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 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)

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, \dots, x n]

of real polynomials in several indeterminates. Thus, the main example is

m = 〈 x_{1}, \dots, x_{n} 〉

being the maximal ideal of

ℝ [x 1, \dots, x n]

. Evidently,

ord (D r n) = r

and

w (D r n) = n

. As well, every other such an algebra

A

of the order

r

can be expressed in a form

where an ideal

i

satisﬁes

m^{r + 1} ⊊ i \subset m^{2}

and is generated by ﬁnite number of polynomials, i.e.

i = 〈 P_{1}, \dots, P_{l} 〉

. The fact

i \subset m^{2}

implies that the width of

A

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

where

j

is an ideal in

D r n

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 \in A

satisfying

φ (a) = a

for all

φ \in Aut A

. Let

be the set of all ﬁxed points of

A

. (We opine that nothing but ﬁxed point subalgebras

A^{G} = {a \in A; φ (a) = a for all φ \in G}

, where

G \subset Aut A

is a ﬁnite group of automorphisms, were studied in detail up to now. See e.g. [2].) It is clear, that

S A

is a subalgebra of

A

. As

(

n_{A}

being the ideal of nilpotent elements of

A

ℝ = ℝ \cdot 1 \subset S A

certainly holds, because every automorphism sends 1 into 1.

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

S A

of a Weil algebra

A

was determined in [5]. Although in the known geometrically motivated examples is usually

S A = ℝ

(such

S A

is called trivial), there are some algebras for which

S A ⊋ ℝ

and we suspect related bundles will have remarkably interesting geometry. Thus, the fundamental problem is a classiﬁcation of algebras having

S A

nontrivial. See [5], [6] for known results up to now. Especially, we underline the fact that

S A

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.

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

The set

G

of generators of

j

(the basis of

j

) can be multifarious. We refer to two important possibilities:

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}, \dots 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} + x y + x z, x y + y^{2} + y z 〉 .

Then

ℬ (A) = {1, x, y, z, x^{2}, x y, 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} x y + k_{7} y^{2} + k_{8} z^{2}

with the following multiplication table.

$\cdot$	$1$	$x$	$y$	$z$	$x^{2}$	$x y$	$y^{2}$	$z^{2}$

$1$	$1$	$x$	$y$	$z$	$x^{2}$	$x y$	$y^{2}$	$z^{2}$
$x$	$x$	$x^{2}$	$x y$	$- x^{2} - x y$	$0$	$0$	$0$	$0$
$y$	$y$	$x y$	$y^{2}$	$- y^{2} - x y$	$0$	$0$	$0$	$0$
$z$	$z$	$- x^{2} - x y$	$- y^{2} - x y$	$z^{2}$	$0$	$0$	$0$	$0$
$x^{2}$	$x^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$x y$	$x y$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$y^{2}$	$y^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$z^{2}$	$z^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Proof. Let

A = D_{2}^{5} ∕ 〈 x^{2} y^{2} + x^{5}, x^{2} y^{2} + y^{5} 〉 .

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

\begin{array}{rcl} 1st component \\ x & \mapsto & x + C_{1, 3} x^{2} + C_{1, 4} x y + C_{1, 6} x^{3} + C_{1, 7} x^{2} y + C_{1, 8} x y^{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} x y^{3} + C_{1, 14} y^{4} + C_{1, 15} x^{4} y + C_{1, 16} x y^{4} \\ y & \mapsto & y + C_{2, 4} x y + C_{2, 5} y^{2} + C_{2, 6} x^{3} + C_{2, 7} x^{2} y + C_{2, 8} x y^{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} x y^{3} + C_{2, 14} y^{4} + C_{2, 15} x^{4} y + C_{2, 16} x y^{4}; \\ 2nd component \\ x & \mapsto & y + C_{1, 4} x y + C_{1, 5} y^{2} + C_{1, 6} x^{3} + C_{1, 7} x^{2} y + C_{1, 8} x y^{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} x y^{3} + C_{1, 14} y^{4} + C_{1, 15} x^{4} y + C_{1, 16} x y^{4} \\ y & \mapsto & x + C_{2, 3} x^{2} + C_{2, 4} x y + C_{2, 6} x^{3} + C_{2, 7} x^{2} y + C_{2, 8} x y^{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} x y^{3} + C_{2, 14} y^{4} + C_{2, 15} x^{4} y + C_{2, 16} x y^{4}; \\ C_{i, j} \in ℝ . \end{array}

The elements of the form

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

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

2.2. Socle elements. If an element

a \in A

has the property

a u = 0

for all

u \in n_{A}

, we call

a

the socle element of

A

Let

a = \sum_{i} r_{i} B_{i}

be an element of

A

expressed as a linear combination of basis elements.

