POINTED HOPF ALGEBRAS ACTING ON QUANTUM POLYNOMIALS

VYACHESLAV A. ARTAMONOV

To the memory of my mother

Abstract. We consider actions of pointed Hopf algebras on general quantum polynomials and their invariants.

Introduction

Let k be a field with a fixed matrix Q = (q_{ij}) ∈ Mat(n, k) whose entries q_{ij} ∈ k^{∗} satisfy relations q_{ii} = q_{ij}q_{ji} = 1 for all i, j. Fix an integer r such that 0 ≤ r ≤ n. Denote by

Λ = k_{Q}[X_{1}^{±1}, …, X_{r}^{±1}, X_{r+1}, …, X_{n}]

the associative k-algebra with a unit element generated by elements X_{1}, …, X_{n}, X_{1}^{−1}, …, X_{r}^{−1} subject to defining relations

X_{i}X_{j} = q_{ij}X_{j}X_{i}, 1 ≤ i, j ≤ n,
X_{i}X_{i}^{−1} = X_{i}^{−1}X_{i} = 1,  1 ≤ i ≤ r.  (1)

The algebra Λ is an algebra of quantum polynomials. The elements q_{ij} are multiparameters. Throughout the paper we shall assume that the algebra Λ is a general algebra of quantum polynomials that is all multiparameters q_{ij}, 1 ≤ i < j ≤ n, are independent in the multiplicative group k^{∗} of the field k and also n ≥ 3.

The algebra Λ can be viewed as a coordinate algebra of a quantum affine space 𝔸_{Q}^{n} if r = 0 [D] and a coordinate algebras of a quantum torus 𝕋_{Q}^{n} if r = n [B]. We shall unify both cases and consider arbitrary r, 0 ≤ r ≤ n. Then Λ can be considered as a coordinate algebra of 𝕋_{Q}^{r} × 𝔸_{Q}^{n−r}. The aim of the present paper is to a study actions of finite quantum groups G on 𝕋_{Q}^{r} × 𝔸_{Q}^{n−r}. Since a coordinate algebra U of G is a finite dimensional Hopf algebra we have to consider a coaction Λ → U ⊗ Λ of U on Λ. Taking a dual Hopf algebra H = U^{∗} we can consider an induced action H ⊗ Λ → Λ of H on Λ. In the paper we shall assume that H is a finite dimensional pointed Hopf algebra and therefore Λ is a left H-module algebra. In Theorem 2.18 we show that there exist linear forms χ_{v}, ω_{v} ∈ H^{∗} where v ∈ ℤ^{n} such that

h · X^{v} = χ_{v}(h)X^{v} + ω_{v}(h)X^{−v}

Research partially supported by INTAS00-566, Grants RFFI 00-15-96128.

for any monomial X^{v}. It is shown that the span F^{∗} = ⟨χ_{v}, ω_{v} | v ∈ ℤ^{n}⟩ in the dual Hopf algebra H^{∗} is a commutative Hopf subalgebra and Λ is a left F -module algebra over F = (F^{∗})^{∗}. Moreover there exists a surjective Hopf algebra homomorphism π : H → F such that the action of H on Λ is a composition of π and the action of F on Λ. If ω_{v} ≠ 0 for some v then there exists a group-like element g ∈ H of order 2 such that F = F_{χ}♯k⟨π(g)⟩, where

F_{χ} = {a ∈ F | ω_{w}(a) = 0 for all w}

and k⟨π(g)⟩ is a group algebra of a cyclic group ⟨π(g)⟩ of order 2. There is given a basis of F_{χ} with (co)multiplication tables.

Particular cases of this situation have been considered in [AC], [OP], [AW], [A1], [ACo]. In [AC] the authors study some automorphism groups and Lie algebras of derivation Der Λ under an assumption that r = 0. Derivation and automorphisms of Λ in the case r = n were considered in [OP]. A systematic study of automorphism group of a general quantum polynomial algebra for any r and for n ≥ 3 was carried out in [AW]. An investigation of group of automorphisms of division rings of fractions of Λ was made in [ACo], [A1]. We shall quote on of these results which will be used in the paper.

Theorem 0.1 ([OP], [AW]). Suppose that γ ∈ Aut Λ and n ≥ 3. Then there exist elements γ_{1}, …, γ_{n} ∈ k^{∗} and an integer ϵ = ±1, such that γ(X_{w}) = γ_{w}X_{w}^{ϵ}, w = 1, …, n. In particular, if r < n, then ϵ = 1 and

Aut Λ ≃ \underbrace{k^{∗} × ⋯ × k^{∗}}_{n times} .

Let G be a finite subgroup of Aut Λ. Then the subalgebra of invariants Λ^{G} is left and right Noetherian and Λ is a finitely generated left and right Λ^{G}-module. Suppose that I is a nonzero left ideal in Λ. Then I ∩ Λ^{G} ≠ 0. In particular if F is the division ring of fractions of Λ, then F^{G} is the division ring of fractions of Λ^{G}.

Notation 0.2. Let ℤ^{r} × (ℕ ∪ 0)^{n−r} be the set of all multi-indices

u = (u_{1}, …, u_{n}), u_{1}, …, u_{n} ∈ ℤ, u_{r+1}, …, u_{n} ≥ 0.  (2)

For a multi-index u from (2) we put

X^{u} = X_{1}^{u_{1}} ⋯ X_{n}^{u_{n}} ∈ Λ.  (3)

A product X^{u} from (3) will be called a monomial. If f is a nonzero element of Λ then supp f is the set of all multi-indices u such that X^{u} occurs in f.

The paper was written during author’s visit to UIA at Antwerpen. The author thanks Prof. Fred van Oystaeyen for hospitality and useful discussions. The author also thanks an unknown referee for valuable remarks and comments.

1. Generalized derivations and automorphisms

The algebra Λ is a left and right Noetherian domain satisfying the Ore condition. In this section we shall consider a Lie algebras of derivation of Λ and some generalizations.

Notation 1.1. For an associative algebra A denote by Der A the Lie algebras of derivations of A. Denote also by Derint A the set of all inner derivations of A. Let also Aut A stand for the groups of automorphisms of A.

Definition 1.2. Let A be an associative algebra and g ∈ Aut A. A linear operator ∂ on A is a g-derivation (or a skew g-derivation) if ∂(xy) = ∂(x)y + g(x)∂(y) for all x, y ∈ A. If u ∈ A, then a linear operator ad_{g} u defined as (ad_{g} u)(x) = ux − g(x)u is an inner g-derivation. The space of all g-derivations of A will be denoted by Der_{g} A.

It is not hard to check that every inner g-derivation is always a gderivation, and ad_{g}(αu_{1} + βu_{2}) = α ad_{g} u_{1} + β ad_{g} u_{2} for all α, β ∈ k. If g is an identical automorphism then every g-derivation is a derivation of A.

Notation 1.3. Throughout the rest of the section we fix an automorphism g ∈ Aut Λ which has a finite order d ≥ 1 such that

g(X_{j}) = ξ_{j}X_{j}^{ϵ}, ξ_{j} ∈ k^{∗}, 1 ≤ j ≤ n, ϵ = ±1,

So ξ_{j}^{d} = 1 for any j = 1, …, n, and ξ_{i} ≠ 1 for some index i if d > 1. By Theorem 0.1 this is always the case if n ≥ 3, n > r and g ∈ Aut Λ.

Proposition 1.4. Let g be from Notation 1.3 with ϵ = 1. Then there is a g-derivation ∂ of Λ such that

∂(X_{i}) = (1 − ξ_{i})X_{i}, i = 1, …, n.

Proof. The algebra Λ is given by defining relations (1). So it suffices to check that

∂(X_{i})X_{j} + g(X_{i})∂(X_{j}) = ∂(X_{i}X_{j}) =
q_{ij}∂(X_{j}X_{i}) = q_{ij} [∂(X_{j})X_{i} + g(X_{j})∂(X_{i})] .

A direct calculation shows that this equation is satisfied. □

The following affirmation will be used throughout the paper.

Proposition 1.5 ([MP]). Let

p = (p_{1}, …, p_{n}), t = (t_{1}, …, t_{n}) ∈ ℤ^{r} × (ℕ ∪ 0)^{n−r}.

Then

X^{p}X^{t} = (∏_{j>i} q_{ji}^{p_{j}t_{i}}) X^{p+t} = (∏_{j>i} q_{ji}^{p_{j}t_{i}−p_{i}t_{j}}) X^{t}X^{p}.

In particular, if r = n then

(X^{p})^{−1} = (∏_{j>i} q_{ji}^{p_{j}p_{i}}) X^{−p}.  (4)

Proof. We have

X^{p}X^{t} = X_{1}^{p_{1}} ⋯ X_{n}^{p_{n}}X_{1}^{t_{1}} ⋯ X_{n}^{t_{n}} =

(∏_{i=1}^{n−1} q_{ni}^{p_{n}t_{i}}) X_{1}^{p_{1}} ⋯ X_{n−1}^{p_{n−1}}X_{1}^{t_{1}} ⋯ X_{n−1}^{t_{n−1}}X_{n}^{p_{n}+t_{n}} =

… = (∏_{j>i} q_{ji}^{p_{j}t_{i}}) X^{p+t}.

Similarly

X^{t}X^{p} = (∏_{j>i} q_{ji}^{t_{j}p_{i}}) X^{t+p}

and the proof follows. □

Corollary 1.6 ([A1], Lemma 3.15). Let X^{p}, X^{t} be from Proposition 1.5. Then the following are equivalent:

a) monomials X^{p}, X^{t} commute;

b) the rank of the matrix

(\begin{matrix}p \\ t \end{matrix})

is at most 1.

We are going to classify elements u_{1}, …, u_{n} ∈ Λ such that

(u_{i}X_{j} − q_{ij}ξ_{j}X_{j}^{ϵ}u_{i}) − (q_{ij}u_{j}X_{i} − ξ_{i}X_{i}^{ϵ}u_{j}) =

α_{ij}X_{i}X_{j} + β_{ij}X_{i}X_{j}^{−1} + γ_{ij}X_{i}^{−1}X_{j} + θ_{ij}X_{i}^{−1}X_{j}^{−1} (5)

for each pair of indices 1 ≤ i, j ≤ n, where α_{ij}, β_{ij}, γ_{ij}, θ_{ij} ∈ k. If r < n, then it is assumed that ϵ = 1 and β_{ij} = γ_{ij} = θ_{ij} = 0.

Proposition 1.7. For all i, j = 1, …, n we have

α_{ij} = −α_{ji}, θ_{ij} = −θ_{ji}, β_{ij} = −q_{ij}^{2}γ_{ji}.

If i = j, then α_{ii} = β_{ii} + γ_{ii} = θ_{ii} = 0

Proof. If we transpose indices i, j in (5), multiply it by −q_{ij} and add to (5), then the left hand side vanishes while the right hand side takes the form

α_{ji}X_{i}X_{j} + q_{ij}^{2}γ_{ij}X_{i}X_{j}^{−1} + q_{ij}^{2}β_{ji}X_{i}^{−1}X_{j} + θ_{ji}X_{i}^{−1}X_{j}^{−1} +
α_{ij}X_{i}X_{j} + β_{ij}X_{i}X_{j}^{−1} + γ_{ij}X_{i}^{−1}X_{j} + θ_{ij}X_{i}^{−1}X_{j}^{−1}.

