Defining relation for semi-invariants of three by three matrix triples

The single defining relation of the algebra of $SL_3\times SL_3$-invariants of triples of $3\times 3$ matrices is explicitly computed. Connections to some other prominent algebras of invariants are pointed out.Comment: 12 page

Rationality of Hilbert series in noncommutative invariant theory

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tensor algebra that arise naturally in the theory of polynomial identities.Comment: Examples both from commutative and noncommutative invariant theory are included, a problem is formulated and references are added. Comments for v3: references added, minor revisio

Noether bound for invariants in relatively free algebras

Let $\mathfrak{R}$ be a weakly noetherian variety of unitary associative algebras (over a field $K$ of characteristic 0), i.e., every finitely generated algebra from $\mathfrak{R}$ satisfies the ascending chain condition for two-sided ideals. For a finite group $G$ and a $d$-dimensional $G$-module $V$ denote by $F({\mathfrak R},V)$ the relatively free algebra in $\mathfrak{R}$ of rank $d$ freely generated by the vector space $V$. It is proved that the subalgebra $F({\mathfrak R},V)^G$ of $G$-invariants is generated by elements of degree at most $b(\mathfrak{R},G)$ for some explicitly given number $b(\mathfrak{R},G)$ depending only on the variety $\mathfrak{R}$ and the group $G$ (but not on $V$). This generalizes the classical result of Emmy Noether stating that the algebra of commutative polynomial invariants $K[V]^G$ is generated by invariants of degree at most $\vert G\vert$

The strong anick conjecture is true

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K〉x, y, z〈 over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of K〈x, y, z〉. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of K〈x, y, z〉. We also find a large new class of wild automorphisms of K〈x, y, z〉 which is not covered by the results of Umirbaev. Finally, we study the lifting problem for automorphisms and coordinates of polynomial algebras, free metabelian algebras and free associative algebras and obtain some interesting new results. © European Mathematical Society 2007.postprin

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

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

Planar trees, free nonassociative algebras, invariants, and elliptic integrals

We consider absolutely free algebras with (maybe infinitely) many multilinear operations. Such multioperator algebras were introduced by Kurosh in 1960. Multioperator algebras satisfy the Nielsen-Schreier property and subalgebras of free algebras are also free. Free multioperator algebras are described in terms of labeled reduced planar rooted trees. This allows to apply combinatorial techniques to study their Hilbert series and the asymptotics of their coefficients. Then, over a field of characteristic 0, we investigate the subalgebras of invariants under the action of a linear group, their sets of free generators and their Hilbert series. It has turned out that, except in the trivial cases, the algebra of elliptic integrals. invariants is never finitely generated. In important partial cases the Hilbert series of the algebras of invariants and the generating functions of their sets of free generators are expressed in terms of elliptic integrals

A Central Polynomial of Low Degree for 4×4 Matrices

AbstractWe have found a central polynomial of degree 13 for the 4 × 4 matrix algebra over a field of characteristic 0. This result agrees with the conjecture that the minimal degree of such polynomials for n × n matrices is (n2 + 3n − 2)/2. The polynomial has been obtained by explicitly exhibiting an essentially weak polynomial identity of degree 9 for 4 × 4 matrices