Let $\mathbb{K}$ be a field of characteristic zero and $B=B_0+B_1$ a finite dimensional associative superalgebra. In this paper we investigate the polynomial identities of the relatively free algebras of finite rank of the variety $\mathfrak V$ defined by the Grassmann envelope of $B$. We also consider the $k$-th Grassmann Envelope of $B$, $G^{(k)}(B)$, constructed with the $k$-generated Grassmann algebra, instead of the infinite dimensional Grassmann algebra. We specialize our studies for the algebra $UT_2(G)$ and $UT_2(G^{(k)})$, which can be seen as the Grassmann envelope and $k$-th Grassmann envelope, respectively, of the superalgebra $UT_2(\mathbb{K}[u])$, where $u^2=1$.
Let $n$ and $s$ be fixed integers such that $n\geq 2$ and $1\leq s\leq \frac{n}{2}$. Let $M_n(\mathbb{K})$ be the ring of all $n\times n$ matrices over a field $\mathbb{K}$. If a map $δ:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$ satisfies that $δ(xy)=δ(x)y+xδ(y)$ for any two rank-$s$ matrices $x,y\in M_n(\mathbb{K})$, then there exists a derivation $D$ of $M_n(\mathbb{K})$ such that $δ(x)=D(x)$ holds for each rank-$k$ matrix $x\in M_n(\mathbb{K})$ with $0\leq k\leq s$.
In this paper we show there are no classical type simple Lie 2-algebras with toral rank odd and we also show that the simple contragredient Lie 2-algebra $G(F_{4, a})$ of dimension 34 has toral rank 4, and we give the Cartan decomposition of $G(F_{4, a})$.
In this article, we introduce Cohn path algebras of higher-rank graphs. We prove that for a higher-rank graph $Λ$, there exists a higher-rank graph $TΛ$ such that the Cohn path algebra of $Λ$ is isomorphic to the Kumjian-Pask algebra of $TΛ$. We then use this isomorphism and properties of Kumjian-Pask algebras to study Cohn path algebras. This includes proving a uniqueness theorem for Cohn path algebras.
We find a basis of the free Malcev algebra on three free generators over a field of characteristic zero. The specialty and semiprimity of this algebra are proved. In addition, we prove the decomposability of this algebra into subdirect sum of the free Lie algebra rank three and the free algebra of rank three of variety of Malcev algebras generated by a simple seven-dimensional Malcev algebra.
In this paper, we study a class of infinitesimal deformations of the centerless higher rank Heisenberg-Virasoro algebras. Explicitly, the universal central extensions, derivations and isomorphism classes of these algebras are determined.
In the present paper we prove the existence outer derivations finite-dimensional solvable Lie algebras with nilradical of maximal rank and complementary subspace to nilardical of dimension less than rank of the nilradical.
We investigate the behavior of finitely generated projective modules over a down-up algebra. Specifically, we show that every noetherian down-up algebra $A(α,β,γ)$ has a non-free, stably free right ideal. Further, we compute the stable rank of these algebras using Stafford's Stable Range Theorem and Kmax dimension.
We construct an analogue of the Nagata automorphism of free nonassociative algebras and free commutative algebras of rank two over a Euclidean domain and prove that it is wild.
We prove that the group of tame automorphisms of a free Lie algebra (as well as of a free anticommutative algebra) rank 3 over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra (as well as of a free anticommutative algebra) of rank 3 over an arbitrary Euclidean ring analogous to the Anick automorphism for free associative algebras.