Comparing coefficients in X_{i}X_{j}, X_{i}X_{j}^{−1}, X_{i}^{−1}X_{j}, X_{i}^{−1}X_{j}^{−1} we complete the proof. □

Proposition 1.8. Let w ∈ Λ and elements u_{1}, …, u_{n} ∈ Λ satisfy (5). Then elements u_{i} − (ad_{g} w)X_{i}, 1 ≤ i ≤ n, also satisfy (5).

Proof. Apply the fact that ad_{g} w is a g-derivation. □

Theorem 1.9. Let r = n and ϵ = −1 in (5). Then θ_{ij} = 0 and

α_{ij} = ξ_{j}^{−1}γ_{ji} − ξ_{i}^{−1}γ_{ij}, ξ_{i}^{−1}γ_{ij} = ξ_{t}^{−1}γ_{tj}

for all different indices i, j, t. Moreover there exists an element w ∈ Λ such that

u_{i} = (ad_{g} w)X_{i} + ξ_{j}^{−1}γ_{ji}X_{i}

for every i = 1, …, n and j ≠ i.

Proof. Fix an index i. Let u_{i} = X_{i}^{−m}u′_{i} + u′′_{i}, where m ≥ 0 and X_{i} does not occur in u′_{i}. Replacing each u_{t} by u_{t} − τ [ad_{g}(X_{i}^{−d+1}u′_{i})] X_{t} for some τ ∈ k and d = m, m − 1, …, 1 we can finally assume that X_{i} occurs in u_{i} with non-negative exponents. If X_{i} occurs in u_{i} with exponent d ≥ 2, then we replace each u_{j} by u_{j} − (ad_{g} w)X_{j} for some w. Thus we can assume that the exponent of each occurrences of X_{i} in u_{i} is equal either to 0 or to 1, i.e.

u_{i} = u_{i0} + u_{i1}X_{i},  (6)

where X_{i} does not occur in u_{i0}, u_{i1} ∈ Λ.

Let j ≠ i be another index. Then

u_{j} = ∑_{s∈ℤ}u_{js}X_{i}^{s}, u_{js} ∈ Λ,

where X_{i} does not occur in u_{js}. It follows from (5) that for any s ∈ ℤ

− q_{ij}u_{j,s−1} + ξ_{i}X_{i}^{−1}u_{j,s+1}X_{i} =

= \begin{cases}0, & if s ≠ −1, 0, 1; \\ γ_{ij}q_{ji}X_{j} + θ_{ij}q_{ij}X_{j}^{−1}, & if s = −1; \\ −u_{i0}X_{j} + q_{ij}ξ_{j}X_{j}^{−1}u_{i0}, & if s = 0; \\ −u_{i1}q_{ij}X_{j} + q_{ij}ξ_{j}X_{j}^{−1}u_{i1}+ \\ α_{ij}q_{ij}X_{j} + β_{ij}q_{ji}X_{j}^{−1}, & if s = 1. \end{cases} (7)

Let u_{jm} be the nonzero constituent in u_{j} with minimal index m. Take s = m − 1 in (7). If either s < −1 or s > 1, then (7) implies u_{jm} = 0, a contradiction. Thus −1 ≤ m − 1 ≤ 1 that is 0 ≤ m ≤ 2.

Similarly let u_{jd} be the nonzero constituent in u_{j} with maximal index d. Set s = d + 1. If either s > 1 or s < −1 then (7) implies u_{jd} = 0, a contradiction. Hence −1 ≤ d + 1 ≤ 1 and therefore −2 ≤ d ≤ 0. Since m ≤ d we conclude that m = d = 0 and u_{j} = u_{j0} does depend on X_{i} for any index j ≠ i. Comparing (6) and (7) for s = −1, s = 0 and s = 1 we obtain, respectively

ξ_{i}X_{i}^{−1}u_{j}X_{i} = γ_{ij}q_{ji}X_{j} + θ_{ij}q_{ij}X_{j}^{−1};  (8)
0 = −u_{i0}X_{j} + q_{ij}ξ_{j}X_{j}^{−1}u_{i0};  (9)
−q_{ij}u_{j} = −u_{i1}q_{ij}X_{j} + q_{ij}ξ_{j}X_{j}^{−1}u_{i1} + α_{ij}q_{ij}X_{j} + β_{ij}q_{ji}X_{j}^{−1}. (10)

By (9)

u_{i0}X_{j} = ξ_{j}q_{ij}X_{j}^{−1}u_{i0}.

Comparing the degrees in X_{j} we deduce u_{i0} = 0 and therefore u_{i} = u_{i1}X_{i}.

The equation (8) is equivalent to

u_{j} = ξ_{i}^{−1} (γ_{ij}X_{j} + θ_{ij}X_{j}^{−1}) .  (11)

The left hand side in (11) does depend on i. Hence for any index t different from j we have

ξ_{i}^{−1}γ_{ij} = ξ_{t}^{−1}γ_{tj}, ξ_{i}^{−1}θ_{ij} = ξ_{t}^{−1}θ_{tj}.  (12)

Moreover in (10) we obtain

− q_{ij}ξ_{i}^{−1} (γ_{ij}X_{j} + θ_{ij}X_{j}^{−1}) =

− u_{i1}q_{ij}X_{j} + q_{ij}ξ_{j}X_{j}^{−1}u_{i1} + α_{ij}q_{ij}X_{j} + β_{ij}q_{ji}X_{j}^{−1}.

Multiplying the last equation by −q_{ji} we can be rewrite the last equality in the form

ξ_{i}^{−1} (γ_{ij}X_{j} + θ_{ij}X_{j}^{−1}) = u_{i1}X_{j} − ξ_{j}X_{j}^{−1}u_{i1} − α_{ij}X_{j} − β_{ij}q_{ji}^{2}X_{j}^{−1}.

Hence

u_{i1}X_{j} − ξ_{j}X_{j}^{−1}u_{i1} =

ξ_{i}^{−1} (γ_{ij}X_{j} + θ_{ij}X_{j}^{−1}) + α_{ij}X_{j} + β_{ij}q_{ji}^{2}X_{j}^{−1}. (13)

The equality (13) means that the minimal and the maximal exponents of X_{j} in u_{i1} are equal to 0. Hence we can deduce that

u_{i1} = α_{ij} + ξ_{i}^{−1}γ_{ij} = −ξ_{j}^{−1} (β_{ij}q_{ji}^{2} + θ_{ij}) ,  (14)

Now take an index t different from j, i and subtitue u_{j}, u_{t} from (11) in (7) where the index i is replaced by t. Then we obtain

ξ_{i}^{−1} [(γ_{it}X_{t} + θ_{it}X_{t}^{−1}) X_{j} − q_{tj}ξ_{j}X_{j}^{−1} (γ_{it}X_{t} + θ_{it}X_{t}^{−1})] −

−ξ_{i}^{−1} [q_{tj} (γ_{ij}X_{j} + θ_{ij}X_{j}^{−1}) X_{t} − ξ_{t}X_{t}^{−1} (γ_{ij}X_{j} + θ_{ij}X_{j}^{−1})] =

α_{tj}X_{t}X_{j} + β_{tj}X_{t}X_{j}^{−1} + γ_{tj}X_{t}^{−1}X_{j} + θ_{tj}X_{t}^{−1}X_{j}^{−1}.

Comparing coefficient in X_{t}X_{j}, X_{t}^{−1}X_{j} we obtain, respectively,

ξ_{i}^{−1}γ_{it} − ξ_{i}^{−1}γ_{ij} = α_{tj}, ξ_{i}^{−1}θ_{it} + ξ_{i}^{−1}ξ_{t}γ_{ij} = γ_{tj}.

From (12) and the last equality we deduce

θ_{it} = ξ_{i}γ_{tj} − ξ_{i}ξ_{t}ξ_{i}^{−1}γ_{ij} = ξ_{i}γ_{tj} − ξ_{i}ξ_{t}ξ_{t}^{−1}γ_{tj} = 0

for all t ≠ i. Moreover

α_{tj} = ξ_{i}^{−1}γ_{it} − ξ_{i}^{−1}γ_{ij} = ξ_{j}^{−1}γ_{jt} − ξ_{t}^{−1}γ_{tj}

and the proof follows. □

Corollary 1.10. Let g be from Notation 1.3 with ϵ = −1. Then any g-derivation is inner.

Proof. Let d be a g-derivation. Put u_{i} = d(X_{i}), 1 ≤ i ≤ n. Then u_{1}, …, u_{n} satisfy (5) with α_{ij} = β_{ij} = γ_{ij} = θ_{ij} = 0 for all i, j. By Theorem 1.9 there exists an element w ∈ Λ such that u_{i} = (ad_{g} w)X_{i}, 1 ≤ i ≤ n. □

Theorem 1.11. Let ϵ = 1 in (5) and n ≥ 3. Then θ_{ij} = 0 for all i, j = 1, …, n. Moreover there exists an element w ∈ Λ and elements λ_{i}, μ_{i} ∈ k such that u_{i} = (ad_{g} w)X_{i} + λ_{i}X_{i} + μ_{i}X_{i}^{−1} where λ_{i}, μ_{i} ∈ k and

λ_{i}(1 − ξ_{j}) − λ_{j}(1 − ξ_{i}) = α_{ij}, μ_{i}(ξ_{j} − q_{ji}^{2}) = β_{ji}.

for i, j = 1, …, n. If i > r then μ_{i} = 0.

Proof. We need

Lemma 1.12. Let u be a monomial which is not a power of X_{i}. Then there exists an element θ ∈ k^{∗} such that (θu)X_{i} − ξ_{i}X_{i}(θu) = uX_{i}.

Proof. By the assumption u = X^{m}, where, m = (m_{1}, …, m_{n}) and m_{j} ≠ 0 for some j ≠ i. Suppose that w = θX^{m}, θ ∈ k. By Proposition 1.5

wX_{i} − ξ_{i}X_{i}w = θ (∏_{j>i} q_{ji}^{m_{j}} − ξ_{i} ∏_{i>j} q_{ij}^{m_{j}}) X^{m+e_{i}}  (15)

where

e_{i} = (0, …, 0, 1^{i}, 0, …, 0).  (16)

If

∏_{j>i} q_{ji}^{m_{j}} = ξ_{i} ∏_{i>j} q_{ij}^{m_{j}}.  (17)

then we can raise (17) to d-th power and obtain

∏_{j>i} q_{ji}^{dm_{j}} = ∏_{i>j} q_{ij}^{dm_{j}}

since ξ_{i}^{d} = 1. But this is impossible because m_{j} ≠ 0 for some j ≠ i. Hence the equation (15) determines θ completely. □

Corollary 1.13. Let u ∈ Λ such that supp f ∩ ℤe_{i} = ∅. There exists an element w ∈ Λ such that wX_{i} − ξ_{i}X_{i}w = uX_{i} and supp w = supp u.

