Related papers: Monomial algebras defined by Lyndon words

Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially…

Nichols algebras naturally appear in the classification of finite dimensional pointed Hopf algebras. Assuming only that the base field has characteristic zero several new finite dimensional rank 2 Nichols algebras of diagonal type are…

We study quadratic algebras over a field $\textbf{k}$. We show that an $n$-generated PBW algebra $A$ has finite global dimension and polynomial growth \emph{iff} its Hilbert series is $H_A(z)= 1 /(1-z)^n$. Surprising amount can be said when…

We mainly study the growth and Gelfand-Kirillov dimension (GK-dimension) of generalized Weyl algebra (GWA) $A=D(\sigma,a)$ where $D$ is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for…

Lie algebra $\mathfrak{sl}(2)$ can be realised by vector fields on $\mathbb{R}^1\ni x$ with polynomial coefficients $1$, $-2x$, $-x^2$; their Wronskian determinants yield the Lie bracket. Likewise, the monomials $1$, $\ldots$, $x^k/k!$,…

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over…

Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive…

We investigate the class of finite dimensional not necessary associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is…

We construct finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (super-polynomial) growth types. This (partially) answers a question raised by the author in a recent…

Let $W$ be a $G$-graded algebra over a field of characteristic zero, where $G$ is a finite group. We develope a theory of generalized $G$-graded polynomial identities satisfied by any finite-dimensional $W$-algebra $A$, by mean of the…

Suppose L is any finite algebraic extension of either the ordinary rational numbers or the p-adic rational numbers. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms…

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological…

Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~))$-type, then $A$…

We study prime monomial algebras. Our main result is that a prime finitely presented monomial algebra is either primitive or it has GK dimension one and satisfies a polynomial identity. More generally, we show this result holds for the…

In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincar\'e algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the…

Let W_n(K) be the Lie algebra of derivations of the polynomial algebra K[X]:=K[x_1,...,x_n] over an algebraically closed field K of characteristic zero. A subalgebra L of W_n(K) is called polynomial if it is a submodule of the K[X]-module…

Let $A$ be a $W$-algebra over a field $F$ of characteristic zero, where $W$ is any $F$-algebra. We first develop a comprehensive theory of generalized identities independent of the algebraic structure of $W$, using the multiplier algebra of…

Let $R$ be a semiartinian (von Neumann) regular ring with primitive factors artinian. The dimension sequence $\mathcal D _R$ is an invariant that captures the various skew-fields and dimensions occurring in the layers of the socle sequence…

An algebra $\A$ is said to be an independence algebra if it is a matroid algebra and every map $\al:X\to A$, defined on a basis $X$ of $\A$, can be extended to an endomorphism of $\A$. These algebras are particularly well behaved…

Smoktunowicz, Lenagan, and the second-named author recently gave an example of a nil algebra of Gelfand-Kirillov dimension at most three. Their construction requires a countable base field, however. We show that for any field $k$ and any…