Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 25, 2007, 187–196 \begin{matrix}c \\ ◯ \end{matrix} Miroslav Kureš and David Sehnal

Miroslav Kureš and David Sehnal

THE ORDER OF ALGEBRAS WITH NONTRIVIAL FIXED POINT SUBALGEBRAS

(submitted by M. A. Malakhaltsev)

Abstract. The paper represents an advancement of research the fundamental problem of which is a classification of algebras A (Weil algebras primarily) having a nontrivial fixed point subalgebra (with respect to \underline{all} algebra automorphisms). The main result is the determination of the algebra order allowing a nontrivial fixed point subalgebra. Moreover, an autonomous importance of some results about socle elements of A and the unipotency of algebra automorphisms is highlighted.

1. Introduction

We consider local commutative ℝ-algebra A with identity, the nilpotent ideal 𝔫_{A} of which has a finite dimension as a vector space and A/𝔫_{A} = ℝ. We call the order of A the minimum ord(A) of the integers r satisfying 𝔫_{A}^{r+1} = 0 and the width w(A) of A the dimension dim_{ℝ}(𝔫_{A}/𝔫_{A}^{2}). One can assume A is expressed as a finite dimensional factor ℝ-algebra of the algebra ℝ[x_{1}, …, x_{n}] of real polynomials in several indeterminates. Thus, the main example is

𝔻_{n}^{r}: = ℝ[x_{1}, …, x_{n}]/𝔪^{r+1},

𝔪 = ⟨x_{1}, …, x_{n}⟩ being the maximal ideal of ℝ[x_{1}, …, x_{n}]. Evidently, ord(𝔻_{n}^{r}) = r and w(𝔻_{n}^{r}) = n. As well, every other such an algebra A of the order r can be expressed in a form

A = ℝ[x_{1}, …, x_{n}]/𝔦 + 𝔪^{r+1},

elling of mechatronic systems”.

2000 Mathematical Subject Classification. 13H99, 16W20. Secondary 58A32.

Key words and phrases. Local algebra, automorphism.

Published results were acquired using the subsidization of the Ministry of Education, Youth and Sports of the Czech Republic, research plan MSM 0021630518 “Simulation mod-

where an ideal 𝔦 satisfies 𝔪^{r+1} ⫋ 𝔦 ⊂ 𝔪^{2} and is generated by finite number of polynomials, i.e. 𝔦 = ⟨P_{1}, …, P_{l}⟩. The fact 𝔦 ⊂ 𝔪^{2} implies that the width of A is n, too. It is evident, that such expressions of algebras in question are not unique after all. Clearly, A can be expressed also in a form

A = 𝔻_{n}^{r}/𝔧,

where 𝔧 is an ideal in 𝔻_{n}^{r} with analogous conditions as above mentioned 𝔦.

In differential geometry, we talk about Weil algebras, see e.g. [3]. In this paper, we shall call them shortly algebras. Notwithstanding that many of our results are not binded with the real field only, we investigate this as our main case here.

Let Aut A be a group of automorphisms of the algebra A. By a fixed point of A we mean every a ∈ A satisfying ϕ(a) = a for all ϕ ∈ Aut A. Let

SA = {a ∈ A; ϕ(a) = a for all ϕ ∈ Aut A}

be the set of all fixed points of A. (We opine that nothing but fixed point subalgebras A^{G} = \{a ∈ A; ϕ(a) = a for all ϕ ∈ G\}, where G ⊂ Aut A is a \underline{finite} group of automorphisms, were studied in detail up to now. See e.g. [2].) It is clear, that SA is a subalgebra of A. As

A = ℝ · (1 ⊕ 𝔫_{A})

(𝔫_{A} being the ideal of nilpotent elements of A), ℝ = ℝ · 1 ⊂ SA certainly holds, because every automorphism sends 1 into 1.

