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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Eggert's Conjecture for 2-Generated Nilpotent Algebras

Miroslav Korbelar*

Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovska, Czech Republic

Corresponding Author:
Miroslav Korbelar
Faculty of Mathematics and Physics
Department of Algebra, Charles University
Sokolovska 83, 186 75 Prague 8, Czech Republic
Tel: +420 224 491 111
E-mail: [email protected]

Received date: July 24, 2015 Accepted date: August 03, 2015 Published date: August 31, 2015

Citation: Korbelar M (2015) Eggert’s Conjecture for 2-Generated Nilpotent Algebras. J Generalized Lie Theory Appl S1:001. doi:10.4172/1736-4337.S1-001

Copyright: © 2015 Korbelar M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Let A be a commutative nilpotent finitely-dimensional algebra over a field F of characteristic p > 0. A conjecture of Eggert says that p. dim A(p) dim A, where A(p) is the subalgebra of A generated by elements ap , a ∈ A. We show that the conjecture holds if A(p) is at most 2-generated.

Keywords

Nilpotent algebra; Eggert's conjecture; Commutative nilpotent ring; Polynomial bases

Introduction

Let F be a field of characteristic p>0 and A a commutative (associative) nilpotent finite-dimensional algebra over F . Let A(p) be the subalgebra generated by the set {ap| a ∈ A}. N. Eggert [1] conjectured that

p· dim A(p) ≤ dim A.

This conjecture gives an answer to the problem, when a finite abelian group is isomorphic to the adjoint group of some finite commutative nilpotent F-algebra. Recall that the adjoint group of A is the set A with the operation x○ y=x + y + x y for every x, y ∈ A.

Validity of this hypothesis would also have influence on an estimation of a (Prüfer) rank of a product of two (abelian) p-groups.

N. Eggert proved his conjecture only when dim A(p)≤ 2. Five years later, R. Bautista [2] proved it when dim A(p)= 3. C. Stack confirmed this results in Stack et al. [3,4], but provided shorter proofs. Finally, Amberg and Kazarin [5] proved the conjecture for the case dim A(p) ≤ 4.

Another type of results presented by McLean [6,7]. He showed that this conjecture is true if the algebra A is either radical of a group algebra of a finite abelian group or A is graded and at least one of the following conditions is fulfilled:

(i) p = 2 and (A(p))4 = 0.

(ii) A(p) is 2-generated.

(iii) (A(p))3 = 0.

(iv) n < 3p and 3 ≤ s - 1 ≤ p, where n is the number of generators of A(p) and s is the index of nilpotence of A(p).

We also should mention the result of Gorlov [8]. He proved the conjecture for nilpotent algebras A with a metacyclic adjoint group.

One paper concerning Eggert's conjecture appeared in 2002 and the author L. Hammoudi [9] claimed he proved it. But, as Amberg [10] and McLean [7] have shown, his proof was incorrect.

In this short note we sketch out the main steps of the proof that Eggert's conjecture is true if the subalgebra A(p) has at most two generators. For the details, the reader is referred to Korbelar [11].

Since we will deal with nilpotency and commutativity only, we point out that the word 'algebra' will mean a commutative one and not necessary possesing a unit.

For an algebra A and a subset X ⊆ A we denote ([X], resp.) the algebra (vector space, resp.) generated by X.

An algebra A is called nilpotent if Am=0 for some m ∈ N.

Through this paper let always F be a field of characteristic p > 0 and R = F [x, y] be the ring of polynomials over the variables x, y and the field F.

We start with the remark, that the number of any minimal generating set of a finite generated nilpotent F -algebra A is equal to dim A/A2. This implies the following:

Lemma 1.1. Suppose that Eggert's conjecture holds for every nilpotent 2-generated F -algebra. Then it also holds for every nilpotent F -algebra A such that A(p) is a 2-generated F -algebra.

In the rest we deal with 2-generated nilpotent algebras.

Bases of Nilpotent Algebras

We will use the well-known concept of monomial ordering and standard bases.

For image put

image

Denote image the multiplicative monoid with the lexicographical ordering ≤ such that

image

and

x(i,j) ≤ 0

for everyimage

For image

image

Finally, f will be called normal iff λα0 = 1, where m(f) = xα 0, and m(f) < π xα implies λα = 0 for every

image

This function m: F [x, y] → [X]0 has common properties of a valuation:

(i) m(fg) = m(f) m(g).

(ii) m(f + g) ≥ min{m(f); m(g) g. Moreover, m(f + g) = m(f) if m(f) < m(g).

(iii) m (f(xp, yp)) = m(f)p.

for every f, g∈ F [x, y].

Finally. a set image will be called upper (lower, resp.) if image implies image

Definition 2.1. Let A be a nilpotent F -algebra generated by {a1, a2}. Put

image

and

Proposition 2.2. Let A be a nilpotent F -algebra generated by {a1, a2}. Then

(i)image

(ii) image is a lower set and 1∈ image

(iii)The set image is a basis of A. In particular, image is finite.

(iv) image

Definition 2.3. Let A be a nilpotent F -algebra generated by {a1, a2}. Denote

image

and

image

for image

Lemma 2.4. Let A be a nilpotent F -algebra generated by {a1, a2}. Then:

image

Eggert's Conjecture for 2-generated Algebras

Let I ⊆ Rx + Ry be an ideal in R such that A = Rx + Ry/I is a nonzero nilpotent F -algebra.

We have A = image and A(p) =image

By definition of image there are image such that mimageare normal.

The main idea of the proof lies in the fact that taking a normal polynomial from I, dividing it by x and then multiplying by some suitable yk, we get again a member of I (3.3). Then, using binomial formula in a suitable way, we obtain a polynomial that will estimate the number image (see 3.4 and the definition of BA(p)(a1p,a2p)

Lemma 3.1. image

image

Definition 3.2. Denote

wA = max BAA(x + I, y + I).

Forimage denote

image

the least integer such that image Put

image

Following lemma is obtained using induction.

Lemma 3.3. Let 1≤ i ≤ n0 + 1 and 0 ≠ f ∈ I be such that m(f) xi. Thenimage

The proof of the next proposition uses only the binomial formula. It finds the particular polynomial the we need to make an estimation of the numbers Di and thus of the dimension of A(p).

Proposition 3.4.

image

Now, only exploring carefully the previous cases for i and li we get the following interesting claim. It says that the inequality image holds for almost every i.

Theorem 3.5. One of the following cases takes place:

image

And our main result is just an easy corollary of this and 1.1.

Theorem 3.6. Let A be a nilpotent F -algebra, char F=p>0, such that A(p) is 2-generated. Then p·dim A(p) dim A.

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