Fix an index 1. Suppose X_{1} is not invertible in Λ. Then r = 0 and u_{1} = u_{10} + u_{11} + u_{12}X_{1}, where u_{10}, u_{11}, u_{12} ∈ Λ, each monomial in u_{10} is a power of X_{1}, X_{1} does not occur in each monomial in u_{11} and each monomial in u_{12} is not a power of X_{1}. By Corollary 1.13 there exists an element v ∈ Λ such that u_{11}X_{i} = (ad_{g} v)X_{1}. By Lemma 1.8 we can replace each u_{j} by u_{j} − (ad_{g} v)X_{j}. Hence without loss of generality we can assume that u_{1} = u_{10} + u_{11}. Then (5) has the form

(u_{1}X_{j} − q_{1j}ξ_{j}X_{j}u_{1}) − (q_{1j}u_{j}X_{1} − ξ_{1}X_{1}u_{j}) = α_{1j}X_{1}X_{j}

and therefore

q_{1j}u_{j}X_{1} − ξ_{1}X_{1}u_{j} = (u_{10}X_{j} − q_{1j}ξ_{j}X_{j}u_{10}) + (u_{11}X_{j} − q_{1j}ξ_{j}X_{j}u_{11}) − α_{1j}X_{1}X_{j}. (18)

It follows from (18) that u_{11}X_{j} − q_{1j}ξ_{j}X_{j}u_{11} = 0. Let λX^{m} be a monomial occurring in u_{11}, where m = (0, m_{2}, …, m_{n}) ∈ (ℕ ∪ 0)^{n}. Then

λX^{m}X_{j} = (λ ∏_{t≥2} q_{tj}^{m_{t}}) X_{j}X^{m} = λq_{1j}ξ_{j}X_{j}X^{m}.

If λ ≠ 0, then

∏_{t≥2} q_{tj}^{m_{t}} = q_{1j}ξ_{j}

which is impossible as in the proof of Lemma 1.12. Hence u_{11} = 0 and u = u_{10} = u(X_{1}). Let

u_{j} = ∑_{m} τ_{m}X^{m}, m ∈ (ℕ ∪ 0)^{n}, τ ∈ k.

Then

q_{1j}u_{j}X_{1} − ξ_{1}X_{1}u_{j} = ∑_{m} τ_{m} [q_{1j} ∏_{i>1} q_{1i}^{m_{i}} − ξ_{1}] X_{1}X^{m}.

Thus (18) can be written in the form

∑_{m} τ_{m} [q_{1j} ∏_{i>1} q_{1i}^{m_{i}} − ξ_{1}] X_{1}X^{m} =

u_{1}(X_{1})X_{j} − q_{1j}ξ_{j}u_{1}(q_{j1}X_{1})X_{j} − α_{1j}X_{1}X_{j}.

Moreover as in the proof of Lemma 1.12

q_{1j} ∏_{i>1} q_{1i}^{m_{i}} − ξ_{1} ≠ 0.

Hence if τ_{m} ≠ 0, then X_{1}X^{m} is equal to one of monomials X_{1}^{s}X_{j}, s ∈ ℕ ∪ 0, or to X^{m} = X_{1}^{s−1}X_{j}. Then

u_{j} = u′_{j}(X_{1})X_{1}X_{j} + λ_{j}X_{j}, u′_{j}(X_{1}) ∈ Λ, λ_{j} ∈ k.

By Corollary 1.13 there exists an element w = w(X_{1}) ∈ Λ such that (ad_{g} w)X_{j} = u′_{j}(X_{1})X_{1}X_{j}. Replacing each u_{t} by u_{t} − (ad_{g} w)X_{t} we can assume that u′_{j} = 0 and therefore u_{j} = λ_{j}X_{j}. Now for any i in (5) we have

u_{i}X_{j}−q_{ij}ξ_{j}X_{j}u_{i} = (q_{ij}λ_{j}−ξ_{i}X_{i}λ_{j}X_{j})+α_{ij}X_{i}X_{j} = (λ_{j}−ξ_{i}λ_{j}+α_{ij})X_{i}X_{j}.

Suppose that

u_{i} = ∑_{m} ζ_{m}X^{m}, m ∈ (ℕ ∪ 0)^{n}, ζ_{m} ∈ k.

As above we have

u_{i}X_{j} − q_{ij}ξ_{j}X_{j}u_{i} = ∑_{m} ζ_{m} (1 − q_{ij}ξ_{j} ∏_{t} q_{jt}^{m_{t}}) X^{m}X_{j} =

(λ_{j} − ξ_{i}λ_{j} + α_{ij})X_{i}X_{j}. (19)

If

1 − q_{ij}ξ_{j} ∏_{t} q_{jt}^{m_{t}} = 0

then m = (m_{1}, …, m_{n}) = e_{i}. Otherwise X^{m}X_{j} = X_{i}X_{j} by (19), which means that m = e_{i} and u_{i} = λ_{i}X_{i}.

Suppose now that r > 0. Then u_{1} = u_{10} + u_{11}X_{1}, where each monomial in u_{11} is not a power of X_{1} and every monomial in u_{10} is a power of X_{1}. By Corollary 1.13 there exists an element w ∈ Λ such that u_{11}X_{1} = (ad_{g} w)X_{1}. Hence replacing each u_{j} by u_{j} − (ad_{g} w)X_{j} we can assume that u_{11} = 0 and u_{1} = u_{1}(X_{1}).

For any index j > 1 we have in (5)

(u_{1}X_{j} − q_{1j}ξ_{j}X_{j}u_{1}) − (q_{1j}u_{j}X_{1} − ξ_{1}X_{1}u_{j}) =

α_{1j}X_{1}X_{j} + β_{1j}X_{1}X_{j}^{−1} + γ_{1j}X_{1}^{−1}X_{j} + θ_{1j}X_{1}^{−1}X_{j}^{−1}. (20)

Let λX^{m} be a monomial occurring in u_{j}, where m = (m_{1}, …, m_{n}) ∈ ℤ^{r} × (ℕ ∪ 0)^{n−r}. Then by (20) and Proposition 1.5

λq_{1j}X^{m}X_{1} − λξ_{1}X_{1}X^{m} = λ (q_{1j} ∏_{i>1} q_{1i}^{−m_{i}} − ξ_{1}) X_{1}X^{m}

occurs in

u_{1}X_{j} − q_{1j}ξ_{j}X_{j}u_{1} − α_{1j}X_{1}X_{j} − β_{1j}X_{1}X_{j}^{−1} − γ_{1j}X_{1}^{−1}X_{j} − θ_{1j}X_{1}^{−1}X_{j}^{−1}.

It follows that either

q_{1j} ∏_{i>1} q_{1i}^{−m_{i}} − ξ_{1} = 0

i.e. m = e_{j} or X_{1}X^{m} coincides with one of monomials X_{1}^{l}X_{j}, l ∈ ℤ, X_{1}^{±1}X_{j}^{−1}. Anyway in all cases X^{m} coincides with one of monomials X_{1}^{s}X_{j}, s ∈ ℤ, X_{1}^{−2}X_{j}^{−1}, X_{j}^{−1}. Thus

u_{j} = v_{j}(X_{1})X_{j} + λ_{j}X_{1}^{−2}X_{j}^{−1} + μ_{j}X_{j}^{−1} + τ_{j}X_{j},

where the constant term of v_{j} vanishes and λ_{j}, μ_{j}, τ_{j} ∈ k. By Corollary 1.13 there exists an element w ∈ Λ such that v_{j}X_{j} = (ad_{g} w)X_{j}. Hence replacing each u_{t} by u_{t} − (ad_{g} w)X_{t} we can assume that v_{j} = 0 and

u_{j} = τ_{j}X_{1}^{−2}X_{j}^{−1} + μ_{j}X_{j}^{−1} + λ_{j}X_{j},  (21)

Let

u_{1} = ∑_{s∈ℤ}ν_{s}X_{1}^{s}, ν_{s} ∈ k.

Then

u_{1}X_{j} − q_{1j}ξ_{i}X_{j}u_{1} = ∑_{s∈ℤ}ν_{s}X_{1}^{s}X_{j} − q_{1j}ξ_{j}X_{j} ∑_{s∈ℤ}ν_{s}X_{1}^{s} =

∑_{s∈ℤ}ν_{s}(1 − ξ_{j}q_{j1}^{s−1})X_{1}^{2}X_{j}.

Therefore in (20) we have

[∑_{s∈ℤ}ν_{s}(1 − ξ_{j}q_{j1}^{s−1})X_{1}^{s}X_{j}] −

[q_{1j} (τ_{j}X_{1}^{−2}X_{j}^{−1} + μ_{j}X_{j}^{−1} + λ_{j}X_{j}) X_{1} −
ξ_{1}X_{1} (τ_{j}X_{1}^{−2}X_{j}^{−1} + μ_{j}X_{j}^{−1} + λ_{j}X_{j})] =

α_{1j}X_{1}X_{j} + β_{1j}X_{1}X_{j}^{−1} + γ_{1j}X_{1}^{−1}X_{j} + θ_{1j}X_{1}^{−1}X_{j}^{−1}. (22)

It follows from (22) that if s ≠ ±1, then ν_{s}(1 − ξ_{j}q_{j1}^{s−1}) = 0. If 1 − ξ_{j}q_{j1}^{s−1} = 0 then 1 = ξ_{j}q_{j1}^{s−1}. Let ξ_{j}^{d} = 1. Then 1 = q_{j1}^{d(s−1)} which is impossible since Λ is a general algebra. Hence ν_{s} = 0 and

u_{1} = λ_{1}X_{1} + μ_{1}X_{1}^{−1}.  (23)

Take an index i ≠ j. By (23) and (21) we get in (5)

[(τ_{i}X_{1}^{−2}X_{i}^{−1} + μ_{i}X_{i}^{−1} + λ_{i}X_{i}) X_{j}−
q_{ij}ξ_{j}X_{j} (τ_{i}X_{1}^{−2}X_{i}^{−1} + μ_{i}X_{i}^{−1} + λ_{i}X_{i})] −
[q_{ij} (τ_{j}X_{1}^{−2}X_{j}^{−1} + μ_{j}X_{j}^{−1} + λ_{j}X_{j}) X_{i} −
ξ_{i}X_{i} (τ_{j}X_{1}^{−2}X_{j}^{−1} + μ_{j}X_{j}^{−1} + λ_{j}X_{j})] =
α_{ij}X_{i}X_{j} + β_{ij}X_{i}X_{j}^{−1} + γ_{ij}X_{i}^{−1}X_{j} + θ_{ij}X_{i}^{−1}X_{j}^{−1}  (24)

Comparing coefficients in X_{1}^{−2}X_{i}^{−1}X_{j} we obtain

τ_{i} − q_{ij}ξ_{i}τ_{i}q_{j1}^{−2}q_{ji}^{−1} = τ_{i}(1 − q_{ij}^{2}q_{j1}^{−2}ξ_{i}) = 0.

If 1 = q_{ij}^{2}q_{j1}^{−2}ξ_{i} and ξ_{i}^{d} = 1, then 1 = q_{ij}^{2d}q_{j1}^{−2d}. But the indices q, i, j are distinct. So by the assumption on Λ we get a contradiction. Thus τ_{i} = 0. Comparing in (24) coefficients in X_{i}X_{j}, X_{i}X_{j}^{−1} X_{i}^{−1}X_{j}^{−1} we prove the final statement. □

Corollary 1.14. Let g be a non-identical automorphism from Notation 1.3 with ϵ = 1 and d a g-derivation. Then there exist elements w ∈ Λ and ω ∈ k such that d = ad_{g} w + ω∂, where ∂ is from Proposition 1.4.