Remark 1. We remark that SA itself is a subalgebra of

S^{Fin}A: = {a ∈ A having a finite orbit with respect to all ϕ ∈ Aut A},

which is another interesting subalgebra of A apt for a computer algorithmization, as we plan explicate in another paper in future.

As to an original motivation of this research, the bijection between all natural operators lifting vector fields from m-dimensional manifolds to bundles of Weil contact elements and the subalgebra of fixed points SA of a Weil algebra A was determined in [5]. Although in the known geometrically motivated examples is usually SA = ℝ (such SA is called trivial), there are some algebras for which SA ⫌ ℝ and we suspect related bundles will have remarkably interesting geometry. Thus, the fundamental problem is a classification of algebras having SA nontrivial. See [5], [6] for known results up to now. Especially, we underline the fact that SA is trivial whenever the ideal 𝔧 is homogeneous. Our new results are facilitated after a sort of a computer working, too; we submit the algorithm aspect in the next joint-work. Nevertheless, one can use pencil paper methods (like in [5], [6]) as well.

Remark 2. As our paper is concentrated on the order of A, we remark that in the original paper of André Weil, [8], the order was called the height of A. Further, the term Loevy length is also sometimes used (= ord(A) + 1).

2. Socle elements and uni potent automorphisms

2.1. The basis of the ideal 𝔧 and the basis of the vector space A. We take an algebra A in a form

A = 𝔻_{n}^{r}/𝔧.

The set G of generators of 𝔧 (the basis of 𝔧) can be multifarious. We refer to two important possibilities:

(i) Groebner basis, very important in computer algebra and implemented in most of computer algebra systems

(ii) elementary polynomial basis outgoing from the minimalization of the length (i.e. number of monomials) of the longest generator among P_{1}, …, P_{l} and the follow-up minimalization of the number of generators with such length, the idea was described in [6]

Clearly, A is a real vector space. Its basis (naturally, fully independent on the mentioned basis of 𝔧) is constituted by equivalence classes [1], [x_{1}], …, [x_{n}], plus classes containing higher monomials in x_{1}, … x_{n}: every such a monomial is contained in one equivalence class, but the choice of the representative is not unique. We shall take the least monomial (with coefficient 1) as to the graded lexicographical order. After a fixing of this basis (denoted B(A) hereafter), we can write elements of A as a linear combination of the basis elements (written ordinarily with omitted brackets [ ]) by a unique way. Of course, the multiplication rules are needful to be added, best in the form of a table.

Example 1. Let

A = 𝔻_{3}^{2}/⟨x^{2} + xy + xz, xy + y^{2} + yz⟩.

Then

B(A) = {1, x, y, z, x^{2}, xy, y^{2}, z^{2}},

hence dim_{ℝ}(A) = 8 and elements of A can be written in the form

k_{1} + k_{2}x + k_{3}y + k_{4}z + k_{5}x^{2} + k_{6}xy + k_{7}y^{2} + k_{8}z^{2}

with the following multiplication table.

\begin{matrix}· & 1 & x & y & z & x^{2} & xy & y^{2} & z^{2} \\ 1 & 1 & x & y & z & x^{2} & xy & y^{2} & z^{2} \\ x & x & x^{2} & xy & −x^{2} − xy & 0 & 0 & 0 & 0 \\ y & y & xy & y^{2} & −y^{2} − xy & 0 & 0 & 0 & 0 \\ z & z & −x^{2} − xy & −y^{2} − xy & z^{2} & 0 & 0 & 0 & 0 \\ x^{2} & x^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ xy & xy & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ y^{2} & y^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ z^{2} & z^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

Proposition 1. Let a ∈ SA, a = ∑_{i} r_{i}B_{i}, r_{i} ≠ 0, B_{i} ∈ B(A). In general, B_{i} may not belong to SA.

Proof. Let

A = 𝔻_{2}^{5}/⟨x^{2}y^{2} + x^{5}, x^{2}y^{2} + y^{5}⟩.

