Maximal Commutative Subalgebras of Steinberg Algebras
We construct large classes of maximal commutative subalgebras in prime Steinberg algebras, generalizing a known result for Leavitt path algebras.
math.RA↗
arXiv subjects
Michal Ziembowski
Publications and source records attributed to Michal Ziembowski.
The structure of subalgebras of full matrix algebras over a field satisfying the identity [x_1, y_1][x_2, y_2] ... [x_q, y_q] = 0
A subalgebra of the full matrix algebra Mn(K), K a field, satisfying the identity [x1, y1][x2, y2]...[xq, yq] = 0 is called a Dq subalgebra of Mn(K). In the paper we deal with the structure, conjugation and isomorphism problems of maximal Dq subalgebras of Mn(K). We show that a maximal Dq subalgebra A of Mn(K) is conjugated with a block triangular subalgebra of Mn(K) with maximal commutative diagonal blocks. By analysis of conjugations, the sizes of the obtained diagonal blocks are uniquely determined. It reduces the problem of conjugation of maximal Dq subalgebras of Mn(K) to the analogous problem in the class of commutative subalgebras of Mn(K). Further examining conjugations, in case A is contained in the upper triangular matrix algebra Un(K), we prove that A is already in a block triangular form. We consider the isomorphism problem in a certain class of maximal Dq subalgebras of Mn(K) which contain all Dq subalgebras of Mn(K) with maximum dimension. In case K is algebraically closed, we invoke Jacobson's characterization of maximal commutative subalgebras of Mn(K) with maximum (K-)dimension to show that isomorphic subalgebras in this class are already conjugated. To illustrate it, we invoke results from [17] and find all isomorphism (equivalently conjugation) classes of Dq subalgebras of Mn(K) with maximum possible dimension, in case K is algebraically closed.
math.RA↗
Lie properties in associative algebras
Let K be a field, then we exhibit two matrices in the full nxn matrix algebra M_{n}(K) which generate M_{n}(K) as a Lie K-algebra with the commutator Lie product. We also study Lie centralizers of a not necessarily commutative unitary algebra and obtain results which we hope will eventually be a step in the direction of, firstly, proving that a Lie-nilpotent K-subspace (or a sub Lie K-algebra) of a finite-dimensional associative algebra over K of index k (say) generates a Lie-nilpotent associative subalgebra of much higher nilpotency index, and secondly, in the light of the sharp upper bound for the maximum (K-)dimension of a Lie-nilpotent K-subalgebra of M_{n}(K) of index k (obtained earlier), finding an upper bound for the maximum dimension of a Lie-nilpotent (of index k) sub Lie K-algebra of M_{n}(K). Finally, the constructive elementary proof of the Skolem-Noether theorem for the matrix algebra M_{n}(K) (appeared in the American Math. Monthly), in conjunction with the well-known characteization of Lie automorphisms of M_{n}(K) (if the characteristic of K is different from 2 and 3) in terms of, amongst others, automorphisms and anti-automorhisms of M_{n}(K), leads us to a unifying approach to constructively describe automorphisms and anti-automorphisms of M_{n}(K).
math.RA↗
On Radicals of Ore Extensions and Related Questions
We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If $R$ is prime radical and $δ$ is a derivation of $R$, then the differential polynomial ring $R[X;δ]$ is locally nilpotent. This answers an open question posed in by Nielsen and Ziembowski. The nil radical of a differential polynomial ring $R[X;δ]$ takes the form $I[X;δ]$ for some ideal $I$ of $R$, provided that the base field is infinite. This answers an open question posed by Hong, Kim, Lee and Nielsen for algebras over infinite fields. If $R$ is a graded algebra generated in degree $1$ over a field of characteristic zero and $δ$ is a grading preserving derivation on $R$, then the Jacobson radical of $R$ is $δ$-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp. Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent. The power series ring $R[[X;σ,δ]]$ is well-defined whenever $δ$ is a locally nilpotent $σ$-derivation; this answers a conjecture by Bergen and Grzeszczuk and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on $δ$.
math.RA↗
Differential poynomial rings over locally nilpotent rings need not be Jacobson radical
We answer a question by Shestakov on the Jacobson radical in differential polynomial rings. We show that if R is a locally nilpotent ring with a derivation D then R[X;D] need not be Jacobson radical. We also show that J(R[X;D])\cap R is a nil ideal of R in the case where D is a locally nilpotent derivation and R is an algebra over an uncountable field.
math.RA↗