Proof. Let u_{i} = d(X_{i}), 1 ≤ i ≤ n. Then (5) is satisfied with α_{ij} = β_{ij} = γ_{ij} = θ_{ij} = 0. By Theorem 1.11 each u_{i} has the form u_{i} = (ad_{g} w)X_{i} + λ_{i}X_{i} where λ_{i}(1 − ξ_{j}) = λ_{j}(1 − ξ_{i}) for all i, j. Since g is non-identical ξ_{i} ≠ 1 for some i. It follows that λ_{j} = λ(1 − ξ_{j}) for all j and for some λ ∈ k. □

We shall apply the previous results to the study of the Lie algebra of derivations of Λ. There exist standard derivations ∂_{1}, …, ∂_{n} in Der Λ such that ∂_{j}(X_{i}) = δ_{ij}X_{i}. Observe that

[∂_{i}, ∂_{j}](X_{t}) = ∂_{i}∂_{j}(X_{t}) − ∂_{j}∂_{i}(X_{t}) = ∂(δ_{jt}X_{t}) − ∂_{j}(δ_{it}X_{t}) = (δ_{it}δ_{jt} − δ_{jt}δ_{it})X_{t} = 0.

Thus [∂_{i}, ∂_{j}] = 0. Moreover if char k = p > 0 then ∂_{i}^{p} = ∂_{i}. Hence the span of ∂_{1}, …, ∂_{n} is an Abelian Lie algebra L of dimension n.

Theorem 1.15. There is a direct decomposition of vector spaces

Der Λ = Derint Λ ⊕ L.

Proof. Let ∂ be a derivation of Λ. Put u_{i} = ∂(X_{i}), i = 1, …, n. Then u_{1}, …, u_{n} ∈ Λ satisfy (5) where g is the identical automorphism and therefore ϵ = ξ_{1} = ⋯ = ξ_{n} = 1, and where α_{ij} = β_{ij} = γ_{ij} = θ_{ij} = 0 for all i, j = 1, …, n. By Theorem 1.11 we can conclude that u_{i} = λ_{i}X_{i}+(ad w)(X_{i}), where w ∈ Λ. Hence ∂ = λ_{1}∂_{1}+⋯+λ_{n}∂_{n}+ad w. □

Observe that for each element u ∈ Λ

[∂_{i}, ad u] = ad(∂_{i}(u)).  (25)

It follows that Derint Λ is an ideal in the Lie algebra Der Λ. If r = n then the algebras Λ is simple [MP] with the center k. Hence by [H] the Lie algebra Derint is a simple Lie algebra. If we consider a special Jordan algebra Λ^{+} with multiplication a ∘ b = ab + ba then the Jordan algebra Λ^{+} is again simple [H]

2. Actions of pointed Hopf algebras

We shall apply the results from the previous section to the study of actions of pointed Hopf algebras H on Λ with n ≥ 3.

Definition 2.1. A Hopf algebra H over a field k is pointed if all simple co-subalgebras in H have dimension 1. By Δ, S, ε we shall denote comultiplication, an antipode and a counity in H, respectively.

Let G = G(H) be the group of group-like elements in the pointed Hopf algebra H. Elements of G act as automorphisms of Λ.

Notation 2.2. Denote by G^{+} the set of all elements γ ∈ G such that γ(X_{i}) = ξ_{i}X_{i}, 1 ≤ i ≤ n, where ξ_{i} ∈ k^{∗}.

It is clear that G^{+} is a normal subgroup in G of the index |G : G^{+}| ≤ 2. If r < n then G = G^{+} by Theorem 0.1. If G ≠ G^{+} and γ ∈ G \ G^{+}, then G = G^{+} ∪ γG^{+}.

The next theorem is due to E.J. Taft and R.L. Wilson, [M, Theorem 5.4.1]

Theorem 2.3. Let H be a pointed Hopf algebra. Then there is a Hopf algebra filtration H_{0} ⊆ H_{1} ⊆ … on H such that

1) H_{−1} = 0;

2) H_{0} = kG, where G is the set of all group-like elements in H;

3) if x ∈ H_{m}, m ≥ 1, then x = ∑_{h,f∈G} x_{h,f}, and

Δ(x_{h,f}) = x_{h,f} ⊗ f + h ⊗ x_{h,f} + w ∈ H ⊗ H,  (26)

where w ∈ H_{m−1} ⊗ H_{m−1}.

Notation 2.4. By P_{h,f}^{(m)} we shall denote the set of all elements x_{h,f} ∈ H_{m} satisfying (26) such that ε(u) = 0.

It is not hard to see that P_{h,f}^{(m)} is a subspace in H_{m}.

Proposition 2.5. Let x ∈ P_{g,f}^{(m)}. Then xf^{−1} ∈ P_{gf^{−1},1}^{(m)}.

Proof. In fact

Δ(xf^{−1}) = Δ(x)Δf^{−1} = Δ(x)(f^{−1} ⊗ f^{−1}) =

(x ⊗ f + h ⊗ x + w)(f^{−1} ⊗ f^{−1}) = xf^{−1} ⊗ 1 + gf^{−1} ⊗ xf^{−1} + wf^{−1}.

□

Now we fix a pointed Hopf algebra H which acts on Λ. Let u ∈ P_{g,1}^{(m)} where g ∈ G is from Notation 1.3. Then

Δ(u) = u ⊗ 1 + g ⊗ u + ∑_{t=1}^{s} h_{t} ⊗ f_{t}  (27)

where h_{t}, f_{t} ∈ H_{m−1}. We shall discuss the following property ♮(m) if h ∈ H_{m} then

h(X_{i}) = χ_{i}(h)X_{i} + ω_{i}(h)X_{i}^{−1}
where χ_{i}(h), ω_{i}(h) ∈ k and ω_{i}(h) = 0 if G = G^{+}.

Proposition 2.6. If the property ♮(m − 1) is satisfied and u ∈ P_{g,1}^{(m)}, where m ≥ 1. Then

u(X_{i}) = (ad_{g} w)X_{i} + λ_{i}X_{i} + μ_{i}^{−1}X_{i}^{−1}

for some elements w ∈ Λ and λ_{i}, μ_{i} ∈ k. Moreover μ_{i} = 0 if either G = G^{+}, or ϵ = −1 in Notation 1.3.

Proof. The equality (27) implies

u(X_{i}X_{j}) = u(X_{i})X_{j} + ξ_{i}X_{i}^{ϵ}u(X_{j}) +
∑_{t} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{j}(f_{t})X_{j} + ω_{j}(f_{t})X_{j}^{−1}) ,

for all 1 ≤ i, j ≤ n. Note that u(X_{i}X_{j}) = q_{ij}u(X_{i}X_{j}) for all i, j and therefore (5) is satisfied where

α_{ij} = ∑_{t} (q_{ij}χ_{j}(h_{t})χ_{i}(f_{t}) − χ_{j}(f_{t})χ_{i}(h_{t}))
β_{ij} = ∑_{t} (q_{ij}ω_{j}(h_{t})χ_{i}(f_{t}) − ω_{j}(f_{t})χ_{i}(h_{t})) ;  (28)
γ_{ij} = ∑_{t} (q_{ij}χ_{j}(h_{t})ω_{i}(f_{t}) − χ_{j}(f_{t})ω_{i}(h_{t})) ;
0 = ∑_{t} (q_{ij}ω_{j}(h_{t})ω_{i}(f_{t}) − ω_{j}(f_{t})ω_{i}(h_{t})) .

If either G = G^{+} or ϵ = −1 then ω_{∗}(h_{t}) = ω_{∗}(f_{t}) = 0 and therefore β_{ij} = γ_{ij} = 0 for all i, j by Proposition 1.7 and (28). Hence μ_{i} = 0 by Theorem 1.11. Applying Theorem 1.9 and Theorem 1.11 we complete the proof. □

Proposition 2.7. Let the property ♮(m) be satisfied and u ∈ H_{m}, m ≥ 1. If i ≤ r, then

u(X_{i}^{−1}) = χ_{−i}X_{i}^{−1} + ω_{−i}X_{i}.

Moreover ω_{−i} = 0 if G = G^{+}.

Proof. We shall proceed by induction on m. The case m = 1 follows from Theorem 0.1. Suppose that the affirmation is proved for m − 1 and u ∈ P_{g,1}^{(m)}. We have 0 = ε(u) = u(1) = u(X_{i}X_{i}^{−1}). Hence by (27) and by induction applied to h_{t}, f_{t} we obtain

0 = u(X_{i})X_{i}^{−1} + g(X_{i})u(X_{i}^{−1}) +
∑_{t} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) =
(χ_{i}(u)X_{i} + ω_{i}(u)X_{i}^{−1}) X_{i}^{−1} + g(X_{i})u(X_{i}^{−1}) +
∑_{t} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) .

Therefore

u(X_{i}^{−1}) = −g(X_{i})^{−1} (χ_{i}(u)X_{i} + ω_{i}(u)X_{i}^{−1}) X_{i}^{−1} −
g(X_{i})^{−1} ∑_{t} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) .

The algebra Λ is a general quantum polynomial algebra algebra with respect to variables X_{1}^{−1}, …, X_{r}^{−1}, X_{r+1}, …, X_{n}. Hence by Theorem 1.9 and Theorem 1.11

u(X_{i}^{−1}) = (ad_{g} v)X_{i}^{−1} + χ_{−i}X_{i}^{−1} + ω_{−i}X_{i}.

Here ω_{−i} = 0 is either G = G^{+} or ϵ = −1. Therefore

(ad_{g}(v)) (X_{i}^{−1}) = −χ_{−i}X_{i}^{−1} − ω_{−i}X_{i} −
ξ_{i}^{−1}X_{i}^{−ϵ} (χ_{i}(u)X_{i} + χ′_{i}(u)X_{i}^{−1}) X_{i}^{−1} −
∑_{t} ξ_{i}^{−1}X_{i}^{−1} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) .

By Proposition 2.10 and Corollary 2.11 we conclude that v = 0. □

Proposition 2.8. Let the property ♮(m − 1) be satisfied and u ∈ P_{g,1}^{(m)}, m ≥ 1. If i ≤ r, then

u(X_{i}^{−1}) = (ad_{g} w)X_{i}^{−1} + χ_{−i}X_{i}^{−1} + ω_{−i}X_{i}.

Here ω_{−i} = 0 if either G = G^{+} or ϵ = −1 Notation 1.3.

Proof. We shall apply the argument from Proposition 2.7. Since 0 = ε(u) = u(1) = u(X_{i}X_{i}^{−1}) Proposition 2.7 and (27) imply

0 = u(X_{i})X_{i}^{−1} + g(X_{i})u(X_{i}^{−1}) +
∑_{t} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) =
((ad_{g} w)X_{i} + λ_{i}X_{i} + μ_{i}X_{i}^{−1}) X_{i}^{−1} + g(X_{i})u(X_{i}^{−1}) +
∑_{t} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) .

It follows that

u(X_{i}^{−1}) = −g(X_{i})^{−1} ((ad_{g} w)X_{i} + λ_{i}X_{i} + μ_{i}X_{i}^{−1}) X_{i}^{−1} −

g(X_{i})^{−1} ∑_{t} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) =

−g(X_{i})^{−1} [(ad_{g} w)X_{i}] X_{i}^{−1} − ξ_{i}^{−1}λ_{i}X_{i}^{−ϵ} − ξ_{i}^{−1}μ_{i}X^{−ϵ−2} −

∑_{t} ξ_{i}^{−1}X_{i}^{−ϵ} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) .

Observe that

− g(X_{i})^{−1} [(ad_{g} w)X_{i}] X_{i}^{−1} = −g(X_{i})^{−1} [wX_{i} − g(X_{i})w] X_{i}^{−1} =

wX_{i}^{−1} − g(X_{i}^{−1})w = (ad_{g} w)X_{i}^{−1}.