Proof. It is clear that if all $B_{i}$ are socle elements, then $a = \sum_{i} r_{i} B_{i}$ 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 $u \in n_{A}$ , $u = \sum_{j} {\bar{r}}_{j} {\bar{B}}_{j}$ ( ${\bar{B}}_{j} \in ℬ (A)$ ), such that $B_{1} u \neq 0$ . However, it implies there exists some ${\bar{B}}_{j}$ , say ${\bar{B}}_{1}$ for which $B_{1} {\bar{B}}_{1} \neq 0$ . We can write ${\bar{B}}_{1}$ as a monomial of basis elements from $n_{A} ∕ n_{A}^{2}$ , i.e. ${\bar{B}}_{1} = x_{n_{1}}^{p_{1}} \dots x_{n_{k}}^{p_{k}}$ , where $p_{1}, \dots, p_{k} \in ℕ$ . Thus $B_{1} x_{n_{1}} \neq 0$ , too. It means $a x_{n_{1}} \neq 0$ and $a$ is not a socle element; a contradiction. □

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 $a \in ℬ (A)$ 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 $a x_{i}$ 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}

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

form a subalgebra

M A

A

used e.g. in [4]. The problem of a relation between

S A

and

M A

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

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 \in Aut A

for which

{id}_{A} - φ

is a nilpotent endomorphism of

A

; alternatively, such

φ \in 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

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

\begin{array}{rcl} 1 & \mapsto & 1 \\ x_{1} & \mapsto & x_{1} + P_{1} \\ \dots \\ x_{n} & \mapsto & x_{n} + P_{n}, \end{array}

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

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:

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} ∕ 〈 x y^{2} + x^{5}, x^{2} y + y^{5} 〉 .

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

\begin{array}{rcl} 1st component \\ x & \mapsto & - x + C_{1, 3} x^{2} + C_{1, 4} x y + C_{1, 6} x^{3} + C_{1, 7} x^{2} y + C_{1, 8} x y^{2} + C_{1, 10} x^{4} + C_{1, 11} y^{4} \\ y & \mapsto & - y + C_{2, 4} x y + C_{2, 5} y^{2} + C_{2, 7} x^{2} y + C_{2, 8} x y^{2} + C_{2, 9} y^{3} + C_{2, 10} x^{4} + C_{2, 11} y^{4}; \\ 2nd component \\ x & \mapsto & x + C_{1, 3} x^{2} + C_{1, 4} x y + C_{1, 6} x^{3} + C_{1, 7} x^{2} y + C_{1, 8} x y^{2} + C_{1, 10} x^{4} + C_{1, 11} y^{4} \\ y & \mapsto & - y + C_{2, 4} x y + C_{2, 5} y^{2} + C_{2, 7} x^{2} y + C_{2, 8} x y^{2} + C_{2, 9} y^{3} + C_{2, 10} x^{4} + C_{2, 11} y^{4}; \\ 3rd component \\ x & \mapsto & - x + C_{1, 3} x^{2} + C_{1, 4} x y + C_{1, 6} x^{3} + C_{1, 7} x^{2} y + C_{1, 8} x y^{2} + C_{1, 10} x^{4} + C_{1, 11} y^{4} \\ y & \mapsto & y + C_{2, 4} x y + C_{2, 5} y^{2} + C_{2, 7} x^{2} y + C_{2, 8} x y^{2} + C_{2, 9} y^{3} + C_{2, 10} x^{4} + C_{2, 11} y^{4}; \\ 4th component \\ x & \mapsto & x + C_{1, 3} x^{2} + C_{1, 4} x y + C_{1, 6} x^{3} + C_{1, 7} x^{2} y + C_{1, 8} x y^{2} + C_{1, 10} x^{4} + C_{1, 11} y^{4} \\ y & \mapsto & y + C_{2, 4} x y + C_{2, 5} y^{2} + C_{2, 7} x^{2} y + C_{2, 8} x y^{2} + C_{2, 9} y^{3} + C_{2, 10} x^{4} + C_{2, 11} y^{4}; \\ 5th component \\ x & \mapsto & - y + C_{1, 4} x y + C_{1, 5} y^{2} + C_{1, 7} x^{2} y + C_{1, 8} x y^{2} + C_{1, 9} y^{3} + C_{1, 10} x^{4} + C_{1, 11} y^{4} \\ y & \mapsto & - x + C_{2, 3} x^{2} + C_{2, 4} x y + C_{2, 6} x^{3} + C_{2, 7} x^{2} y + C_{2, 8} x y^{2} + C_{2, 10} x^{4} + C_{2, 11} y^{4}; \\ 6th component \\ x & \mapsto & y + C_{1, 4} x y + C_{1, 5} y^{2} + C_{1, 7} x^{2} y + C_{1, 8} x y^{2} + C_{1, 9} y^{3} + C_{1, 10} x^{4} + C_{1, 11} y^{4} \\ y & \mapsto & - x + C_{2, 3} x^{2} + C_{2, 4} x y + C_{2, 6} x^{3} + C_{2, 7} x^{2} y + C_{2, 8} x y^{2} + C_{2, 10} x^{4} + C_{2, 11} y^{4}; \\ 7th component \\ x & \mapsto & - y + C_{1, 4} x y + C_{1, 5} y^{2} + C_{1, 7} x^{2} y + C_{1, 8} x y^{2} + C_{1, 9} y^{3} + C_{1, 10} x^{4} + C_{1, 11} y^{4} \\ y & \mapsto & x + C_{2, 3} x^{2} + C_{2, 4} x y + C_{2, 6} x^{3} + C_{2, 7} x^{2} y + C_{2, 8} x y^{2} + C_{2, 10} x^{4} + C_{2, 11} y^{4}; \\ 8th component \\ x & \mapsto & y + C_{1, 4} x y + C_{1, 5} y^{2} + C_{1, 7} x^{2} y + C_{1, 8} x y^{2} + C_{1, 9} y^{3} + C_{1, 10} x^{4} + C_{1, 11} y^{4} \\ y & \mapsto & x + C_{2, 3} x^{2} + C_{2, 4} x y + C_{2, 6} x^{3} + C_{2, 7} x^{2} y + C_{2, 8} x y^{2} + C_{2, 10} x^{4} + C_{2, 11} y^{4}; \\ C_{i, j} \in ℝ . \end{array}

