Ideals on the Quantum Plane's jet space

The goal of this paper is to introduce some rings that play the role of the jet spaces of the quantum plane and unlike the quantum plane itself possess interesting nontrivial prime ideals. We will prove some results (theorems 1-4) about the prime spectrum of these rings.


Introduction
According to the classical perception of plane geometry the affine plane corresponds to the algebra freely generated by two variables x and y subject to the trivial commutation relation yx = xy. When the commutation relation yx = xy is replaced by yx = qxy the resulting algebra is called the quantum plane [1], [2].
Objects like "planes" are expected to possess some analogue of "curves". But the quantum plane possesses very few prime ideals. The idea of the paper is to look at certain rings that play the role of jet spaces of the quantum planes. This is done by introducing new "jet" variables in the style of Kolchin's differential algebra [3] and by considering commutation relations among these variables which are compatible with the action of the natural derivations on these rings. These are the multiplicative relations unlike the ones of Weyl type considered in particular in [4]. It turns out these new rings possess plenty of prime ideals which are related to the (commutative) geometry of P n × P n , n ≥ 1 2. Background and Motivation.
2.1. Quantum Symmetry (basic example). It is well-known that symmetry plays an important role in modern mathematics and theoretical physics. Gauge symmetry leads to the Standard Model in high energy physics; crystallographic space symmetry is fundamental to solid state physics, conformal symmetry is crucial to string theory and critical phenomenon. In a sense, the progress of modern physics is accompanied by study of symmetry.
The mathematical counterpart of all the above mentioned symmetries and other popular ones is group. Recently there has been great interest in the study of quantum group and quantum symmetry. Quantum group in contrast to its literal meaning is not a group, even not a semi-group. However a quantum group is the deformation of the universal enveloping algebra of a finite-dimensional semi-simple Lie algebra introduced by Drinfeld [5]and Jimbo [6] in their study of the Yang-Baxter equation.
The word "quantum" in quantum group is from the Yang-Baxter equation: solutions of the classical Yang-Baxter equation are closely related with classical or semi-simple groups, while solutions of the quantum Yang-Baxter equation related with quantum groups. The quantum thus really differs from the canonical quantization and possesses different meanings for different systems. Conventionally, quantum group is a Hopf algebra which is neither commutative nor cocommutative. A Hopf algebra is endowed with the algebra homomorphisms: comultiplication ∆ and counit ε, and the algebra anti-homomorphism: the antipode S.
Quantum group theory has been developed in different directions. In the quantum space approach [1], the initial object is a quadratic algebra which is considered being as the polynomial algebra on a quantum linear space. Quantum group appears like a group of automorphisms of the quantum linear space.
The basic example is a Quantum Group GL q (2). Let k be a ground field, q ∈ k * . By definition, the ring of polynomial functions F = F [GL q (2)] is a Hopf algebra which can be described in the following way. As a k−algebra, it is generated by a, b, c, d and a formal inverse of the central element where a, b, c, d satisfy the following commutation relations: c d where the tensor product in the r.h.s. denotes the usual product of matrices in which products like ab are replaced by a ⊗ b . The counit is given by It can be checked directly that all these structures are well defined and satisfy the Hopf algebra axioms.
3. Quantum plane: Gauss Polynomials and the q-Binomial Formula.
Definition. Let k be a field . Let q = 1 be an invertible element of the ground field k and let I q be the two-sided ideal of the free algebra k < x, y > of noncommutative polynomials in x and y generated by the element f = yx − qxy. The quantum plane is defined as the quotient algebra k<x,y> Iq . For future developments, we need to compute the powers of x + y in the quantum plane. To this end we have to consider Gauss polynomials..
Gauss polynomials are polynomials in one variable q whose values at q = 1 are equal to the classical binomial coefficients. For any integer n > 0 , set (n) q = 1 + q + q 2 + ... + q n−1 = q n − 1 q − 1 Define the q−factorial of n by (0)! q = 1 and The q−factorial is a polynomial in q with integral coefficients and with value at q = 1 equal to the usual factorial n!. We define the Gauss polynomials for 0 ≤ k ≤ n by n k q = (n)! q (k)! q (n − k)! q with following properties: a). n k q is a polynomial in q with integral coefficients and with value at q = 1 equal to the binomial coefficient n k . b).q−Pascal identity Finally for all n > 0, In particular, if q is a root of unity of order p > 0, Let z be a variable commuting with q.The q−exponential can be defined as following formal series: In this paper q will eventually be assumed not a root of unity. However some of the results can be extended to the case when q is a root of unity in which case the q−binomial formulae become relevant.

Quantum plane and Quantum Group.
A more conceptual approach to GL q (2) consists in introducing quantum plane k<x,y> Iq and obtaining the commutation relations of GL q (2) from the following matrix relations: such that x ′ , y ′ and x ′′ , y ′′ are on Quantum Plane and In this way, GL q (2) emerges merely as a quantum automorphism group of noncommutative linear space.