Thus

u(X_{i}^{−1}) = (ad_{g} w)X_{i}^{−1} − ξ_{i}^{−1}λ_{i}X_{i}^{−ϵ} − ξ_{i}^{−1}μ_{i}X^{−ϵ−2} −
∑_{t} ξ_{i}^{−1}X_{i}^{−ϵ} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) .

The algebra Λ is a general quantum polynomial algebra algebra with respect to variables X_{1}^{−1}, …, X_{r}^{−1}, X_{r+1}, …, X_{n}. Hence by Theorem 1.9 and Theorem 1.11 we obtain

u(X_{i}^{−1}) = (ad_{g} v)X_{i}^{−1} + χ_{−i}X_{i}^{−1} + ω_{−i}X_{i},

where ω_{−i} = 0 is either G = G^{+} or ϵ = −1. Therefore

(ad_{g}(w − v)) (X_{i}^{−1}) = η_{i}X_{i}^{−1} + η′_{i}X_{i} +
∑_{t} ξ_{i}^{−1}X_{i}^{−ϵ} (χ_{i}(h_{t})X_{i} + ω_{i}(h_{t})X_{i}^{−1}) (χ_{−i}(f_{t})X_{i}^{−1} + ω_{−i}(f_{t})X_{i}) .

By Proposition 2.10 and Corollary 2.11 we conclude that w = v. □

Proposition 2.9. Let Δ_{m} : H → H^{⊗m} be an iterated comultiplication, Δ_{2} = Δ. If u ∈ P_{g,1}^{(m)} then

Δ_{m}(u) = ∑_{j=0}^{m−1} \underbrace{g ⊗ ⋯ ⊗ g}_{j} ⊗u ⊗ 1 ⊗ … ⊗ 1 + ∑ h_{i_{1}} ⊗ ⋯ ⊗ h_{i_{m}}

where h_{i_{1}}, …, h_{i_{m}} ∈ H_{m−1}.

Proof. Apply coassociativity of Δ and induction on m. □

In order to prove the next theorem we need some auxiliary statements.

Proposition 2.10. Let v, m ∈ ℤ^{r} × (ℕ ∪ 0)^{n−r} be multi-indices. Then

(ad_{g} X^{m})X^{v} =

\begin{cases}[∏_{j>i} q_{ji}^{v_{j}m_{i}} − ξ_{1}^{v_{1}} ⋯ ξ_{n}^{v_{n}} ∏_{j>i} q_{ji}^{v_{i}m_{j}}] X^{m+v}, & if ϵ = 1; \\ (∏_{j>i} q_{ji}^{v_{j}m_{i}}) X^{m+v} − ξ_{1}^{v_{1}} ⋯ ξ_{n}^{v_{n}} ∏_{j>i} q_{ji}^{v_{j}(v_{i}−m_{i})}X^{m−v}, & if ϵ = −1. \end{cases}

Proof. As in the proof of Lemma 1.12 in the case ϵ = 1 we have

(ad_{g}X^{m})X^{v} = X^{m}X^{v} − g(X^{v})X^{m} =

(∏_{j>i} q_{ji}^{v_{j}m_{i}}) X^{m+v} − ξ_{1}^{v_{1}} ⋯ ξ_{n}^{v_{n}} (∏_{j>i} q_{ji}^{v_{i}m_{j}}) X^{m+v} =

[∏_{j>i} q_{ji}^{v_{j}m_{i}} − ξ_{1}^{v_{1}} ⋯ ξ_{n}^{v_{n}} ∏_{j>i} q_{ji}^{v_{i}m_{j}}] X^{m+v}.

If ϵ = −1, then

(ad_{g} X^{m})X^{v} = X^{m}X^{v} − ξ_{1}^{v_{1}} ⋯ ξ_{n}^{v_{n}}X_{n}^{−v_{n}} ⋯ X_{1}^{−v_{1}}X^{m} =
(∏_{j>i} q_{ji}^{v_{j}m_{i}}) X^{m+v} − ξ_{1}^{v_{1}} ⋯ ξ_{n}^{v_{n}} ∏_{j>i} q_{ji}^{v_{j}(v_{i}−m_{i})}.
□

Corollary 2.11. Let p, t ∈ ℤ^{r} × (ℕ ∪ 0)^{n−r} and

rank (\begin{matrix}p \\ t \end{matrix}) = 2.  (29)

Then (ad_{g} X^{p})X^{t} = θX^{p+t}, where θ ∈ k^{∗}.

Proof. By Proposition 2.10 we can assume that ϵ = 1. If θ = 0, then

1 = ξ_{1}^{t_{1}} ⋯ ξ_{n}^{t_{n}} ∏_{j>i} q_{ji}^{t_{j}p_{i}−t_{i}p_{j}},

where ξ_{1}, …, ξ_{n} are roots of 1. As in the proof of Lemma 1.12 we can conclude that t_{j}p_{i} − t_{i}p_{j} = 0 for all j > i and therefore for all j, i. It leads to a contradiction with the assumption (29). □

Proposition 2.12. Let σX^{p}, p > 0, σ ∈ k, be the leading monomial in w. Suppose that t be a multi-index such that the rank

rank (\begin{matrix}p \\ t \end{matrix}) = 2.

Then the leading term of u(X^{t}) = ζX^{p+t}, ζ ∈ k^{∗}.

Proof. Apply Proposition 2.9 and Corollary 2.11. □

Theorem 2.13. Let and ♮(m − 1) be satisfied. Suppose that for every u ∈ P_{g,1}^{(m)} there exists a nonzero polynomial f(T) ∈ k[T] such that f(u) = 0. Then ♮(m) holds.

Proof. Suppose p = (p_{1}, …, p_{n}) ∈ supp w and p_{i} > 0 for some i. Rearranging variables X_{1}, …, X_{n} we can assume that i = 1. Then w > 0. Suppose that θX^{p} is the leading term of w. Let

f(T) = c_{0} + c_{1}T + ⋯ + c_{s}T^{s} ∈ k[T], c_{i} ∈ k, c_{s} ≠ 0.

By Proposition 2.12 there exists a monomial X^{t} such that the leading term of f(u)X^{t} is equal to θ_{s}X^{sp+t}, a contradiction.

Hence p_{i} ≤ 0 for all i. If r = 0 then the proof is over. Suppose that r > 0 and p_{i} < 0 for some i. As we have mentioned above the algebra Λ is a general quantum algebra with respect to X_{1}^{−1}, …, X_{r}^{−1}, X_{r+1}, …, X_{n}. By Proposition 2.8 and Proposition 2.12 we can conclude that p_{i} ≥ 0 for i ≤ r. Hence p = 0 and w ∈ k. In this case (ad_{g} w)X_{i} = 0 and u(X_{i}) = λ_{i}X_{i} + μ_{i}X_{i}^{−1}. □

Corollary 2.14. Let H be a finite dimensional pointed Hopf algebra. Then the property ♮(m) is satisfied for any m.

Starting from now we shall assume that H is a finite dimensional pointed Hopf algebra acting on Λ. Consider the coaction ρ : Λ → H^{∗} ⊗ Λ related to the action of H on Λ [M, Lemma 1.6.4], where H^{∗} = Hom(H, k) – a convolution algebra with convolutive multiplication ∗. The unit element of H^{∗} is the counity ε. By Corollary 2.14

ρ(X_{i}) = χ_{i} ⊗ X_{i} + ω_{i} ⊗ X_{i}^{−1}, χ_{i}, ω_{i} ∈ H^{∗},

where h(X_{i}) = χ_{i}(h)X_{i} + ω_{i}(h)X_{i}^{−1} for any h ∈ H. Note that ω_{i} = 0 if G = G^{+}. Recall that by Proposition 2.8 if i ≤ r then

h(X_{i}^{−1}) = χ_{−i}(h)X_{i}^{−1} + ω_{−i}(h)X^{,}

that is

ρ(X_{i}^{−1}) = χ_{−i} ⊗ X_{i}^{−1} + ω_{−i} ⊗ X_{i}, χ_{−i}, ω_{−i} ∈ H^{∗}.

Proposition 2.15. The following equalities are satisfied in H^{∗} if i ≤ r:

χ_{i} ∗ ω_{−i} = ω_{−i} ∗ χ_{i} = χ_{−i} ∗ ω_{i} = ω_{i} ∗ χ_{−i} = 0;
χ_{i} ∗ χ_{−i} + ω_{i} ∗ ω_{−i} = χ_{−i} ∗ χ_{i} + ω_{−i} ∗ ω_{i} = ε.  (30)

Moreover if 1 ≤ i ≠ j ≤ n and π, σ = ±1 then

χ_{πi} ∗ ω_{σj} = ω_{σj} ∗ χ_{πi} = 0;  (31)
χ_{πi} ∗ χ_{σj} = χ_{σj} ∗ χ_{πi}, ω_{σj} ∗ ω_{πi} = ω_{πi} ∗ ω_{σj}.  (32)

Proof. The coaction ρ is an algebra homomorphism. Hence by Corollary 2.14

1 ⊗ 1 = ρ(1) = ρ(X_{i})ρ(X_{i}^{−1}) =
(χ_{i} ⊗ X_{i} + ω_{i} ⊗ X_{i}^{−1})(χ_{−i} ⊗ X_{i}^{−1} + ω_{−i} ⊗ X_{i}) =
(χ_{i} ∗ ω_{−i}) ⊗ X_{i}^{2} + (χ_{i} ∗ χ_{−i} + ω_{i} ∗ ω_{−i}) ⊗ 1 + (ω_{i} ∗ χ_{−i}) ⊗ X_{i}^{−2}.(33)

Comparing in (33) coefficients in X_{i}^{2}, 1, X_{i}^{−2} we can partially prove (30). Similarly considering 1⊗1 = ρ(X_{i}^{−1})ρ(X_{i}) we complete the proof of (30).

We have to prove (32) only in the case when ω_{j} ≠ 0 and therefore r = n. Then Λ is a general quantum polynomial algebra with respect to the variables

X_{1}, …, X_{i−1}, X_{i}^{π}X_{j}^{σ}, X_{i+1}, …, X_{n}.

Then

ρ(X_{i}^{π}X_{j}^{σ}) = λ ⊗ X_{i}^{π}X_{j}^{σ} + λ′ ⊗ X_{j}^{−σ}X_{i}^{−π} = ρ(X_{i}^{π})ρ(X_{j}^{σ}) =
(χ_{πi} ⊗ X_{i}^{π} + ω_{πi} ⊗ X_{i}^{−π})(χ_{σj} ⊗ X_{j}^{σ} + ω_{σj} ⊗ X_{j}^{−σ}) =
(χ_{πi} ∗ χ_{σj}) ⊗ X_{i}^{π}X_{j}^{σ} + (χ_{πi} ∗ ω_{σj}) ⊗ X_{i}^{π}X_{j}^{−σ} +
(ω_{πi} ∗ χ_{σj}) ⊗ X_{i}^{−π}X_{j}^{σ} + (ω_{πi} ∗ ω_{σj}) ⊗ X_{i}^{−π}X_{j}^{−σ} =
(ω_{πi} ∗ χ_{σj}) ⊗ X_{i}^{−π}X_{j}^{σ} + q_{ij}^{πσ}(ω_{πi} ∗ ω_{σj}) ⊗ X_{j}^{−σ}X_{i}^{−π}.

and (31) follows. Moreover

ρ(X_{i}^{π}X_{j}^{σ}) = (χ_{πi} ∗ χ_{σj}) ⊗ X_{i}^{π}X_{j}^{σ} + q_{ij}^{πσ}(ω_{πi} ∗ ω_{σj}) ⊗ X_{j}^{−σ}X_{i}^{−π}.

