Alexei Krasilnikov

The additive group of a Lie nilpotent associative ring

Abstract Let Z 〈 X 〉 be the free unital associative ring freely generated by an infinite countable set X = { x 1 , x 2 , … } . Define a left-normed commutator [ x 1 , x 2 , … , x n ] by [ a , b ] = a b − b a , [ a , b , c ] = [ [ a , b ] , c ] . For n ⩾ 2 , let T ( n ) be the two-sided ideal in Z 〈 X 〉 generated by all commutators [ a 1 , a 2 , … , a n ] ( a i ∈ Z 〈 X 〉 ) . It can be easily seen that the additive group of the quotient ring Z 〈 X 〉 / T ( 2 ) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z 〈 X 〉 / T ( 3 ) is also free abelian. In the present note we show that this is not the case for Z 〈 X 〉 / T ( 4 ) . More precisely, let T ( 3 , 2 ) be the ideal in Z 〈 X 〉 generated by T ( 4 ) together with all elements [ a 1 , a 2 , a 3 ] [ a 4 , a 5 ] ( a i ∈ Z 〈 X 〉 ) . We prove that T ( 3 , 2 ) / T ( 4 ) is a non-trivial elementary abelian 3-group and the additive group of Z 〈 X 〉 / T ( 3 , 2 ) is free abelian.

Relations in universal Lie nilpotent associative algebras of class 4

ABSTRACT Let K be a unital associative and commutative ring and let K⟨X⟩ be the free unital associative K-algebra on a non-empty set X of free generators. Define a left-normed commutator inductively by , . For n≥2, let T(n) be the two-sided ideal in K⟨X⟩ generated by all commutators . It can be easily seen that the ideal T(2) is generated (as a two-sided ideal in K⟨X⟩) by the commutators . It is well known that T(3) is generated by the polynomials and . A similar generating set for T(4) contains 3 types of polynomials in xi∈X if and 5 types if . In the present article, we exhibit a generating set for T(5) that contains 8 types of polynomials in xi∈X.

The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3

Abstract Let Z 〈 X 〉 be the free unital associative ring freely generated by an infinite countable set X = { x 1 , x 2 , … } . Define a left-normed commutator [ a 1 , a 2 , … , a n ] inductively by [ a , b ] = a b − b a , [ a 1 , a 2 , … , a n ] = [ [ a 1 , … , a n − 1 ] , a n ] ( n ≥ 3 ). For n ≥ 2 , let T ( n ) be the two-sided ideal in Z 〈 X 〉 generated by all commutators [ a 1 , a 2 , … , a n ] ( a i ∈ Z 〈 X 〉 ). Let T ( 3 , 2 ) be the two-sided ideal of the ring Z 〈 X 〉 generated by all elements [ a 1 , a 2 , a 3 , a 4 ] and [ a 1 , a 2 ] [ a 3 , a 4 , a 5 ] ( a i ∈ Z 〈 X 〉 ). It has been recently proved in [22] that the additive group of Z 〈 X 〉 / T ( 4 ) is a direct sum A ⊕ B where A is a free abelian group isomorphic to the additive group of Z 〈 X 〉 / T ( 3 , 2 ) and B = T ( 3 , 2 ) / T ( 4 ) is an elementary abelian 3-group. A basis of the free abelian summand A was described explicitly in [22] . The aim of the present article is to find a basis of the elementary abelian 3-group B.

Products of commutators in a Lie nilpotent associative algebra

Abstract Let F be a field and let F 〈 X 〉 be the free unital associative algebra over F freely generated by an infinite countable set X = { x 1 , x 2 , … } . Define a left-normed commutator [ a 1 , a 2 , … , a n ] recursively by [ a 1 , a 2 ] = a 1 a 2 − a 2 a 1 , [ a 1 , … , a n − 1 , a n ] = [ [ a 1 , … , a n − 1 ] , a n ] ( n ≥ 3 ). For n ≥ 2 , let T ( n ) be the two-sided ideal in F 〈 X 〉 generated by all commutators [ a 1 , a 2 , … , a n ] ( a i ∈ F 〈 X 〉 ). Let F be a field of characteristic 0. In 2008 Etingof, Kim and Ma conjectured that T ( m ) T ( n ) ⊂ T ( m + n − 1 ) if and only if m or n is odd. In 2010 Bapat and Jordan confirmed the “if” direction of the conjecture: if at least one of the numbers m, n is odd then T ( m ) T ( n ) ⊂ T ( m + n − 1 ) . The aim of the present note is to confirm the “only if” direction of the conjecture. We prove that if m = 2 m ′ and n = 2 n ′ are even then T ( m ) T ( n ) ⊈ T ( m + n − 1 ) . Our result is valid over any field F.