The problem and the main results
The family of prime ideals of the quantum plane has a simple structure as we shall presently review. Recall that an ideal P is prime if P = (1) and if from ab ∈ P it follows that a ∈ P or b ∈ P. We denote by Spec B the set of prime ideals in any ring B. Spec k<x,y> Iq consists of the following prime ideals: {< 0 >, < x, y >, < x − α, y >, < x, y − β >} where α, β ∈ k × and < S > denotes the two-sided ideal generated by set S.
Due to the commutation relation yx = qxy the above set of ideals can be rewritten as {< 0 >< x, y >, < x − α >, < y − β >} since, for example, The fact that the ring structure of the quantum plane is so trivial prevents us from considering "curves" on it . That is a motivation to attempt to introduce new rings that play the role of the jet spaces of the quantum plane and possess interesting nontrivial prime ideals. Let us consider the noncommutative ring ...and δy = y ′ , δy ′ = y ′′ , ... Recall that a k−derivation δ is a k−linear map satisfying the usual Leibniz rule: δ(F G) = δF G + F δG Define the following elements of B (1) : y and, more generally, define the following elements of B (n) : For further purposes we define the commutative ring A which is the ring of the usual(commutative) polynomials. Any monomial ..(y (n) ) jn has the bi-degree (i, j) where the total degree in x, x ′ , ..., x (n) is i = i 0 +i 1 +...+i n and the total degree in y, y ′ , y ′′ , ...y (n) is j = j 0 +j 1 +...+j n Lemma. There is a unique k−linear bijective map A (n) ∼ = A c such that for any two bi-homogeneous polynomials of bi-degrees (i, j) and (k, l) respectively, The bijection in the Lemma is not an isomorphism of rings. From now on we shall identify A (n) c and A (n) as sets via above bijection. Note that A (n) c is bi-graded in the usual way. In the following let q be not a root of unity. Our main results about Spec A (n) can be presented as the following theorems 1-4.

Theorem 1.
If 0 = P ⊂ A (n) is a prime ideal then P contains a non-zero bi-homogeneous polynomial which as an element of A (n) c is irreducible.
Theorem 3. Any prime ideal P ⊂ A (n) not containing any of the ideals < x, x ′ , ..., x (n) > or < y, y ′ , ....y (n) > is of the form P =< T >, where T is the family of all bi-homogeneous polynomials in P.

δ−prime ideals
Let us recall the previously defined derivation δ : B (n−1) → B (n) . Let A = lim − → A (n) . Then δ induces a k−derivation δ : A → A. For each n we have x (n) = δx (n−1) and y (n) = δy (n−1) Define a δ−prime ideal to be a prime ideal P such that δP ⊂ P. As in Theorem 3 let T = {f ∈ P |f is bi-homogeneous } so P =< T > We can prove the following Proposition: Since δT ⊂ T , then δf ∈< T >= P

Appendix
For the proofs of theorems 1-4 we need the following definition of the lexicographical ordering in N 2 Let's consider a polynomial g ∈ A (n) Write g = ij g ij , such that g ij is bi-homogeneous of bi-degree (i, j). Let's consider the set Γ g = {(i, j), g ij = 0} ⊂ N 2 . The size of a polynomial g in A (n) is size(g)= #Γ g , number of points in Γ g . If g ij has the bi-degree (i, j) then g ij x has the bi-degree (i + 1, j) and yg ij has bi-degree (i, j + 1). The size of gx and yg will stay the same as the size of g. Lemma 1. If h = yg − q ν gy and (ν, µ) ∈ Γ g , then the size of Γ h will be strictly less than the size of Γ g . Indeed, .
It follows that all points of Γ g with the first coordinate equal to ν will disappear in Γ h and the size of Γ h will be strictly less than the size of Γ g .
Similarly, if h = gx − q µ xg and (ν, µ) ∈ Γ g , then the size of Γ h will be strictly less than the size of Γ g . 7.1. Proof of Theorem 1. We start by showing the following claim: there exists a nonzero bi-homogeneous polynomial in P. Indeed take 0 = g ∈ P of smallest possible size. We claim that size(g) = 1 which means g is bi-homogeneous. Assume that size(g) ≥ 2 Case 1. g is not homogeneous in x, x ′ , ..., x (n) . Let's consider g = yg − q ν gy ∈ P such that there is at least one term with total degree in x, x ′ , ..., x (n) equal to ν. Since g is not homogeneous in x, x ′ , ..., x (n) , g = 0. On the other hand by the Lemma we have #Γ g < #Γ g which contradicts the minimality of size(g).
Case 2. g is homogeneous in x, x ′ , ..., x (n) but not in y, y ′ , ....y (n) . Let's considerĝ = gx − q µ xg ∈ P such that there is at least one term in g with the total degree in y, y ′ , ....y (m) equal to µ. Since g is not homogeneous in y, y ′ , ....y (n) ,ĝ = 0. On the other hand by the Lemma we have #Γĝ < #Γ g which contradicts with minimality of size(g) This proves our claim. To conclude the proof of the Theorem 1, using our claim one can pick a nonzero bi-homogeneous polynomial f ∈ P of smallest bi-degree (i * , j * ) with respect to lexicographical order among the nonzero bi-homogeneous polynomials in P . We Note the following properties of bi-degrees: If (i 1 , j 1 ) < (i 0 , j 0 ) and (k 1 , l 1 ) ≤ (k 0 , l 0 ) then (i 1 + k 1 , j 1 + l 1 ) < (i 0 + k 0 , j 0 + l 0 ) Let (i 0 , j 0 ) be the highest element of Γ g with respect to lexicographical order, (k 0 , l 0 ) be the highest element of Γ h with respect to lexicographical order. and let (i 1 , j 1 ) be the lowest element of Γ g , (k 1 , l 1 ) be the lowest element of Γ h Then the highest element of Γ g ·ch will be (i 0 +k 0 , j 0 +l 0 ) and the lowest element of Γ g ·ch will be (i 1 + k 1 , j 1 + l 1 ). Since f = g · c h we have (i 0 + k 0 , j 0 + l 0 ) = (i 1 + k 1 , j 1 + l 1 ) = (i * , j * ) Since i * = i 0 + k 0 = i 1 + k 1 and i 0 ≥ i 1 it follows that i 0 = i 1 because if i 0 > i 1 then k 0 has to be less then k 1 which contradicts with the choice of k 0 . It immediately follows that k 0 = k 1 . Similarly, j 0 = j 1 and l 0 = l 1 , so g and h are both bi-homogeneous of degrees less than (i * , j * ).
Since P is a prime ideal, at least one of them belongs to P. This contradicts the choice of f.

