1.
What can Hecke Algebra be interpreted as?
Correct Answer
B. Algebra of double cosets
Explanation
Hecke algebra can be interpreted as an algebra of double cosets. This means that it is a mathematical structure that describes the relationships between elements in a group when they are partitioned into double cosets. Double cosets are a way of organizing the elements of a group based on their relationships with two specific subgroups. The Hecke algebra provides a framework for studying these relationships and understanding the algebraic properties that arise from them.
2.
Which of these is a deformation of coxeter groups?
Correct Answer
C. Iwahori-Hecke Algebra
Explanation
The Iwahori-Hecke Algebra is a deformation of the Hecke Algebra, which in turn is a deformation of the group algebra of the symmetric group, also known as the Coxeter group. Deformation refers to a mathematical process of modifying a structure while preserving certain properties. In this case, the Iwahori-Hecke Algebra is a modified version of the Hecke Algebra, designed to study representations of certain groups called Iwahori-Hecke algebras. Therefore, the Iwahori-Hecke Algebra is a deformation of Coxeter groups.
3.
Hecke Algebra are quotient of which type of ring?
Correct Answer
C. Artin Braid group
Explanation
The Hecke Algebra is a quotient of the Artin Braid group. The Artin Braid group is a mathematical structure that represents the ways in which strands can be braided together. The Hecke Algebra, on the other hand, is a generalization of the symmetric group algebra that arises in the study of representations of finite groups. The connection between the two lies in the fact that the Hecke Algebra can be obtained by quotienting the group algebra of the Artin Braid group by certain relations. Therefore, the correct answer is Artin Braid group.
4.
In Hecke Algebra of coxeter group, what does R have?
Correct Answer
B. Identity
Explanation
In the Hecke Algebra of a Coxeter group, the element R refers to the identity element. This means that when R is multiplied with any other element in the algebra, it does not change the value of that element. The identity element is an important concept in algebra as it serves as the neutral element for multiplication.
5.
The element of natural basis is which of the following?
Correct Answer
A. Multiplicative
Explanation
The element of natural basis is multiplicative because a natural basis is a set of vectors that spans a vector space and can be used to express any vector in that space as a linear combination of the basis vectors. In this context, the term "multiplicative" refers to the fact that the basis vectors are combined using multiplication, rather than addition or any other operation.
6.
Who discovered the quantum groups?
Correct Answer
D. Michio Jimbo
Explanation
Michio Jimbo is credited with the discovery of quantum groups. Quantum groups are a mathematical concept that emerged in the 1980s as an extension of the theory of Lie groups. They have applications in various areas of mathematics and physics, including representation theory, knot theory, and statistical mechanics. Jimbo's work on quantum groups has had a significant impact on the field and has led to further developments and applications in the years since its discovery.
7.
How many relation does the multiparameter Hecke Algebra have?
Correct Answer
C. 2
Explanation
The multiparameter Hecke Algebra has 2 relations.
8.
Who proposed Hecke Algebra to be the foundation of Topological Quantum computation?
Correct Answer
C. Michael Freedman
Explanation
Michael Freedman proposed Hecke Algebra to be the foundation of Topological Quantum computation.
9.
What does R stand for in Hecke Algebra of a coxeter group?
Correct Answer
D. Commutative ring
Explanation
The correct answer is "Commutative ring." In the Hecke Algebra of a Coxeter group, the letter R stands for "Commutative ring." This means that the algebraic structure associated with the Coxeter group is a commutative ring, where multiplication is commutative and satisfies certain properties. This is an important property in the study of Coxeter groups and their associated algebras.
10.
Which of these has been an advantage of the representation of Hecke Algebra?
Correct Answer
A. Discovery of Quantum groups
Explanation
The representation of Hecke Algebra has been advantageous in the discovery of Quantum groups. This is because Hecke Algebras, which are a type of algebraic structure, have been used to study and understand the properties of Quantum groups. Quantum groups are a generalization of Lie groups and Lie algebras, and they have important applications in various areas of mathematics and physics, such as quantum mechanics and theoretical physics. Therefore, the representation of Hecke Algebra has played a significant role in the advancement and exploration of Quantum groups.