A JUST NON-FINITELY BASED VARIETY OF BIGROUPS

A bigroup is a pair (H, π) consisting of a group H and an idempotent endomorphism π of H. One can consider π as a unary operation on H so a bigroup is a universal algebra. The aim of our paper is to construct the first example of a just non-finitely based variety of bigroups i.e. a variety which is non-finitely based but all whose proper subvarieties are finitely based. There is a close similarity between varieties of bigroups and varieties of groups so we hope that our result could help to construct a just non-finitely based variety of groups. *The first author was supported by NSERC, Canada.

Limit T-subspaces and the central polynomials in n variables of the Grassmann algebra

Abstract Let F 〈 X 〉 be the free unitary associative algebra over a field F on the set X = { x 1 , x 2 , … } . A vector subspace V of F 〈 X 〉 is called a T-subspace (or a T-space) if V is closed under all endomorphisms of F 〈 X 〉 . A T-subspace V in F 〈 X 〉 is limit if every larger T-subspace W ≩ V is finitely generated (as a T-subspace) but V itself is not. Recently Brandao Jr., Koshlukov, Krasilnikov and Silva have proved that over an infinite field F of characteristic p > 2 the T-subspace C ( G ) of the central polynomials of the infinite dimensional Grassmann algebra G is a limit T-subspace. They conjectured that this limit T-subspace in F 〈 X 〉 is unique, that is, there are no limit T-subspaces in F 〈 X 〉 other than C ( G ) . In the present article we prove that this is not the case. We construct infinitely many limit T-subspaces R k ( k ⩾ 1 ) in the algebra F 〈 X 〉 over an infinite field F of characteristic p > 2 . For each k ⩾ 1 , the limit T-subspace R k arises from the central polynomials in 2k variables of the Grassmann algebra G.

A SIMPLE EXAMPLE OF A NON-FINITELY BASED SYSTEM OF POLYNOMIAL IDENTITIES

ABSTRACT A system of polynomial identities is called finitely based if it is equivalent to some finite system of polynomial identities. Every system of polynomial identities in associative algebras over a field of characteristic 0 is finitely based: this is a celebrated result of Kemer. The first non-finitely based systems of polynomial identities in associative algebras over a field of a prime characteristic have been constructed recently by Belov, Grishin and Shchigolev. These systems of identities are relatively complicated. In the present paper we construct a simpler example of such a system and give a simple self-contained proof of the fact that the system is non-finitely based.

Products of several commutators in a Lie nilpotent associative algebra

Let F be a field of characteristic ≠2, 3 and let A be a unital associative F-algebra. Define a left-normed commutator [a1,a2,…,an] (ai ∈ A) recursively by [a1,a2] = a1a2 − a2a1, [a1,…,an−1,an] = [[a1,…,an−1],an] (n ≥ 3). For n ≥ 2, let T(n)(A) be the two-sided ideal in A generated by all commutators [a1,a2,…,an] (ai ∈ A). Define T(1)(A) = A. Let k,l be integers such that k > 0, 0 ≤ l ≤ k. Let m1,…,mk be positive integers such that l of them are odd and k − l of them are even. Let Nk,l =∑i=1km i − 2k + l + 2. The aim of the present note is to show that, for any positive integers m1,…,mk, in general, T(m1)(A)…T(mk)(A) ⊈ T(1+Nk,l)(A). It is known that if l < k (that is, if at least one of mi is even), then T(m1)(A)…T(mk)(A) ⊆ T(Nk,l)(A) for each A so our result cannot be improved if l < k. Let Nk =∑i=1km i − k + 1. Recently, Dangovski has proved that if m1,…,mk are any positive integers then, in general, T(m1)(A)…T(mk)(A) ⊈ T(1+Nk)(A). Since Nk,l = Nk − (k − l − 1), Dangovski’s result is stronger than ours i...

Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

AbstractLet K be a field and let ℕ = {1, 2, . . .} be the set of all positive integers. Let Rn = K[xij| 1 ≤ i ≤ n, j ∈ ℕ] be the ring of polynomials in xij (1 ≤ i ≤ n, j ∈ ℕ) over K. Let Sn = Sym({1, 2, . . . , n}) and Sym(ℕ) be the groups of permutations of the sets {1, 2, . . . , n} and ℕ, respectively. Then Sn and Sym(ℕ) act on Rn in a natural way: τ (xij) = xτ(i)j and σ(xij) = xiσ(j), for all i ∈ {1, 2, . . . , n} and j ∈ ℕ, τ ∈ Sn and σ ∈ Sym(ℕ). Let R¯