Proof of the Theorem 2. Assume f is irreducible in
and bi-homogeneous of bi-degree (i, j).
We prove by induction on the total degree N in x, x ′ , ..., x (n) , y, y ′ , ....y (n) that if f has a total degree N then from g · h ∈< f >it follows that g or h ∈< f > If N = 0 the theorem is clear. Assume the theorem is true for total degree less or equal to N − 1.
Let N be the total degree of f. We have that from g · h ∈< f >it follows that g · h = i α i f β i where α i and β i belong to A (n) .We may assume that α i and β i are bihomogeneous.
Let (i 0 , j 0 ) be the highest element of Γ g with respect to lexicographical order, (k 0 , l 0 ) be the highest element of Γ h and (m 0 , n 0 ) be the highest element of Γ γ . Then for some t and s. Since f is irreducible in the commutative ring A (n) c , it follows that g i 0 j 0 = η · c f = q −li η ·f (f is bi-homogeneous and the bi-degree of η is (k, l) ) or h k 0 l 0 = η · c f = q −li η · f (bi-degree of η is k, l ) Assume, for example, the former is the case. From gh = γf we get (g − g Since the total degree in x, x ′ , ..., x (n) , y, y ′ , ....y (m) of g ′ · h is less or equal to N − 1, by the induction hypothesis either g ′ ∈< f >and g = g ′ + q w η · f ∈< f >or h ∈< f > and we are done. 7.3. Proof of the theorem 3. It is obvious that < T >⊂ P.
To prove < T >⊃ P assume on the contrary that P does not belong to < T > . Let f ∈ P \ < T > be of minimal size. Since by this assumption f cannot be bi-homogeneous the size(f ) > 1. There are two cases.
There exists a pair (i, j) ∈ Γ f such that i = k otherwise f ought to be homogeneous in x, x ′ , ..., x (n) .
Let h = yf − q i f y ∈ P.
Then by the Lemma 1 size(h) < size(f ). It follows that h ∈< T > so h can be written as where B l ∈ P and bi-homogeneous Let us pick out the bi-homogeneous components of bi-degree (k, l + 1). Then λ ki ·f kl ·y = γ 1 B 1 + γ 2 B 2 + ... + γ m B m ∈< T > where γ 1 , γ 2, ...., γ m are bi-homogeneous. Since λ ki = 0 because i = k, we have f kl · y ∈< T > . So f kl · y ∈ P Similarly let h (s) = y (s) f − q i f y (s) . As above we get f kl · y (s) ∈ P for all s. Since < y, y ′ , ....y (n) > is not contained in P it follows that at least one of y (s) / ∈ P. Because P is prime, f kl ∈ P. But f kl is obviously bi-homogeneous so f kl ∈< T > Since the pair (k, l) is arbitrary it follows that f = st f st ∈< T > -a contradiction.
Let h = f x − q j xf ∈ P. Then by the Lemma 1 size(h) < size(f ). It follows that h ∈< T > so h can be written as h = γ 1 B 1 + γ 2 B 2 + ... + γ m B m where B l ∈ P and bi-homogeneous .