We study relatively free associative algebras $F^{(n)}_r$ of ranks $r=2,3$ with the identity $[x_1,\dots, x_n]=0$ of Lie nilpotency of step $n\geqslant 3$ over a field $K$ of characteristic $\neq 2,3$. First we prove a Theorem on the inclusion $ T^{(m)}T^{(n)}\subseteq T^{(m+n-1)}$ for an associative algebra $A$ of rank $3$, where $T^{(n)}=T^{(n)}(A)$ is a T-ideal of $A$ generated by the commutator $[x_1,\dots, x_n]$; the restriction on rank is essential. Further, we describe 3-variable identities of the algebra $F^{(n)}$. In particular, the obtained description implies that $Z(F^{(n)}_r)=(T^{…
We show that $E$ is a finite graph with no sinks if and only if the Leavitt path algebra $L_R(E)$ is isomorphic to an algebraic Cuntz-Krieger algebra if and only if the $C^*$-algebra $C^*(E)$ is unital and $rank(K_0(C^*(E)))=rank(K_1(C^*(E)))$. When $k$ is a field and $rank(k^{\times})< \infty$, we show that the Leavitt path algebra $L_k(E)$ is isomorphic to an algebraic Cuntz-Krieger algebra if and only if $L_k(E)$ is unital and $rank(K_1(L_k(E)))=(rank(k^{\times})+1)rank(K_0(L_k(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cu…
Simple Lie algebras of finite dimension over an algebraically closed field of characteristic 0 or $p> 3$ were recently classified. However, the problem over an algebraically closed field of characteristics 2 or 3 there exist only partial results. The first result on the problem of classification of simple Lie algebra of finite dimension over an algebraically closed field of characteristic 2 is that these algebras have absolute toral rank greater than or equal to 2. In this paper we show that there are not simple Lie 2-algebras with toral rank 3 over an algebraically closed field of characteris…
A linear locally nilpotent derivation of the polynomial algebra $K[X_m]$ in $m$ variables over a field $K$ of characteristic 0 is called a Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants $K[X_{m}]^δ$ of a Weitzenböck derivation $δ$ is finitely generated. Assume that $δ$ acts on the polynomial algebra $K[X_{2d}]$ in $2d$ variables as follows: $δ(x_{2i})=x_{2i-1}$, $δ(x_{2i-1})=0$, $i=1,\ldots,d$. The Nowicki conjecture states that the algebra $K[X_{2d}]^δ$ is generated by $x_1,x_3.\ldots,x_{2d-1}$, and $x_{2i-1}x_{2j}-x_{2i}x_{2j-…
In order to be able to use methods of Universal Algebra for investigating posets, we assign to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset a certain algebra (based on a commutative directoid or on a lambda-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. It turns out that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such pose…
We classify all graded compatible left-symmetric algebraic structures on high rank Witt algebras, and classify all non-graded ones satisfying a minor condition. Furthermore, graded compatible left-symmetric algebraic structures on high rank Virasoro algebras are also classified.
In this short note, we study the rank of a restricted Lie algebra $(\mathfrak{g},[p])$ and give some applications, which concerns the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra $K(n)$.
Milliet asks the following question: given two prime numbers $p\neq q$, is there a division algebra of characteristic $p$ which is of dp-rank $q^2$ and of dimension $q^2$ over its center? We answer in the affirmative. We also give an example of a finite burden central division algebra over some ultraproduct of $p$-adic numbers. As a conclusion we revisit an example of Albert to prove that there exists non-cyclic division algebras of finite dp-rank.
The additive primitive length of an element $f$ of a relatively free algebra $F_d$ in a variety of algebras is equal to the minimal number $\ell$ such that $f$ can be presented as a sum of $\ell$ primitive elements. We give an upper bound for the additive primitive length of the elements in the $d$-generated polynomial algebra over a field of characteristic 0, $d>1$. The bound depends on $d$ and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free $d$-generated nilpotent-by-abelian Lie algebras is bounded by 5 for $d=3$ …
Motivated by the structure of certain modules over the loop Virasoro Lie conformal algebra and the Lie structures of Schrodinger-Virasoro algebras, we construct a class of infinite rank Lie conformal algebras CSV (a, b), where a, b are complex numbers. The conformal derivations of CSV (a, b) are uniformly determined. The rank one conformal modules and Z-graded free intermediate series modules over CSV (a, b) are classified. Corresponding results of the conformal subalgebra CHV (a, b) of CSV (a, b) are also presented.
We study algebraic properties of full rank 1 algebras in a general framework and derive a method to verify if one such matrix polynomial sub-algebra is bispectral. We give two examples illustrating the method. In the first one, we consider the eigenvalue to be scalar-valued, whereas, in the second one, we assume it to be matrix-valued. In the former example, we put forth a Pierce decomposition of that algebra.