The basis is B(A) = {1, x, y, x^{2}, xy, y^{2}, x^{3}, x^{2}y, xy^{2}, y^{3}, x^{4}, x^{3}y, x^{2}y^{2}, xy^{3}, y^{4}, x^{4}y, xy^{4}}. Aut A has two connected components:

1st component
x ↦ x + C_{1,3}x^{2} + C_{1,4}xy + C_{1,6}x^{3} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,9}y^{3} + C_{1,10}x^{4} +
C_{1,11}x^{3}y + C_{1,12}x^{2}y^{2} + C_{1,13}xy^{3} + C_{1,14}y^{4} + C_{1,15}x^{4}y + C_{1,16}xy^{4}
y ↦ y + C_{2,4}xy + C_{2,5}y^{2} + C_{2,6}x^{3} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,9}y^{3} + C_{2,10}x^{4} +
C_{2,11}x^{3}y + C_{2,12}x^{2}y^{2} + C_{2,13}xy^{3} + C_{2,14}y^{4} + C_{2,15}x^{4}y + C_{2,16}xy^{4};
2nd component
x ↦ y + C_{1,4}xy + C_{1,5}y^{2} + C_{1,6}x^{3} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,9}y^{3} + C_{1,10}x^{4} +
C_{1,11}x^{3}y + C_{1,12}x^{2}y^{2} + C_{1,13}xy^{3} + C_{1,14}y^{4} + C_{1,15}x^{4}y + C_{1,16}xy^{4}
y ↦ x + C_{2,3}x^{2} + C_{2,4}xy + C_{2,6}x^{3} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,9}y^{3} + C_{2,10}x^{4} +
C_{2,11}x^{3}y + C_{2,12}x^{2}y^{2} + C_{2,13}xy^{3} + C_{2,14}y^{4} + C_{2,15}x^{4}y + C_{2,16}xy^{4};
C_{i,j} ∈ ℝ.

The elements of the form

c_{1} + c_{2}x^{2}y^{2} + c_{3}(x^{4}y + xy^{4})

are fixed for all c_{1}, c_{2}, c_{3} ∈ ℝ; however, x^{4}y ∈ B(A) is not fixed and xy^{4} ∈ B(A) is not fixed. □

2.2. Socle elements. If an element a ∈ A has the property au = 0 for all u ∈ 𝔫_{A}, we call a the socle element of A.

Let a = ∑_{i} r_{i}B_{i} be an element of A expressed as a linear combination of basis elements.

Lemma 1. a is a socle element if and only if all B_{i} are socle elements.

Proof. It is clear that if all B_{i} are socle elements, then a = ∑_{i} r_{i}B_{i} is also a socle element. Let a be a socle element of A and let us suppose that some B_{i}, say B_{1}, is not a socle element. It follows there is u ∈ 𝔫_{A}, u = ∑_{j} r̄_{j}B̄_{j} (B̄_{j} ∈ B(A)), such that B_{1}u ≠ 0. However, it implies there exists some B̄_{j}, say B̄_{1} for which B_{1}B̄_{1} ≠ 0. We can write B̄_{1} as a monomial of basis elements from 𝔫_{A}/𝔫_{A}^{2}, i.e. B̄_{1} = x_{n_{1}}^{p_{1}} … x_{n_{k}}^{p_{k}}, where p_{1}, …, p_{k} ∈ ℕ. Thus B_{1}x_{n_{1}} ≠ 0, too. It means ax_{n_{1}} ≠ 0 and a is not a socle element; a contradiction. □

Lemma 2. Every algebra A has non-zero socle elements.

