Math Paper A (XLC)

Duration: 4 hours

Use of calculator disallowed.

We propose to study the algebras of the remarkable endomorphisms of vector spaces of infinite dimension.

**Preamble**

An root of unity is called **primitive** if it generates the group of roots of unity.

In this problem, all vector spaces are over the base field of complex numbers field .

If **ε** is a vector space, the algebra of the endomorphisms of ε is denoted by **L(ε)**, and the group of the automorphisms of ε is denoted by **GL(ε)**.

denotes the identity mapping of ε.

If u ∈ L(ε), denotes the sub-algebra of L(ε) of the Polynomials in u.

denotes the vector space of the functions of to .

If ƒ is the function of to , **supp(ƒ)** denotes the set of κ ∈ such that ƒ(κ) ≠0

We call this set the **support** of ƒ.

Throughout the problem, **V **denotes the set of functions of to of which the support is a finite set.

I – **Operators on the functions with finite support**

1. a. Prove that **V** is a vector subspace of .

Given ƒ ∈ ,

we define E(ƒ) ∈

by

E(ƒ)(κ)= ƒ(κ+1), κ ∈

1. b. Show that E∈ L( ) and V is stable by E.

In the following, E denotes uniquely the endomophism of V induced.

2. Show that E ∈ GL(V).

3. For , we define by:

3.a. Prove that the family is the base of V.

3.b. Calculate E().

Let ,

we define the respective linear mappings

by:

and

4. Prove that

if and only if for all

**In the remaining of Section I (but not in the following Sections), we asume the conditions in question 4 are verified.**

5. Prove that

if and only if

6.a. Prove that for the vector space generated by has finite dimension.

6.b. Deduce that a vector subspace non-reduced to {0} of V, stable by H, contains at least one of the .

**In the remaining of Section I (but not in the following Sections), we asume the conditions in question 5 are verified and **

7.a. Prove that

7.b. Prove that E and F are not of finite order in the group GL(V).

7.c. Calculate the kernel of H and prove that

8. denotes the polynomials with complex number coefficients in one indeterminate X.

8.a. Prove that is isomorphic (as Algebra) to .

8.b. Prove that is isomorphic (as Algebra) to .

8.c. Prove that is isomorphic (as Algebra) to .

**II – Interlude**

In all the rest of the problem, we fix an odd interger and** q **a primitive root of unity.

9. Prove that is a primitive root of unity.

Let

10. Consider the element is :

10.a. Calculate .

Prove that

10.b. Let **b** root of **a**.

Calacute the eigenvectors of and the associated eigenvalues in function of b, q and .

Let’s define a linear mapping by

, and

define r and p respectively the residue and the quotient of the euclidian division of i by ℓ; ie:

11. Prove that is a projector of image .

**III – Quantum Operators**

12. Prove that

if and only if

**In the following problem, we asume the conditions in question 12 are verified and **

13. Prove that .

14. Prove that

if and only if

**In the following problem, we asume the conditions in question 14 are verified.**

15.a. Prove that are periodic over , of periods dividing .

15.b. Prove that the period of .

15.c. Prove that the period of is also equal to .

16. Let with being inverse of H.

16.a. Prove that

.

16.b. For , prove that is an eigenvector of C.

16.c. Deduce that C is a homothety of v of which we calculate the ratio of in function of .

16.d. Let’s fix . Prove that the mapping

is a bijection of onto .

16.e. Let’s fix . Prove that the mapping

is a surjection of onto but not a bijection.

**IV – Modular Quantum Operators**

Let like in the Section II. We say an element of L(V) is **compatible** with if

17.a. Prove that if is commutative with ,then is compatible with .

17.b. Prove that are compatible with .

Let the set of endomorphisms which are compatible with .

18. Prove that is a sub-algebra of L(V).

19. Prove that .

20.a. Show that there exists an unique morphism of algebras such that:

20.b. Prove that is contained in the kernel of if and only if the image of is a vector subspace of v generated by the vectors where is the euclidian division of .

21. Let’s study in this question .

21.a. Determine .

21.b. Deduce .

21.c. Calculate the dimension of the vector subspace of

21.d. Calculate the eigenvectors of

22. Let W a non zero sub-space of stable by .

22.a. Show that W contains at least one of the vectors .

22.b. What do you say if W is in addition stable by ?

23. Give a necessary and sufficient condition on in order for the operator to be nilpotent.

—End—

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