In the same time

ρ(X_{i}^{π}X_{j}^{σ}) = q_{ij}^{πσ}ρ(X_{j}^{σ}X_{i}^{π}) =
q_{ij}^{πσ}(χ_{σj} ∗ χ_{πi}) ⊗ X_{j}^{σ}X_{i}^{π} + q_{ij}^{πσ}(ω_{σj} ∗ ω_{πi}) ⊗ X_{j}^{−σ}X_{i}^{−π} =
(χ_{σj} ∗ χ_{πi}) ⊗ X_{i}^{π}X_{j}^{σ} + q_{ij}^{πσ}(ω_{πi} ∗ ω_{σj}) ⊗ X_{j}^{−σ}X_{i}^{−π}

and (32) follows. □

Corollary 2.16. If i ≤ r then χ_{−i} = S(χ_{i}).

Proof. Apply (30). □

Using Corollary 2.16 we can expand the action of H on Λ to the action of H on the localized algebra Λ_{X_{1}···X_{n}}, i.e. we can assume starting from now that r = n.

Corollary 2.17. For any i = 1, …, n and π = ±1 we have

χ_{πi} ∗ ω_{πi} = ω_{πi} ∗ χ_{πi} = 0.

Proof. We can assume that π = 1 and ω_{i} ≠ 0. If j ≠ i then by Proposition 2.15

(χ_{i} ∗ ω_{i}) ∗ χ_{j} = χ_{i} ∗ (ω_{i} ∗ χ_{j}) = 0;

(χ_{i} ∗ ω_{i}) ∗ ω_{j} = χ_{i} ∗ (ω_{i} ∗ ω_{j}) = χ_{i} ∗ (ω_{j} ∗ ω_{i}) = (χ_{i} ∗ ω_{j}) ∗ ω_{i} = 0.

18 VYACHESLAV A. ARTAMONOV

By (30) 0 = (χ_{i} ∗ ω_{i}) ∗ χ_{j} ∗ χ_{−j} + (χ_{i} ∗ ω_{i}) ∗ ω_{j} ∗ ω_{−j} = (χ_{i} ∗ ω_{i}) ∗ ε = χ_{i} ∗ ω_{i}.

□

The next theorem generalizes Proposition 2.15. Theorem 2.18. Let H be a finite dimensional pointed Hopf algebra acting on Λ with n ≥ 3. If h ∈ H then for any monomial X^{v} we have

h(X^{v}) = χ_{v}(h)X^{v} + ω_{v}(h)X^{−v}  (34)

where χ_{v}, ω_{v} ∈ H^{∗}. Moreover if v, t ∈ ℤ^{n} are nonzero multi-indices then

χ_{v} ∗ ω_{t} = ω_{v} ∗ χ_{t} = 0;  (35)
χ_{v} ∗ χ_{t} = χ_{t} ∗ χ_{v}, ω_{v} ∗ ω_{t} = ω_{t} ∗ ω_{v}.  (36)

and if t ≠ −v then

χ_{v} ∗ χ_{t} = χ_{v+t}, ω_{v} ∗ ω_{t} = ω_{v+t}.  (37)

In addition

χ_{v} ∗ χ_{−v} + ω_{v} ∗ ω_{−v} = ε.  (38)

If G = G^{+}, then ω_{v} = 0 for all v. Proof. Without loss of generality we can assume that

v = (v_{1}, …, v_{n}) ∈ ℤ^{n}, v_{1}, …, v_{n} ≥ 0.

Then

ρ(X^{v}) = ρ(X_{1})^{v_{1}} ⋯ ρ(X_{n})^{v_{n}} = ∏_{i=1}^{n}(χ_{i} ⊗ X_{i} + ω_{i} ⊗ X_{i}^{−1})^{v_{i}}.

Applying Corollary 2.17 and Proposition 2.15 we prove (34). Thus by Proposition 1.5 ρ(X^{v}X^{t}) = ρ(X^{v})ρ(X^{t}) =

(χ_{v} ⊗ X^{v} + ω_{v} ⊗ X^{−v}) (χ_{t} ⊗ X^{t} + ω_{t} ⊗ X^{−t}) =

(∏_{i>j} q_{ji}^{v_{j}t_{i}}) (χ_{v} ∗ χ_{t}) ⊗ X^{v+t} + (∏_{j>i} q_{ji}^{−v_{j}t_{i}}) (χ_{v} ∗ ω_{t}) ⊗ X^{v−t}+

(∏_{i>j} q_{ji}^{−v_{j}t_{i}}) (ω_{v} ∗ χ_{t}) ⊗ X^{−v+t} + (∏_{i>j} q_{ji}^{v_{j}t_{i}}) (ω_{v} ∗ ω_{t}) ⊗ X^{−v−t}. (39)

On the other hand

ρ(X^{v}X^{t}) = (∏_{i>j} q_{ji}^{v_{j}t_{i}}) ρ(X^{v+t}) =

(∏_{i>j} q_{ji}^{v_{j}t_{i}}) χ_{v+t} ⊗ X^{v+t} + (∏_{i>j} q_{ji}^{v_{j}t_{i}}) ω_{v+t} ⊗ X^{−v−t}. (40)

Suppose first that v ≠ ±t. Then the multi-indices

v + t, v − t, −v + t, −v − t

are pairwise different. Comparing coefficients in

X^{v+t}, X^{v−t}, X^{−v+t}, X^{−v−t}

in (39) and in (40) we obtain (35) and (37) if v ≠ ±t.

Now if v ≠ 0 then 2v ≠ ±v. Hence by (37)

χ_{v} ∗ χ_{v} = χ_{v} ∗ χ_{2v} ∗ χ_{−v} = χ_{3v} ∗ χ_{−v} = χ_{2v};
ω_{v} ∗ ω_{v} = ω_{v} ∗ ω_{2v} ∗ ω_{−v} = ω_{3v} ∗ ω_{−v} = ω_{2v};
χ_{v} ∗ ω_{v} = χ_{v} ∗ ω_{2v} ∗ ω_{−v} = 0 = ω_{v} ∗ χ_{v};
χ_{v} ∗ ω_{−v} = χ_{v} ∗ ω_{v} ∗ ω_{−2v} = 0 = ω_{v} ∗ χ_{−v};
χ_{v} ∗ χ_{−v} = χ_{2v} ∗ χ_{−v} ∗ χ_{−v} = χ_{−v} ∗ χ_{−v} ∗ χ_{2v} = χ_{−v} ∗ χ_{v}
ω_{v} ∗ ω_{−v} = ω_{2v} ∗ ω_{−v} ∗ ω_{−v} = ω_{−v} ∗ ω_{v}

and therefore (37) holds if t ≠ −v and (35), (36) hold for all nonzero v, t.

Similarly as in (39) we have

1 ⊗ 1 = ρ (X^{v}(X^{v})^{−1}) = (∏_{i>j} q_{ji}^{v_{j}v_{i}}) ρ(X^{v}X^{−v}) =

(∏_{i>j} q_{ji}^{v_{j}v_{i}}) (χ_{v} ⊗ X^{v} + ω_{v} ⊗ X^{−v}) (χ_{−v} ⊗ X^{−v} + ω_{−v} ⊗ X^{v}) =

(∏_{i>j} q_{ij}^{v_{i}v_{j}}) {(∏_{i>j} q_{ji}^{−v_{i}v_{j}}) [(χ_{v} ∗ χ_{−v}) + (ω_{v} ∗ ω_{−v})] ⊗ 1+
(∏_{i>j} q_{ij}^{v_{i}v_{j}}) [(χ_{v} ∗ ω_{−v}) ⊗ X^{2v} + (ω_{v} ∗ χ_{−v}) ⊗ X^{−2v}]} .

Now (38) follows. □

Corollary 2.19. If v, t are nonzero multi-indices then

(χ_{v} ∗ χ_{−v}) ∗ χ_{t} = χ_{t}, (ω_{v} ∗ ω_{−v}) ∗ ω_{t} = ω_{t}.

In particular χ_{v} ∗ χ_{−v} = χ_{t} ∗ χ_{−t} and ω_{v} ∗ ω_{−v} = ω_{t} ∗ ω_{−t}.

Proof. By (38) and (35) we have

(χ_{v} ∗ χ_{−v}) ∗ χ_{t} = (ε − ω_{v} ∗ ω_{−v}) ∗ χ_{t} = ε ∗ χ_{t} = χ_{t}.

Similarly we prove the second equality. The last two equalities follows.

□

Proposition 2.20. Let v be a multi-index. Then

Δ(χ_{v}) = χ_{v} ⊗ χ_{v} + ω_{−v} ⊗ ω_{v}; Δ(ω_{v}) = ω_{v} ⊗ χ_{v} + χ_{−v} ⊗ ω_{v}. (41)

Proof. For any h, u ∈ H we have in H^{∗}

Δ(χ_{v})(h ⊗ u) = χ_{v}(hu), Δ(ω_{v})(h ⊗ u) = ω_{v}(hu).

So

χ_{v}(hu)X^{v} + ω_{v}(hu)X^{−v} = (hu)X^{v} =

h(u(X^{v})) = h (χ_{v}(u)X^{v} + ω_{v}(u)X^{−v}) =

χ_{v}(u) (χ_{v}(h)X^{v} + ω_{v}(h)X^{−v}) + ω_{v}(u) (χ_{−v}(h)X^{−v} + ω_{−v}(h)X^{v}) =

[χ_{v}(h)χ_{v}(u) + ω_{−v}(h)ω_{v}(u)] X^{v} + [ω_{v}(h)χ_{v}(u) + χ_{−v}(h)ω_{v}(u)] X^{−v}.

Hence

χ_{v}(hu) = χ_{v}(h)χ_{v}(u) + ω_{−v}(h)ω_{v}(u),
ω_{v}(hu) = ω_{v}(h)χ_{v}(u) + χ_{−v}(h)ω_{v}(u)

and (41) follows. □

Proposition 2.21. Let v be a nonzero multi-index. Then

ε(χ_{v}) = 1, ε(ω_{v}) = 0, S(χ_{v}) = χ_{−v}, S(ω_{v}) = ω_{v}.

Proof. By Proposition 2.20, (35) and (36) we have ε(χ_{v}) = χ_{v}(1) = 1 and ε(ω_{v}) = ω_{v}(1) = 0. It follows that χ_{0} = ε, ω_{0} = 0 and therefore S(χ_{0}) = ε, S(ω_{0}) = 0. So it suffices to prove that if v ≠ 0 then S(χ_{v}) = χ_{−v}, S(ω_{v}) = ω_{v}. In fact by (38) and Proposition 2.20 we have

χ_{v} ∗S(χ_{v})+ω_{−v} ∗S(ω_{v}) = χ_{v} ∗χ_{−v} +ω_{−v} ∗ω_{v} = χ_{v} ∗χ_{−v} +ω_{v} ∗ω_{−v} = ε.