Proof. If ord A = 0, then A = ℝ and 𝔫_{A} = {0}: that is why all elements of A are socle elements. If ord A = r > 0, then it is sufficient look for socle elements amidst elements of B(A) (cf. Lemma 1). We take an element a ∈ B(A) and multiply it by all x_{1}, …, x_{n}: if all products equal zero, then a is a socle element, if not, we take some non-zero product ax_{i} and multiply it by all x_{1}, …, x_{n} repeatedly; the number of such multiplications (till then a socle element is found) is maximally r as 𝔫_{A}^{r+1} = 0. □

We can consult the Example 1 anew and find that the multiplication table is an elegant tool for the identification of socle elements. Easily, we have observed that all socle elements constitute an ideal called the socle of A and denoted by soc(A). The following assertion specifies the relation between soc(A) and 𝔫_{A}.

Lemma 3. For every algebra A with ord(A) = r,

𝔫_{A}^{r} ⊂ soc(A).

Proof. The lemma is very transparent: every ”maximal power” is a socle element. It is also clear, that there exist algebras, for which 𝔫_{A}^{r} ⫋ soc(A). □

Moreover, elements of A in the form

r_{1} + r_{2}a, r_{1}, r_{2} ∈ ℝ, a ∈ soc(A)

form a subalgebra MA of A used e.g. in [4]. The problem of a relation between SA and MA is still open.

2.3. Unipotent automorphisms. We denote by G_{A} the connected identity component of the group Aut A of automorphisms A. Further, we have the morphism

ϵ_{A}: Aut A → GL(𝔫_{A}/𝔫_{A}^{2}).

The kernel of ϵ_{A} is denoted by U_{A} and it is a subgroup of Aut A having all elements unipotent. A unipotent automorphism is such ϕ ∈ Aut A for which id_{A} −ϕ is a nilpotent endomorphism of A; alternatively, such ϕ ∈ Aut A for which all eigenvalues (over ℂ) of its matrix representation are equal 1. Of course, there are also unipotent automorphisms not belonging to U_{A}, in general. The subgroup U_{A} is connected and the inclusions

id_{A} ∈ U_{A} ⊂ G_{A} ⊂ Aut A

always hold. For the properties of U_{A} see [1] and [7].

Immediately, we have the following assertion.

Proposition 2. Let A be a Weil algebra of the order r. If U_{A} = Aut A, then all socle elements of A belonging to 𝔫_{A}^{r} belong to SA, too.

Proof. The assertion is clear: all automorphisms have a form

1 ↦ 1
x_{1} ↦ x_{1} + P_{1}
…
x_{n} ↦ x_{n} + P_{n},

where P_{i} are polynomials without absolute and linear terms. □

(We shall not write 1 ↦ 1 below in the description of automorphisms.)

Taking Lemma 2 into consideration, we have obtained immediately:

Corollary 1. If ord A > 0 and U_{A} = Aut A, then SA is nontrivial.

We will not remind the trivial case ord(A) = 0 below. Within a time of this research, some conjectures about an effect of the property U_{A} = G_{A} have occurred. In particular, we have proved:

Proposition 3. If U_{A} = G_{A}, then SA can be both nontrivial and trivial (i.e. U_{A} = G_{A} is not a sufficient condition for a nontrivial SA).

Proof. Of course, the nontrivial case is included e.g. in the previous proposition. Examples of algebras with U_{A} = Aut A are in the proof of Proposition 4. The trivial case comes in the algebra

A = 𝔻_{2}^{5}/⟨xy^{2} + x^{5}, x^{2}y + y^{5}⟩.

The basis is B(A) = {1, x, y, x^{2}, xy, y^{2}, x^{3}, x^{2}y, xy^{2}, y^{3}, x^{4}, y^{4}}. Aut A has eight connected components:

1st component

x ↦ −x + C_{1,3}x^{2} + C_{1,4}xy + C_{1,6}x^{3} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,10}x^{4} + C_{1,11}y^{4}
y ↦ −y + C_{2,4}xy + C_{2,5}y^{2} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,9}y^{3} + C_{2,10}x^{4} + C_{2,11}y^{4};

