Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

Let L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L.

A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions

We generalize Contou-Carrère symbols to higher dimensions. To an ( n + 1 ) (n+1) -tuple f 0 , … , f n ∈ A ( ( t 1 ) ) ⋯ ( ( t n ) ) × f_0,\dots ,f_n \in A((t_1))\cdots ((t_n))^{\times } , where A A denotes an algebra over a field k k , we associate an element ( f 0 , … , f n ) ∈ A × (f_0,\dots ,f_n) \in A^{\times } , extending the higher tame symbol for k = A k = A , and earlier constructions for n = 1 n = 1 by Contou-Carrère, and n = 2 n = 2 by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic K K -theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols.

Symmetric polynomials of algebras related with 2 × 2 generic traceless matrices

Let [Formula: see text] and [Formula: see text] be two generic traceless matrices of size [Formula: see text] with entries from a commutative associative polynomial algebra over a field [Formula: see text] of characteristic zero. Consider the associative unitary algebra [Formula: see text], and its Lie subalgebra [Formula: see text] generated by [Formula: see text] and [Formula: see text] over the field [Formula: see text]. It is well known that the center [Formula: see text] of [Formula: see text] is the polynomial algebra generated by the algebraically independent commuting elements [Formula: see text], [Formula: see text], [Formula: see text]. We call a polynomial [Formula: see text] symmetric, if [Formula: see text]. The set of symmetric polynomials is equal to the algebra [Formula: see text] of invariants of symmetric group [Formula: see text]. Similarly, we define the Lie algebra [Formula: see text] of symmetric polynomials in the Lie algebra [Formula: see text]. We give the description of the algebras [Formula: see text] and [Formula: see text], and we provide finite sets of free generators for [Formula: see text], and [Formula: see text] as [Formula: see text]-modules.

Weak polynomial identities and their applications

Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

Growth functions of lie algebras associated with associative algebras

Let [Formula: see text] be a finitely generated associative algebra over a field [Formula: see text] of characteristic [Formula: see text] and let [Formula: see text] be its associated Lie algebra. In this paper, we investigate relations between the growth functions of [Formula: see text] and the Lie algebra [Formula: see text]. We prove that if A is generated by a finite collection of nilpotent elements, then the growth functions are asymptotically equivalent.

On the Dimension of Non-Abelian Tensor Squares of $n$-Lie Algebras

Let $L$ be an $n$-Lie algebra over a field $\F$. In this paper, we introduce the notion of non-abelian tensor square $L\otimes L$ of $L$ and define the central ideal $L\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\square L\cong L^{ab}\square L^{ab}$. Also, we establish the short exact sequence\[0\lra\M(L)\lra\frac{L\otimes L}{L\square L}\lra L^2\lra0\]and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.

On the Lie structure of locally matrix algebras

Let $A$ be a unital locally matrix algebra over a field $\mathbb{F}$ of characteristic different from $2.$ We find a necessary and sufficient condition for the Lie algebra $A\diagup\mathbb{F}\cdot 1$ to be simple and for the Lie algebra of derivations $\text{Der}(A)$ to be topologically simple. The condition depends on the Steinitz number of $A$ only.