Inner automorphisms of Lie algebras related with generic 2 × 2 matrices

Let Fm = Fm(var(sl₂(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl₂(K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion Fmˆ of Fm with respect to the formal power series topology. Our results are more precise for m = 2 when F₂ is isomorphic to the Lie algebra L generated by two generic traceless 2×2 matrices. We give a complete description of the group of inner automorphisms of Lˆ. As a consequence we obtain similar results for the automorphisms of the relatively free algebra Fm / Fm c⁺¹ = Fm(var(sl₂(K)) ∩ Nc

Gr\"obner-Shirshov bases for LL-algebras

In this paper, we firstly establish Composition-Diamond lemma for $\Omega$-algebras. We give a Gr\"{o}bner-Shirshov basis of the free $L$-algebra as a quotient algebra of a free $\Omega$-algebra, and then the normal form of the free $L$-algebra is obtained. We secondly establish Composition-Diamond lemma for $L$-algebras. As applications, we give Gr\"{o}bner-Shirshov bases of the free dialgebra and the free product of two $L$-algebras, and then we show four embedding theorems of $L$-algebras: 1) Every countably generated $L$-algebra can be embedded into a two-generated $L$-algebra. 2) Every $L$-algebra can be embedded into a simple $L$-algebra. 3) Every countably generated $L$-algebra over a countable field can be embedded into a simple two-generated $L$-algebra. 4) Three arbitrary $L$-algebras $A$, $B$, $C$ over a field $k$ can be embedded into a simple $L$-algebra generated by $B$ and $C$ if $|k|\leq \dim(B*C)$ and $|A|\leq|B*C|$, where $B*C$ is the free product of $B$ and $C$.Comment: 22 page

On subgroups in division rings of type 22

Let $D$ be a division ring with center $F$. We say that $D$ is a {\em division ring of type $2$} if for every two elements $x, y\in D,$ the division subring $F(x, y)$ is a finite dimensional vector space over $F$. In this paper we investigate multiplicative subgroups in such a ring.Comment: 10 pages, 0 figure

Computing SL(2,C) Central Functions with Spin Networks

Let G=SL(2,C) and F_r be a rank r free group. Given an admissible weight in N^{3r-3}, there exists a class function defined on Hom(F_r,G) called a central function. We show that these functions admit a combinatorial description in terms of graphs called trace diagrams. We then describe two algorithms (implemented in Mathematica) to compute these functions.Comment: to appear in Geometriae Dedicat

Subexponential estimations in Shirshov's height theorem (in English)

In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that F_{2, m} is a 2-generated associative ring with the identity x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential growth?" We show that the nilpotency degree of l-generated associative algebra with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l (nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let l, n and d>n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Psi(n,d,l) are either n-divided or contain d-th power of subword, where a word W is n-divided, if it can be represented in the following form W=W_0 W_1...W_n such that W_1 >' W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov proved that the set of non n-divided words over alphabet of cardinality l has bounded height h over the set Y consisting of all the words of degree <n. Original Shirshov's estimation was just recursive, in 1982 double exponent was obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation. We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our proof uses Latyshev idea of Dilworth theorem application.Comment: 21 pages, Russian version of the article is located at the link arXiv:1101.4909; Sbornik: Mathematics, 203:4 (2012), 534 -- 55

Inner and outer automorphisms of relatively free lie algebras

Let Fm(var G) = Lm/I(G) be the relatively free Lie algebra of rank m in the variety of Lie algebras generated by a Lie algebra G over a field K of characteristic 0. We describe the groups of inner and outer automorphisms of the free metabelian nilpotent of class c algebra Lm/(L'' m + Lc+1 m) and the inner automorphisms of the relatively free algebra of rank 2 in the variety varsl2(K) ? n{fraktur}c. To obtain the results we first describe the group of inner automorphisms of the completion of the relatively free Lie algebras Lm/L'' m and F2(varsl2(K)) with respect to the formal power series topology. In the metabelian case we describe also the group of continuous outer automorphisms of the completion

The Nowicki conjecture for free metabelian Lie algebras

Let K[Xd] = K[x1,...,xd] be the polynomial algebra in d variables over a field K of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations (known as Weitzenböck derivations), the algebra of constants K[Xd] is finitely generated. When the Weitzenböck derivation acts on the polynomial algebra K[Xd,Yd] in 2d variables by (yi) = xi, (xi) = 0, i = 1,...,d, Nowicki conjectured that K[Xd,Yd] is generated by Xd and xiyj - yixj for all 1 ? i < j ? d. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free d-generated metabelian Lie algebra Fd, with few trivial exceptions, the algebra Fd is not finitely generated. However, the vector subspace (Fd') of the commutator ideal Fd' of Fd is finitely generated as a K[Xd]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the K[Xd,Yd]-module (F2d'). © 2020 World Scientific Publishing Company