Introduction to elliptic algebras Exercises 1 A family of associative ...

Introduction to elliptic algebras
Exercises
1 A family of associative algebras
Let p, q ∈ C, p, q = 0 and n ∈ N. Let F (n)
m (for m = 1, 2, 3, ...) be a linear space of symmetric
polynomials in m variables of degree less than n in each variable. For f(z1, ..., zm) ∈ F (n)
m and
g(z1, ..., zl) ∈ F (n)
l define a function f ∗ g in m + l variables by the formula:
f ∗ g(z1, . . .,zm+l) =
= 1
m!l!
σ∈Sm+l
f(zσ1, . . .,zσm)g(p m zσm+1, . . .,p m zσm+l )
where λ(x, y) = x−qy
x−y and Sm+l is the symmetric group.
1 Prove that f ∗ g ∈ F (n)
m+l .
1 ≤ i ≤ m
m + 1 ≤ j ≤ m + l
2 Prove that (f ∗ g) ∗ h = f ∗ (g ∗ h) for all f ∈ F (n)
m , g ∈ F (n)
l , h ∈ F (n)
k .
Does f ∗ g = g ∗ f hold for all f, g?
λ(zσi , zσj )
Let F (n)
q,p be the algebra with underlying linear space F (n)
1 ⊕ F (n)
2 ⊕ . . . and product ∗.
3 Prove that F (n)
q,p is generated by F (n)
1 for generic q, p.
Let ei ∈ F (n)
1 where i = 0, ..., n − 1 be defined as follow: ei(z) = zi . Let ei,j ∈ F (n)
2 be
defined as follow: ei,j(z1, z2) = zi 1z j
2 + z j
4 Calculate ei ∗ ej, ei ∗ ej,k and ei,j ∗ ek.
1z i 2 and so on.
5 Prove that F (n)
q,p ⊂ F (n+1)
q,p as associative algebras.
6 Find quadratic relations (if any) between e0 and e1.
7 Find quadratic relations (if any) between e0, e1 and e2.
8 Find quadratic relations (if any) between e0, e1, e2 and e3.
9 Find quadratic relations (if any) between e0, ..., en−1.
10 For which q, p is the algebra F (2)
q,p commutative?
11 For which q, p does the algebra F (4)
q,p have central elements of degree 2? Find these
elements.
1

12 For which q, p does the algebra F (3)
q,p have central elements of degree 3? Find these
elements.
13 Classify representations of the algebra F (n)
q,p with basis v1, v2, ... and action of the form
eivj = xi,jvj+1 where xi,j are nonzero complex numbers.
14 Classify representations of the algebra F (n)
q,p with basis vj,k where j, k = 1, 2, ... and action
of the form eivj,k = xi,j,kvj+1,k + yi,j,kvj,k+1.
15 Generalize exercises 13, 14 for representations with basis vi1,...,il .
16 Prove that the algebra F (n)
q,p is generated by F (n)
1 if and only if q is not root of unity.
17 Prove that if q N = 1 then F (n)
N
2 θ-functions in one variable
is not generated by F (n)
1 .
Let n ∈ N and c, τ ∈ C where Im τ > 0. Let Γ be the integral lattice in C generated by 1 and
τ. We denote by Θn,c(τ) the vector space of holomorphic functions φ : C → C such that
φ(z + 1) = φ(z), φ(z + τ) = (−1) n exp (−2πi(nz − c)) φ(z).
1 Prove that dim Θn,c(τ) = n.
2 Prove that every function f ∈ Θn,c(τ) has exactly n zeros modulo Γ (counted with
multiplicities).
3 Prove that the sum of zeros of every function f ∈ Θn,c(τ) modulo Γ is equal to c.
Let θ(z) be a nonzero element of Θ1,0(τ).
4 Find the Fourier decomposition of θ(z).
5 Prove that θ(0) = 0, and this is the only zero modulo Γ.
6 Prove that θ(−z) = −e −2πiz θ(z).
7 Prove that any element f(z) of Θn,c(τ) has the form λθ(z−u1)...θ(z−un) where λ, u1, ..., un ∈
C and u1 + ... + un = c.
8 Prove that the following functions θ(z −u)θ(z +u), θ(z −v)θ(z +v) and θ(z −w)θ(z +w)
are linearly dependent as functions in z. Find linear equation between them.
Let us introduce the following linear operators T 1
n
of one variable:
T 1 f(z) = f
n
z + 1
n
and T 1
τ acting on the space of functions
n
1 n−1
−
, T 1 τf(z) = e2πi(z+ 2n 2n
n τ)
f z + 1
n τ
.
9 Prove that the space Θ n−1
n, (τ) is invariant with respect to the operators T 1
2
n
2
and T 1
n τ.

10 Prove that T 1 T 1
τ = e2πi n T 1 1 τT n n n n
Θ n−1
n, (τ) satisfy the relations T
2
n 1 = T
n
n 1 = 1. τ n
and the restriction of these operators to the space
11 Prove that there is a basis {θα; α ∈ Z/nZ} in the space Θ n−1
n, (τ) in which our operators
2
α
2πi act as follows: T 1 θα = e nθα, and T 1
n
nτθα = θα+1. Prove that this basis is unique up to
multiplication by a common constant.
12 Find the zeroes of the functions θα(z).
13 Find the Fourier decomposition of the functions θα(z).
14 Prove the following identity:
θ0(z1) θ1(z1) . . . θn−2(z1) θn−1(z1)
θ0(z2) θ1(z2) . . . θn−2(z2) θn−1(z2)
=
. . . . .
θ0(zn) θ1(zn) . . . θn−2(zn) θn−1(zn)
µ exp(2πi(z2 + 2z3 + . . . + (n − 1)zn))θ(z1 + ... + zn −
where µ is a constant.
3 An algebra with three generators
n − 1
)
2
θ(zi − zj),
1≤i

4 Skew polynomials
Consider the associative algebra defined by n generators x1, ..., xn and the following quadratic
relations xjxi = qi,jxixj, where i < j and qi,j = 0. This algebra is called the algebra of skew
polynomials.
1 Prove that the monomials {x α1
1 . . .x αn
n ; α1, . . .,αn ∈ Z≥0} form a basis of the algebra of
skew polynomials.
2 Find a formula for the product in this basis.
4