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

of $A$ ( $k_{i} \in ℝ$ ) we ﬁnd $S A$ 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}, x y^{2} 〉 .

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

\begin{array}{rcl} x & \mapsto & C_{2, 2}^{4} x + C_{1, 3} x^{2} + C_{1, 4} x y + (C_{2, 2}^{6} - C_{2, 2}^{3}) y^{2} + C_{1, 6} x^{2} y + C_{1, 7} y^{3} \\ y & \mapsto & (C_{2, 2}^{4} - C_{2, 2}^{3}) x + C_{2, 2}^{3} y + C_{2, 3} x^{2} + C_{2, 4} x y + C_{2, 5} y^{2} + C_{2, 6} x^{2} y + C_{2, 7} y^{3} \\ C_{i, j} \in ℝ; C_{2, 2} \neq 0 . \end{array}

3. The order theorem

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 ${\hat{A}}_{4}$ is $ℬ ({\hat{A}}_{4}) = {1, x, y, x^{2}, x y, y^{2}, x^{2} y, x y^{2}}$ .

\begin{array}{rcl} 1st component \\ x & \mapsto & - ∣ C_{2, 2} ∣^{3} x + \frac{1}{128} (2 ∣ C_{2, 2} ∣^{3} - 80 C_{1, 4} + 3 C_{2, 2}^{4} - C_{2, 2}^{6} + 96 C_{2, 2}^{2} C_{2, 5} + \\ 48 ∣ C_{2, 2} ∣ (C_{1, 4} - 4 C_{2, 4} - 2 C_{2, 5})) x^{2} + C_{1, 4} x y - \frac{3}{8} (∣ C_{2, 2} ∣^{3} + C_{2, 2}^{4}) y^{2} + \\ C_{1, 6} x^{2} y + C_{1, 7} x y^{2} \\ y & \mapsto & - \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} x y + C_{2, 5} y^{2} + C_{2, 6} x^{2} y + \\ C_{2, 7} x y^{2}; \\ 2nd component \\ x & \mapsto & ∣ C_{2, 2} ∣^{3} x + \frac{1}{128} (- 2 ∣ C_{2, 2} ∣^{3} - 80 C_{1, 4} + 3 C_{2, 2}^{4} - C_{2, 2}^{6} + 96 C_{2, 2}^{2} C_{2, 5} - \\ 48 ∣ C_{2, 2} ∣ (C_{1, 4} - 4 C_{2, 4} - 2 C_{2, 5})) x^{2} + C_{1, 4} x y + \frac{3}{8} (∣ C_{2, 2} ∣^{3} - C_{2, 2}^{4}) y^{2} + \\ C_{1, 6} x^{2} y + C_{1, 7} x y^{2} \\ y & \mapsto & \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} x y + C_{2, 5} y^{2} + C_{2, 6} x^{2} y + \\ C_{2, 7} x y^{2}; \\ C_{i, j} \in ℝ . \end{array}