2nd component

x ↦ x + C_{1,3}x^{2} + C_{1,4}xy + C_{1,6}x^{3} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,10}x^{4} + C_{1,11}y^{4}
y ↦ −y + C_{2,4}xy + C_{2,5}y^{2} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,9}y^{3} + C_{2,10}x^{4} + C_{2,11}y^{4};

3rd component

x ↦ −x + C_{1,3}x^{2} + C_{1,4}xy + C_{1,6}x^{3} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,10}x^{4} + C_{1,11}y^{4}
y ↦ y + C_{2,4}xy + C_{2,5}y^{2} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,9}y^{3} + C_{2,10}x^{4} + C_{2,11}y^{4};

4th component

x ↦ x + C_{1,3}x^{2} + C_{1,4}xy + C_{1,6}x^{3} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,10}x^{4} + C_{1,11}y^{4}
y ↦ y + C_{2,4}xy + C_{2,5}y^{2} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,9}y^{3} + C_{2,10}x^{4} + C_{2,11}y^{4};

5th component

x ↦ −y + C_{1,4}xy + C_{1,5}y^{2} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,9}y^{3} + C_{1,10}x^{4} + C_{1,11}y^{4}
y ↦ −x + C_{2,3}x^{2} + C_{2,4}xy + C_{2,6}x^{3} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,10}x^{4} + C_{2,11}y^{4};

6th component

x ↦ y + C_{1,4}xy + C_{1,5}y^{2} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,9}y^{3} + C_{1,10}x^{4} + C_{1,11}y^{4}
y ↦ −x + C_{2,3}x^{2} + C_{2,4}xy + C_{2,6}x^{3} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,10}x^{4} + C_{2,11}y^{4};

7th component

x ↦ −y + C_{1,4}xy + C_{1,5}y^{2} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,9}y^{3} + C_{1,10}x^{4} + C_{1,11}y^{4}
y ↦ x + C_{2,3}x^{2} + C_{2,4}xy + C_{2,6}x^{3} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,10}x^{4} + C_{2,11}y^{4};

8th component

x ↦ y + C_{1,4}xy + C_{1,5}y^{2} + C_{1,7}x^{2}y + C_{1,8}xy^{2} + C_{1,9}y^{3} + C_{1,10}x^{4} + C_{1,11}y^{4}
y ↦ x + C_{2,3}x^{2} + C_{2,4}xy + C_{2,6}x^{3} + C_{2,7}x^{2}y + C_{2,8}xy^{2} + C_{2,10}x^{4} + C_{2,11}y^{4};

C_{i,j} ∈ ℝ.

By a direct application of automorphisms of the whole group Aut A to a general element

k_{1} + k_{2}x + k_{3}y + k_{4}x^{2} + k_{5}xy + k_{6}y^{2} + k_{7}x^{3} + k_{8}x^{2}y + k_{9}xy^{2} + k_{10}y^{3} + k_{11}x^{4} + k_{12}y^{4}

of A (k_{i} ∈ ℝ) we find SA trivial. □

Remark 3. We remark that in the list of algebras having unipotent G_{A} (in [1], Theorem 3.11, the case c.ii) is an error. Surely, let us evaluate automorphisms for

A = 𝔻_{2}^{4}/⟨x^{3} + y^{4}, x^{2}y + y^{4}, xy^{2}⟩.

The basis is B(A) = {1, x, y, x^{2}, xy, y^{2}, x^{2}y, y^{3}}. Aut A consists of only one connected component:

x ↦ C_{2,2}^{4}x + C_{1,3}x^{2} + C_{1,4}xy + (C_{2,2}^{6} − C_{2,2}^{3})y^{2} + C_{1,6}x^{2}y + C_{1,7}y^{3}
y ↦ (C_{2,2}^{4} − C_{2,2}^{3})x + C_{2,2}^{3}y + C_{2,3}x^{2} + C_{2,4}xy + C_{2,5}y^{2} + C_{2,6}x^{2}y + C_{2,7}y^{3}
C_{i,j} ∈ ℝ; C_{2,2} ≠ 0.