Similarly S(χ_{v}) ∗ χ_{−v} + S(ω_{−v}) ∗ ω_{v} = χ_{−v} ∗ χ_{v} + ω_{−v}ω_{v} = ε. Moreover by (35)

ω_{v} ∗ S(χ_{v}) + χ_{−v} ∗ S(ω_{v}) = ω_{v} ∗ χ_{−v} + χ_{−v} ∗ ω_{v} = 0,
S(ω_{v}) ∗ χ_{v} + S(χ_{−v}) ∗ ω_{v} = ω_{v} ∗ χ_{−v} + χ_{v} ∗ ω_{v} = 0,
□

Corollary 2.22. For any h ∈ H and a nonzero multi-index v we have χ_{v}(S(h)) = χ_{−v}(h), ω_{v}(S(h)) = ω_{v}(h).

Since the algebra H^{∗} has finite dimension there exists a minimal positive integer m such that

χ_{v_{m}} + γ_{m−1}χ_{v_{m−1}} + ⋯ + γ_{1}χ_{v_{1}} = 0  (42)

for some γ_{m−1}, …, γ_{1} ∈ k, where v_{1}, …, v_{m} are nonzero distinct multiindices. It is pretty obvious that γ_{m−1}, …, γ_{1} ≠ 0 and the elements χ_{v_{m−1}}, …, χ_{v_{1}} are linearly independent over the basic field k.

Proposition 2.23. We have m = 2 and γ_{1} = −1.

Proof. Applying comultiplication Δ to (42) we obtain by (41)

(χ_{v_{m}} ⊗ χ_{v_{m}} + ω_{−v_{m}} ⊗ ω_{v_{m}}) + ∑_{i=1}^{m−1} γ_{i} (χ_{v_{i}} ⊗ χ_{v_{i}} + ω_{−v_{i}} ⊗ ω_{v_{i}}) = 0.

Choose a multi-index v_{0} ≠ 0 such that v_{0} +v_{i} ≠ 0 for all i. Multiplying the last equation by χ_{v_{0}} ⊗ χ_{v_{0}} and applying (35) we obtain

χ_{v_{m}+v_{0}} ⊗ χ_{v_{m}+v_{0}} + ∑_{i=1}^{m−1} γ_{i} (χ_{v_{i}+v_{0}} ⊗ χ_{v_{i}+v_{0}}) = 0.  (43)

We claim that χ_{v_{m−1}+v_{0}}, …, χ_{v_{1}+v_{0}} are also linearly independent. In fact if μ_{m−1}χ_{v_{m−1}+v_{0}} +⋯+μ_{1}χ_{v_{1}+v_{0}} = 0, where μ_{m−1}, …, μ_{1} ∈ k, then multiplying this equality by χ_{−v_{0}} we obtain by (36) that

μ_{m−1}χ_{v_{m−1}} + ⋯ + μ_{1}χ_{v_{1}} = 0,

and therefore μ_{m−1} = … = μ_{1} = 0. Now

χ_{v_{m}+v_{0}} = χ_{v_{m}} ∗ χ_{v_{0}} = − ∑_{i=1}^{m−1} γ_{i}χ_{v_{i}} ∗ χ_{v_{0}} = − ∑_{i=1}^{m−1} γ_{i}χ_{v_{i}+v_{0}}.

Thus (42) implies

χ_{v_{m}+v_{0}} ⊗ χ_{v_{m}+v_{0}} = ∑_{i,j=1}^{m−1} γ_{i}γ_{j} (χ_{v_{i}+v_{0}} ⊗ χ_{v_{j}+v_{0}}) .

Hence (43) can be written in the form

∑_{i,j=1}^{m−1} γ_{i}γ_{j} (χ_{v_{i}+v_{0}} ⊗ χ_{v_{j}+v_{0}}) + ∑_{i=1}^{m−1} γ_{i} (χ_{v_{i}+v_{0}} ⊗ χ_{v_{i}+v_{0}}) = 0.  (44)

If i ≠ j then γ_{i}γ_{j} = 0. If i = j = 1 then γ_{1}^{2} + γ_{1} = 0 and therefore γ_{1} = −1. Since all γ_{i} ≠ 0 we can conclude that m = 2 □

Proposition 2.24. Let v, t be nonzero distinct multi-indices and χ_{v} = χ_{t} then ω_{t} = ζω_{v} for some ζ ∈ k. Moreover either ω_{t}, ω_{v}, ζ ≠ 0 or ω_{t} = ω_{v} = ζ = 0.

Proof. Applying comultiplication Δ to the given identity we get

χ_{v} ⊗ χ_{v} + ω_{−v} ⊗ ω_{v} = χ_{t} ⊗ χ_{t} + ω_{−t} ⊗ ω_{t}.

Then

ω_{−v} ⊗ ω_{v} = ω_{−t} ⊗ ω_{t}  (45)

that is ω_{t} = ζω_{v} where ζ ∈ k. If ω_{t} ≠ 0 then ζ ≠ 0 and ω_{v} ≠ 0. Suppose that ω_{t} = 0 and ω_{v} ≠ 0. Then ω_{−v} ⊗ ω_{v} = 0 by (45) and therefore ω_{−v} = 0. Multiplying by ω_{2v} we obtain ω_{v} = ω_{−v} ∗ ω_{2v} = 0. So in this cases we can take ζ = 0. □

Proposition 2.25. Let v_{1}, …, v_{m} be distinct nonzero multi-indices such that the elements

ω_{v_{1}}, …, ω_{v_{m−1}}  (46)

are linearly independent and

ω_{v_{m}} + γ_{m−1}ω_{v_{m−1}} + ⋯ + γ_{1}ω_{v_{1}} = 0  (47)

where γ_{1}, …, γ_{m−1} ∈ k^{∗}. Then m = 2.

Proof. Applying comultiplication Δ to (47) we obtain

[ω_{v_{m}} ⊗ χ_{v_{m}} + χ_{−v_{m}} ⊗ ω_{v_{m}}] + ∑_{i=1}^{m−1} λ_{i} [ω_{v_{i}} ⊗ χ_{v_{i}} + χ_{−v_{i}} ⊗ ω_{v_{i}}] = 0. (48)

Pick a nonzero multi-index v_{0} such that v_{i}+v_{0} ≠ 0 for all i = 1, …, m− 1. Multiplying (48) by ω_{v_{0}} ⊗ χ_{v_{0}} we obtain by (35) that

ω_{v_{m}+v_{0}} ⊗ χ_{v_{m}+v_{0}} + ∑_{i=1}^{m−1} λ_{i} [ω_{v_{i}+v_{0}} ⊗ χ_{v_{i}+v_{0}}] = 0.  (49)

As in the proof of Proposition 2.23 the elements

ω_{v_{1}+v_{0}}, …, ω_{v_{m−1}+v_{0}}  (50)

are linearly independent and

ω_{v_{m}+v_{0}} + γ_{m−1}ω_{v_{m−1}+v_{0}} + ⋯ + γ_{1}ω_{v_{1}+v_{0}} = 0.

Hence in (49) we have

∑_{i=1}^{m−1} λ_{i}ω_{v_{i}+v_{0}} ⊗ [−χ_{v_{m}+v_{0}} + χ_{v_{i}+v_{0}}] = 0

Since the elements (50) are linearly independent we conclude that χ_{v_{m}+v_{0}} = χ_{v_{i}+v_{0}} and therefore χ_{v_{m}} = χ_{v_{i}} for any i. By Proposition 2.24 we can conclude that ω_{v_{m}} = ζ_{i}ω_{v_{i}} where ζ_{i} ∈ k^{∗}. But the elements (46) are linearly independent, hence m = 2. □

Proposition 2.26. Let v be a nonzero multi-index. Then the following are equivalent:

1) there exists a nonzero multi-index t such that

χ_{t} ∗ χ_{v} = χ_{t};  (51)

2) (51) holds for any multi-index t;

3) χ_{v} = χ_{t} ∗ χ_{−t} for some t.

Proof. Apply Theorem 2.18 and Corollary 2.19. □

Notation 2.27. Denote by U the subset of ℤ^{n} consisting of zero and all non-zero multi-indices v for which Proposition 2.26 holds.

Proposition 2.28. U is a subgroup of a finite index in ℤ^{n} and χ_{v} = χ_{t} where t, v ≠ 0 if and only if v − t ∈ U. In this case ω_{t} = ζω_{v} for some ζ ∈ k. Moreover either ω_{t}, ω_{v}, ζ ≠ 0 or ω_{t} = ω_{v} = ζ = 0. Conversely if ω_{t} ≠ 0 for some t and ω_{s+v} = ξω_{s} for some ξ ∈ k, and some multi-index s. Then v ∈ U.

Proof. Apply Theorem 2.18, Proposition 2.26 and Proposition 2.24. Suppose now that ω_{s+v} = ξω_{s}. If t ≠ s + v we multiply the last equation by ω_{t−s−v}. If t = s + v then we multiply this equation by some ω_{r} such that t ≠ r +s+v and afterwards by ω_{t−r−s−v}. Anyway in all cases we obtain ω_{t} = ξω_{t−v}. Applying Δ to this equation we obtain

ω_{t} ⊗ χ_{t} + χ_{−t} ⊗ ω_{t} = ξ (ω_{t−v} ⊗ χ_{t−v} + χ_{−t+v} ⊗ ω_{t−v}) .

Replacing ω_{t} by ξω_{t−v} we get

ω_{t} ⊗ (χ_{t} − χ_{t−v}) + (χ_{v−t} − χ_{−t}) ⊗ ω_{t} = 0.

Since ω_{t} ≠ 0 we deduce χ_{t} = χ_{t−v} and χ_{v−t} = χ_{−t} which means that v ∈ U by Proposition 2.26. □

Let u ∈ U then for all t, s ∈ ℤ^{n} \ 0 we have ω_{t+u} = ζω_{t} and therefore ω_{t+s+u} = ζω_{t+s}. Hence the element ζ = ζ_{u} depends only on u. Moreover if ζ_{u} ≠ 0 then ζ_{−u} = ζ_{u}^{−1}. In fact if t ∈ ℤ^{n} \ 0, and t ≠ u then ω_{t} = ω_{t−u+u} = ζ_{u}ω_{t−u} and therefore ω_{t−u} = ζ_{u}^{−1}ω_{t}. Similarly ζ_{v+u} = ζ_{v}ζ_{u} for all v, u ∈ U.

Proposition 2.29. Let v ∈ ℤ^{n}. Then

χ_{v} + ζ_{v}ω_{−v} = ε, ω_{v} + ζ_{v}χ_{−v} = ζ_{v}ε  (52)

if and only if v ∈ U.

Proof. Let v ∈ U. For any t ∈ ℤ^{n} \ 0, by Theorem 2.18 and the preceding considerations we have

(χ_{v} + ζ_{v}ω_{−v}) ∗ χ_{t} = χ_{v} ∗ χ_{t} = χ_{t}, (χ_{v} + ζ_{v}ω_{−v}) ∗ ω_{t} = ζ_{v}ω_{−v} ∗ ω_{t} = ω_{t}.

Thus

χ_{v} + ζ_{v}ω_{−v} = (χ_{v} + ζ_{v}ω_{−v}) ∗ ε =
(χ_{v} + ζ_{v}ω_{−v}) ∗ (χ_{t} ∗ χ_{−t} + ω_{t} ∗ ω_{−t}) = χ_{t} ∗ χ_{−t} + ω_{t} ∗ ω_{−t} = ε.

Similarly

(ω_{v} + ζ_{v}χ_{−v}) ∗ χ_{t} = ζ_{v}χ_{t}, (ω_{v} + ζ_{v}χ_{−v}) ∗ ω_{t} = ζ_{v}ω_{t},

and therefore ω_{v} + ζ_{v}χ_{−v} = ζ_{v}ε.

