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Let $k$ be an algebraically closed field of characteristic different from 2. Up to isomorphism, the algebra $operatorname{Mat}_{n times n}(k)$ can be endowed with a $k$-linear involution in one way if $n$ is odd and in two ways if $n$ is even. In this paper, we consider $r$-tuples $A_bullet in operatorname{Mat}_{ntimes n}(k)^r$ such that the entries of $A_bullet$ fail to generate $operatorname{Mat}_{ntimes n}(k)$ as an algebra with involution. We show that the locus of such $r$-tuples forms a closed subvariety $Z(r;V)$ of $operatorname{Mat}_{ntimes n}(k)^r$ that is not irreducible. We describe the irreducible components and we calculate the dimension of the largest component of $Z(r;V)$ in all cases. This gives a numerical answer to the question of how generic it is for an $r$-tuple $(a_1, dots, a_r)$ of elements in $operatorname{Mat}_{ntimes n}(k)$ to generate it as an algebra with involution.
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We give a complete description of quadratic potential and twisted potential algebras on 3 generators as well as cubic potential and twisted potential algebras on 2 generators up to graded algebra isomorphisms under the assumption that the ground field is algebraically closed and has characteristic different from 2 or 3. We also prove that for two generated potential algebra necessary condition of finite-dimensionality is that potential contains terms of degree three, this answers a question of Agata Smoktunowicz and the first named author, formulated in [AN]. We clarify situation in case of arbitrary number of generators as well.
Theorem 7 in Ref. [Linear Algebra Appl., 430, 1-6, (2009)] states sufficient conditions to determine whether a pair generates the algebra of 3x3 matrices over an algebraically closed field of characteristic zero. In that case, an explicit basis for the full algebra is provided, which is composed of words of small length on such pair. However, we show that this theorem is wrong since it is based on the validity of an identity which is not true in general.
We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.
In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.
Kadeishvilis proof of the minimality theorem induces an algorithm for the inductive computation of an $A_infty$-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete $A_infty$-algebra structure after a finite amount of computational work.
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