3. The order theorem

3.1. The width 2. Let r ≥ 4 and let

Â_{r} = 𝔻_{2}^{r}/⟨x^{r−2} + y^{r−1}, x^{r−1} + y^{r}⟩.

We have obtained the following result about Â_{r}.

Proposition 4.

i) SÂ_{4} is trivial.

ii) If r ≥ 5, then SÂ_{r} is nontrivial and dim_{ℝ} SÂ_{r} ≥ r − 2.

Proof. i) The problem was formulated in [5] as the Exercise 1. It is sufficient to find one automorphism for which only constants are fixed. Nevertheless, we can describe the whole group Aut A. The basis of Â_{4} is B(Â_{4}) = {1, x, y, x^{2}, xy, y^{2}, x^{2}y, xy^{2}}.

1st component

x ↦ −|C_{2,2}|^{3}x + \frac{1}{128}(2|C_{2,2}|^{3} − 80C_{1,4} + 3C_{2,2}^{4} − C_{2,2}^{6} + 96C_{2,2}^{2}C_{2,5} +

48|C_{2,2}|(C_{1,4} − 4C_{2,4} − 2C_{2,5}))x^{2} + C_{1,4}xy − \frac{3}{8}(|C_{2,2}|^{3} + C_{2,2}^{4})y^{2} +

C_{1,6}x^{2}y + C_{1,7}xy^{2}

y ↦ −\frac{1}{4}(|C_{2,2}|^{3} + C_{2,2}^{2})x + C_{2,2}^{2}y + C_{2,3}x^{2} + C_{2,4}xy + C_{2,5}y^{2} + C_{2,6}x^{2}y +

C_{2,7}xy^{2};

2nd component

x ↦ |C_{2,2}|^{3}x + \frac{1}{128}(−2|C_{2,2}|^{3} − 80C_{1,4} + 3C_{2,2}^{4} − C_{2,2}^{6} + 96C_{2,2}^{2}C_{2,5} −

48|C_{2,2}|(C_{1,4} − 4C_{2,4} − 2C_{2,5}))x^{2} + C_{1,4}xy + \frac{3}{8}(|C_{2,2}|^{3} − C_{2,2}^{4})y^{2} +

C_{1,6}x^{2}y + C_{1,7}xy^{2}

y ↦ \frac{1}{4}(|C_{2,2}|^{3} − C_{2,2}^{2})x + C_{2,2}^{2}y + C_{2,3}x^{2} + C_{2,4}xy + C_{2,5}y^{2} + C_{2,6}x^{2}y +

C_{2,7}xy^{2}; C_{i,j} ∈ ℝ.

The example of an automorphism precluding nontrivial elements of SÂ_{4} is e.g. (we take C_{2,2} = 2 and all other C_{i,j} = 0 in the 1st component of Aut A)

x ↦ −8x − 9y^{2}
y ↦ −3x + 4y.

ii) For r ≥ 5, polynomials x^{r}, x^{r−1}y, x^{r−2}y^{2}, x^{r−1} +y^{r}, x^{r−2} +y^{r−1}, x^{r−1} +xy^{r−1}, x^{r−2}y + y^{r} are belonging to 𝔧 = ⟨x^{r−2} + y^{r−1}, x^{r−1} + y^{r}⟩. The automorphisms have a form

x ↦ Ax + By + K(x, y)
y ↦ Cx + Dy + L(x, y),

where K and L are polynomials without absolute and linear terms.

The condition x^{r−2} + y^{r−1} = 0 gives

B^{r−2} = 0,

looking at y^{r−2}. It follows B = 0. The same condition gives

CD^{r−2} = 0,

looking at xy^{r−2} (this is obtained as the image of y^{r−1} only and the condition r > 4 is essential). It follows C = 0 (because D ≠ 0 as B = 0). The same condition also gives

A^{r−2} = D^{r−1},

looking at x^{r−2}.