Conversely if (52) holds then for any t ∈ ℤ^{n} \ 0, by Theorem 2.18 χ_{t} ∗ χ_{v} = (ε − ζ_{v}ω_{−v}) = χ_{t}. Apply Proposition 2.26. □

Corollary 2.30. Let Λ^{H} be the subalgebra of invariants of H. Then X^{v} + ζ_{v}X^{−v} ∈ Λ^{H} if and only if v ∈ U. In particular Λ is a left and right Noetherian Λ^{H}-module.

Proof. Consider the image of X^{v} + ζ_{v}X^{−v} under the comultiplication ρ,

ρ(X^{v} + ζ_{v}X^{−v}) =

(χ_{v} ⊗ X^{v} + ω_{v} ⊗ X^{−v}) + ζ_{v}(χ_{−v} ⊗ X^{−v} + ω_{−v} ⊗ X^{v}) =
(χ_{v} + ζ_{v}ω_{−v}) ⊗ X^{v} + (ω_{v} + ζ_{v}χ_{−v}) ⊗ X^{−v}.

Hence ρ(X^{v} + ζ_{v}X^{−v}) = ε ⊗ (X^{v} + ζ_{v}X^{−v}) if and only if (52) holds. Apply Proposition 2.29. □

Denote by F_{χ}^{∗} the linear span of all elements χ_{v} for all multi-indices v. Similarly denote by F_{ω}^{∗} the linear span of elements ω_{v} for all nonzero multi-indices v. By Theorem 2.18, Proposition 2.20, Corollary 2.19, Proposition 2.21 F^{∗} = F_{χ}^{∗} ⊕ F_{ω}^{∗} is a commutative Hopf subalgebra in H^{∗}. Moreover, χ_{v}, v ∈ U, is the unit element of the subalgebra F_{χ}^{∗}. If G ≠ G^{+} then dim F_{ω}^{∗} = dim F_{χ}^{∗} = |ℤ^{n}/U| by Proposition 2.28. The algebra Λ is a left F -module algebra and there exists a surjective Hopf algebra morphism π : H → F = (F^{∗})^{∗} such that the action of H on Λ is a composition of π and the action of F on Λ.

Choose a system of representatives T = \{\overline{v}\} of ℤ^{n}/U in ℤ^{n}. Then a basis of F^{∗} consists of all \{χ\overline{v}, ω\overline{v} | \overline{v} ∈ T \}. Choose a dual basis \{χ\overline{v}, ω\overline{v} | \overline{v} ∈ T \} in F. Then

⟨χ\overline{v}, χ\overline{w}⟩ = ⟨ω\overline{v}, ω\overline{w}⟩ = δ_{\overline{v},\overline{w}}, ⟨χ\overline{v}, ω\overline{w}⟩ = ⟨ω\overline{v}, χ\overline{w}⟩ = 0.

Denote by F_{χ} the linear span of all χ\overline{v}, \overline{v} ∈ T.

Proposition 2.31. The (co)multiplication table for the basis χ\overline{v}, \overline{v} ∈ T of F_{χ} has the form

χ\overline{t}χ\overline{s} = δ_{\overline{t},\overline{s}}χ\overline{t}, Δ(χ\overline{t}) = ∑_{\overline{r}+\overline{s}=\overline{t}} χ\overline{r} ⊗ χ\overline{s},
ε(χ\overline{t}) = δ_{\overline{t},0}, S(χ\overline{t}) = χ^{−\overline{t}},

where \overline{z} + \overline{s} = \overline{t} in ℤ^{n}/U. In particular F_{χ} is a cocommutative and commutative Hopf subalgebra in F.

Proof. By Proposition 2.20 for any \overline{u} ∈ T we have

⟨χ\overline{t}χ\overline{s}, χ\overline{u}⟩ = ⟨χ\overline{t}, χ\overline{u}⟩⟨χ\overline{s}, χ\overline{u}⟩ + ⟨χ\overline{t}, ω\overline{u}⟩⟨χ\overline{u}, ω_{−\overline{u}}⟩ = δ_{\overline{t},\overline{u}}δ_{\overline{s},\overline{u}}.
⟨χ\overline{t}χ\overline{s}, ω\overline{u}⟩ = ⟨χ\overline{t}, ω\overline{u}⟩⟨χ\overline{s}, χ\overline{u}⟩ + ⟨χ\overline{t}, χ_{−\overline{u}}⟩⟨χ\overline{u}, ω\overline{u}⟩ = 0

Hence

χ\overline{t}χ\overline{s} = \begin{cases}0, & if \overline{t} ≠ \overline{s}; \\ χ\overline{s}, & if \overline{t} = \overline{s}. \end{cases}

Now

Δ(χ\overline{t})(χ\overline{s} ⊗ χ\overline{u}) = ⟨χ\overline{t}, χ\overline{s} ∗ χ\overline{u}⟩ = ⟨χ\overline{t}, χ_{\overline{s}+\overline{u}}⟩ = δ_{\overline{t},\overline{s}+\overline{u}}.

Similarly

Δ(χ\overline{t})(χ\overline{s} ⊗ ω\overline{u}) = ⟨χ\overline{t}, χ\overline{s} ∗ ω\overline{u}⟩ = 0 = Δ(χ\overline{t})(ω\overline{s} ⊗ χ\overline{u});
Δ(χ\overline{t})(ω\overline{s} ⊗ ω\overline{u}) = ⟨χ\overline{t}, ω_{\overline{s}+\overline{u}}⟩ = 0.

Hence

Δ(χ\overline{t}) = ∑_{\overline{r}+\overline{s}=\overline{t}} χ\overline{r} ⊗ χ\overline{s}.

In the same way one can calculate the values of ε and S. □

Proposition 2.32. Let g ∈ G \ G^{+}. Then g^{2} = 1 and π(g)F_{χ}π(g)^{−1} = F_{χ}.

Proof. We have g(X_{i}) = ξ_{i}X_{i}^{−1}, ξ_{i} ∈ k^{∗}, for any i = 1, …, n. Then g^{2}(X_{i}) = X_{i} for any i that is g^{2} = 1.

Observe that g(X^{v}) = ω_{v}(g)X^{−v} for any v ∈ ℤ^{n}. Hence π(g) belongs to the linear span of ω\overline{v}, \overline{v} ∈ T. By (41) we have

⟨ω\overline{v}, π(g)χ\overline{w}π(g)⟩ =

⟨ω\overline{v}, π(g)⟩⟨χ\overline{v}, χ\overline{w}π(g)⟩ + ⟨χ_{−\overline{v}}, π(g)⟩⟨ω\overline{v}, χ\overline{w}π(g)⟩ =

⟨ω\overline{v}, π(g)⟩⟨χ\overline{v}, χ\overline{w}⟩⟨χ\overline{v}, π(g)⟩ + ⟨ω\overline{v}, π(g)⟩⟨ω_{−\overline{v}}, χ\overline{w}⟩⟨ω\overline{v}, π(g)⟩+

⟨χ_{−\overline{v}}, π(g)⟩⟨ω\overline{v}, χ\overline{w}⟩⟨χ\overline{v}, π(g)⟩ + ⟨χ_{−\overline{v}}, π(g)⟩⟨χ_{−\overline{v}}, χ\overline{w}⟩⟨ω\overline{v}, π(g)⟩ = 0.

□

Proposition 2.33. π(g)F_{χ}π(g) ∩ F_{χ} = 0.

Proof. Suppose that a, b ∈ F_{χ} and π(g)a = b. Then

0 = ⟨ω\overline{v}, π(g)a⟩ =
⟨ω\overline{v}, π(g)⟩⟨χ\overline{v}, a⟩ + ⟨χ\overline{v}, π(g)⟩⟨ω\overline{v}, a⟩ = ⟨ω\overline{v}, π(g)⟩⟨χ\overline{v}, a⟩.

Hence ⟨χ_{v}, a⟩ = 0. for any \overline{v} and therefore a = b = 0. □

Thus we have proved

Theorem 2.34. F_{χ} is a commutative and cocommutative Hopf subalgebra in F. Moreover F is a smash product F = F_{χ}♯k⟨π(g)⟩, where k⟨π(g)⟩ is a group algebra of a cyclic group of order 2.

In terms of the chosen basis of F the surjective homomorphism of Hopf algebras π : H → F has the form

π(h) = ∑_{v} (χ_{v}(h)χ^{(v)} + ω_{v}(h)ω^{(v)})

for any h ∈ H.

The algebra F^{∗} is pointed as a factor algebra of H [M, Corollary 5.3.5] and cocommutative. Let F_{1} be the irreducible connected component of F containing 1. Since F_{χ} is Hopf subalgebra of F containing 1, we get F_{1} ⊆ F_{χ} and therefore F_{1} is a commutative and cocommutative Hopf algebra. By [M, Corollary 5.6.4] we have a decomposition F = F_{1}♯kΓ, where Γ = G(F). By [M, Corollary 5.3.5] we have π(G) = Γ. Put Γ^{+} = π(G^{+}). If G ≠ G^{+} and g ∈ G \ G^{+}, then G is a semidirect product G = G^{+} ⋋ ⟨g⟩ of a normal subgroup G^{+} and a cyclic group ⟨g⟩ of order 2. Similarly Γ = Γ^{+} ⋋ ⟨π(g)⟩.

Proposition 2.35. Suppose that the field k is perfect and dim H is coprime with the characteristic of k. Then F = kΓ. If k contains a primitive root of 1 of degree dim H, then Γ^{+} ≃ ℤ^{n}/U. If G ≠ G^{+} and g ∈ G \ G^{+}, then G = G^{+} ⋋ ⟨g⟩ and Γ = Γ^{+} ⋋ ⟨π(g)⟩.

Proof. If char k = 0 then F_{1} is a universal enveloping algebra for the Lie algebra P (F) of primitive elements of F. But F has a finite dimension. Hence P (H) = 0 and F is the group algebra kΓ of the group Γ of grouplike elements in F.

Suppose that char k = p > 0. By [M, Corollary 5.6.9] dim F_{1} is a power of p and therefore by Nichols-Zoeller theorem [M, §3.1] dim F = dim F^{∗} is divided by p. But F^{∗} is a Hopf subalgebra in H^{∗}. Hence dim F divides dim H^{∗} = dim H by Nichols-Zoeller theorem.

Let G = G^{+}. Then F_{ω}^{∗} = 0 and G is an Abelian group. In this case F^{∗} = F_{χ}^{∗} is a group algebra of the group ℤ^{n}/U. Under the assumption on k its dual F is isomorphic to F^{∗}. □

Thus we have finally proved

Theorem 2.36. Let the field k be perfect and char k coprime with dim H and G = G(H). Then there exists a finite automorphism group Γ of Λ and an epimorphism of Hopf algebras π : H → kΓ such that the action of H on Λ is a composition of π and the natural of kΓ on Λ. Moreover π(G) = Γ. If k contains a primitive root of 1 of degree dim H, then Γ^{+} ≃ ℤ^{n}/U. If G ≠ G^{+} and g ∈ G\G^{+} then G = G^{+}⋋⟨g⟩ and Γ = Γ^{+} ⋋ ⟨π(g)⟩.

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Department of Algebra, Faculty of Mechanics and Mathematics Moscow State University, Leninsky Gory, 119992, GSP-2, Moscow, RUSSIA

  E-mail address: artamon@mech.math.msu.su