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

\begin{array}{rcl} x & \mapsto & - 8 x - 9 y^{2} \\ y & \mapsto & - 3 x + 4 y . \end{array}

ii) For $r \geq 5$ , polynomials $x^{r}$ , $x^{r - 1} y$ , $x^{r - 2} y^{2}$ , $x^{r - 1} + y^{r}$ , $x^{r - 2} + y^{r - 1}$ , $x^{r - 1} + x y^{r - 1}$ , $x^{r - 2} y + y^{r}$ are belonging to $j = 〈 x^{r - 2} + y^{r - 1}, x^{r - 1} + y^{r} 〉$ . The automorphisms have a form

\begin{array}{rcl} x & \mapsto & A x + B y + K (x, y) \\ y & \mapsto & C x + D y + L (x, y), \end{array}

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

C D^{r - 2} = 0,

looking at $x y^{r - 2}$ (this is obtained as the image of $y^{r - 1}$ only and the condition $r > 4$ is essential). It follows $C = 0$ (because $D \neq 0$ 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 $U_{{\hat{A}}_{r}} = Aut {\hat{A}}_{r}$ and, as to Corollary 1, $S {\hat{A}}_{r}$ is nontrivial. The dimension ${dim}_{ℝ} S {\hat{A}}_{r}$ must be greater or equal to the number of (linearly independent) basis elements of the socle. □

Proof. That was recalled in Introduction, that the fulﬁllment of $j \subset m^{2}$ in the expression $A = D r n ∕ j$ 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 $S A$ , cf. Introduction and [5]. □

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}, x y + z^{3}, y^{2} z + y z^{2} 〉

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

\begin{array}{rcl} 1st component \\ x & \mapsto & - x + C_{1, 4} x^{2} + C_{1, 5} x y + y^{2} + C_{1, 7} x z + C_{1, 10} x z^{2} + C_{1, 11} y z^{2} \\ y & \mapsto & y + C_{2, 4} x^{2} + C_{2, 5} x y + C_{2, 6} y^{2} + C_{2, 7} x z + C_{2, 8} y z + 3 C_{3, 1} z^{2} + \\ C_{2, 10} x z^{2} + C_{2, 11} y^{2} z \\ z & \mapsto & C_{3, 1} x - y - z + C_{3, 4} x^{2} + C_{3, 5} x y + C_{3, 6} y^{2} + C_{3, 7} x z + C_{3, 8} y z + C_{3, 9} z^{2} + \\ C_{3, 10} x z^{2} + C_{3, 11} y^{2} z; \\ 2nd component \\ x & \mapsto & x + C_{1, 4} x^{2} + C_{1, 5} x y + C_{1, 7} x z + C_{1, 10} x z^{2} + C_{1, 11} y z^{2} \\ y & \mapsto & y + C_{2, 4} x^{2} + C_{2, 5} x y + C_{2, 6} y^{2} + C_{2, 7} x z + C_{2, 8} y z - 3 C_{3, 1} z^{2} + C_{2, 10} x z^{2} + \\ C_{2, 11} y^{2} z \\ z & \mapsto & C_{3, 1} x + z + C_{3, 4} x^{2} + C_{3, 5} x y + C_{3, 6} y^{2} + C_{3, 7} x z + C_{3, 8} y z + \\ C_{3, 9} z^{2} + C_{3, 10} x z^{2} + C_{3, 11} y^{2} z; \\ C_{i, j} \in ℝ . \end{array}

The elements of the form

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

are ﬁxed for all $c_{1}, c_{2} \in ℝ$ . □

Proof. It is evident that $G_{A}$ for $A = D_{3}^{3} ∕ 〈 x^{2} + y^{3}, x y + z^{3}, y^{2} z + y z^{2} 〉$ has elements (with $C_{3, 1} \neq 0$ ) 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). □

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) \geq 4

. For all

w (A) > 2

, there exist algebras

A

with a nontrivial ﬁxed point subalgebra if and only if

ord (A) \geq 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 $S A = {c_{1} + c_{2} x^{2} y}$ was proved for $A = D_{2}^{4} ∕ 〈 x^{2} y + y^{4}, x^{3} + x y^{2} 〉$ in [6]. Finally, the case $w (A) > 2$ is solved by Proposition 4, Lemma 4 and Proposition 5 listed above. □

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

1. Guil-Asensio, F., Saorín, M., The group of automorphisms of a commutative algebra, Mathematische Zeitschrift 219, 1995, 31–48

2. Kharchenko, V.K., Automorphisms and Derivations of Associative Rings, Kluwer Academic Publishers, Dordrecht / Boston / London, 1991

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

David Sehnal, Faculty of Informatics, Masaryk University in Brno, Botanická 68a, 60200 Brno, Czech Republic