The condition x^{r−1} + y^{r} = 0 gives

A^{r−1} = D^{r},

looking at x^{r−1}. The last two conditions imply A = D = 1.

We have UÂ_{r} = Aut Â_{r} and, as to Corollary 1, SÂ_{r} is nontrivial. The dimension dim_{ℝ} SÂ_{r} must be greater or equal to the number of (linearly independent) basis elements of the socle. □

Remark 4. As to Â_{4}, Professor Manuel Saorín had proposed in our e-mail communication another way to the squaring up to the group of automorphisms of this algebra: The algebra Â_{4} = 𝔻_{2}^{4}/⟨x^{2} +y^{3}, x^{3} +y^{4}⟩ is isomorphic to the algebra 𝔻_{2}^{4}/⟨X^{2} −Y^{3}, Y^{4}⟩ through the isomorphisms of ideals given by

X ↦ −8x + xy − 3y^{2} + \frac{19}{56}y^{3}
Y ↦ x − 4y − \frac{1}{14}xy + \frac{1}{4}y^{2}.

3.2. The general width.

Lemma 4. If ord(A) ≤ 2, then SA is trivial.

Proof. That was recalled in Introduction, that the fulfillment of 𝔧 ⊂ 𝔪^{2} in the expression A = 𝔻_{n}^{r}/𝔧 is presumed. Thus A = 𝔻_{n}^{2} or 𝔻_{n}^{2} factorized through an ideal generated by homogeneous polynomials of the second order; however, it means that we have only algebras with trivial SA, cf. Introduction and [5]. □

If we know an algebra with nontrivial fixed point subalgebra, then one can construct an algebra with nontrivial fixed point subalgebra of a greater width and the same order as the existing. (It was showed in [6].) If w(A) = 2, then the opening order for the possibility of non-trivial SA is 4; we have just proved in Proposition 4 that for all orders greater than 4 it is possible, too. Consequently, the following Proposition 5 completes the classification.

Proposition 5. There are algebras of the order 3 with nontrivial fixed point subalgebras.

Proof. It follows from Proposition 1 in [6], that we must look for it in the width greater then 2. Truly, for

A = 𝔻_{3}^{3}/⟨x^{2} + y^{3}, xy + z^{3}, y^{2}z + yz^{2}⟩

we have obtained B(A) = {1, x, y, z, x^{2}, xy, y^{2}, xz, yz, z^{2}, xz^{2}, y^{2}z} and Aut A with two connected components:

1st component

x ↦ −x + C_{1,4}x^{2} + C_{1,5}xy + y^{2} + C_{1,7}xz + C_{1,10}xz^{2} + C_{1,11}yz^{2}
y ↦ y + C_{2,4}x^{2} + C_{2,5}xy + C_{2,6}y^{2} + C_{2,7}xz + C_{2,8}yz + 3C_{3,1}z^{2} +

C_{2,10}xz^{2} + C_{2,11}y^{2}z

z ↦ C_{3,1}x − y − z + C_{3,4}x^{2} + C_{3,5}xy + C_{3,6}y^{2} + C_{3,7}xz + C_{3,8}yz + C_{3,9}z^{2} +

C_{3,10}xz^{2} + C_{3,11}y^{2}z; 2nd component

x ↦ x + C_{1,4}x^{2} + C_{1,5}xy + C_{1,7}xz + C_{1,10}xz^{2} + C_{1,11}yz^{2}
y ↦ y + C_{2,4}x^{2} + C_{2,5}xy + C_{2,6}y^{2} + C_{2,7}xz + C_{2,8}yz − 3C_{3,1}z^{2} + C_{2,10}xz^{2} +

C_{2,11}y^{2}z

z ↦ C_{3,1}x + z + C_{3,4}x^{2} + C_{3,5}xy + C_{3,6}y^{2} + C_{3,7}xz + C_{3,8}yz +

C_{3,9}z^{2} + C_{3,10}xz^{2} + C_{3,11}y^{2}z; C_{i,j} ∈ ℝ.

