A rigid representation of planes

    • C. 

      An irreducible representation of an algebraic structure

    • D. 

      A postulate of representation theory

  • 4. 
    "Any finite-dimensional representation is a direct sum of simple representations" is a postulate of which of the following?
    • A. 

      Representation theorem

    • B. 

      Schur's Lemma

    • C. 

      Maschke's theorem

    • D. 

      Reducibility theorem

  • 5. 
    What is semi-simplicity also called?
    • A. 

      Multiplicity

    • B. 

      Complete reducibility

    • C. 

      Complete rigidity

    • D. 

      Triviality

  • 6. 
    Which of these describes every representation of a finite group?
    • A. 

      Semi-simple

    • B. 

      Simple

    • C. 

      Non-simple

    • D. 

      Trivial

  • 7. 
    What is a finite-dimensional representation of a semisimple compact Lie group?
    • A. 

      Absolute

    • B. 

      Semisimple

    • C. 

      Simple

    • D. 

      Rigid

  • 8. 
    Which is a feature of fusion category?
    • A. 

      Flexible

    • B. 

      Trivial

    • C. 

      Monoidal

    • D. 

      Non-simple

  • 9. 
    What is a collection of objects and maps between such objects?
    • A. 

      Category

    • B. 

      Aggregate

    • C. 

      Set

    • D. 

      Concrete

  • 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.
    • A. 

      Schur's Lemma

    • B. 

      Jordan–Hölder theorem

    • C. 

      Holder's Theory of Multiplicity

    • D. 

      Mackshen Theory

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