The elements of the form

c_{1} + c_{2}x^{2}

are fixed for all c_{1}, c_{2} ∈ ℝ. □

The example implies immediately:

Corollary 2. U_{A} = G_{A} is not a necessary condition for a nontrivial SA.

Proof. It is evident that G_{A} for A = 𝔻_{3}^{3}/⟨x^{2} + y^{3}, xy + z^{3}, y^{2}z + yz^{2}⟩ has elements (with C_{3,1} ≠ 0) not belonging to the kernel of ϵ_{A}; nevertheless, all elements of G_{A} are still unipotent here (we left a verification of this fact to the reader). □

Finally, we summarize to the following ”order theorem”.

Theorem. There is no algebra A with w(A) = 1 and with nontrivial fixed point subalgebra. There exist algebras A with w(A) = 2 with a nontrivial fixed point subalgebra if and only if ord(A) ≥ 4. For all w(A) > 2, there exist algebras A with a nontrivial fixed point subalgebra if and only if ord(A) ≥ 3.

Proof. The case w(A) = 1 is trivial. The case w(A) = 2 is solved in [6], Proposition 1 and Proposition 2. In particular, nontrivial SA = {c_{1} + c_{2}x^{2}y} was proved for A = 𝔻_{2}^{4}/⟨x^{2}y + y^{4}, x^{3} + xy^{2}⟩ in [6]. Finally, the case w(A) > 2 is solved by Proposition 4, Lemma 4 and Proposition 5 listed above. □

The result can be recapitulated by the tabular form:

Algebras with a nontrivial fixed point subalgebra

width\^{order} ord(A) = 1 ord(A) = 2 ord(A) = 3 ord(A) = 4 ord(A) = 5 …
w(A) = 1  ∄  ∄  ∄  ∄  ∄  ∄
w(A) = 2  ∄  ∄  ∄  ✓  ✓  ✓
w(A) = 3  ∄  ∄  ✓  ✓  ✓  ✓
w(A) = 4  ∄  ∄  ✓  ✓  ✓  ✓
…  ∄  ∄  ✓  ✓  ✓  ✓

Acknowledgement. The authors thank to Professor Manuel Saorín from the University of Murcia for his friendly sharing of ideas and for challenging oral and electronic discussions upon the topic. The authors also thank to Professor Guillermo Cortiñas from the University of Buenos Aires for his concerned reading of manuscript and helpful comments arising therefrom.

References

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3. Kolář, I., Michor, P.W. and Slovák, J., Natural Operations in Differential Geometry, Springer Verlag 1993

4. Kureš, M., Weil modules and gauge bundles, Acta Mathematica Sinica (English Series) 22 (1), 2006, 271–278

5. Kureš, M., Mikulski, W.M., Natural operators lifting vector fields to bundles of Weil contact elements, Czechoslovak Mathematical Journal 54 (129), 2004, 855–867

6. Kureš, M., Mikulski, W.M., Natural operators lifting 1-forms to bundles of Weil contact elements, Bulletin of the Irish Mathematical Society 49 (2002), 23–41

7. Pollack, R.D., Algebras and their automorphism groups, Communications in Algebra 17 (8), 1989, 1843–1866

8. Weil, A., Théorie des points proches sur les variétés différentiables (French), Géométrie différentielle, Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, 111–117

Miroslav Kureš, Institute of Mathematics, Brno University of Technology, Technická 2, 61669 Brno, Czech Republic

E-mail address: kures@fme.vutbr.cz

David Sehnal, Faculty of Informatics, Masaryk University in Brno, Botanická 68a, 60200 Brno, Czech Republic E-mail address: david.sehnal@gmail.com

Received January 2, 2007