We give an efficient solution to the following problem: Given X1,…Xd and Y some n by n matrices can we determine if Y is in the unital algebra generated by X1,…,Xd as a subalgebra of all n by n ...matrices? The solution also gives an easy method for computing the dimension of this algebra.
Let Mn(F) be the algebra of n×n matrices over an arbitrary field F. We consider linear maps Φ:Mn(F)→Mr(F) preserving matrices annihilated by a fixed polynomial f(x)=(x−a1)⋯(x−am) with m≥2 distinct ...zeroes a1,a2,…,am∈F; namely,f(Φ(A))=0wheneverf(A)=0.
Suppose that f(0)=0, and the zero set Z(f)={a1,…,am} is not an additive group. Then Φ assumes the form(†)A↦S(A⊗D1At⊗D20s)S−1, for some invertible matrix S∈Mr(F), invertible diagonal matrices D1∈Mp(F) and D2∈Mq(F), where s=r−np−nq≥0. The diagonal entries λ in D1 and D2, as well as 0 in the zero matrix 0s, are zero multipliers of f(x) in the sense that λZ(f)⊆Z(f).
In general, assume that Z(f)−a1 is not an additive group. If Φ(In) commutes with Φ(A) for all A∈Mn(F), or if f(x) has a unique zero multiplier λ=1, then Φ assumes the form (†).
The above assertions follow from the special case when f(x)=x(x−1)=x2−x, for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map Φ:Mn(F)→Mr(F) sending disjoint rank one idempotents to disjoint idempotents always assume the above form (†) with D1=Ip and D2=Iq, unless Mn(F)=M2(Z2).
An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and ...isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.
Gradings on block-triangular matrix algebras Diniz, Diogo; Galdino da Silva, José; Koshlukov, Plamen
Proceedings of the American Mathematical Society,
01/2024, Letnik:
152, Številka:
1
Journal Article
Recenzirano
Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial ...identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura Arch. Math. (Basel) 110 (2018), pp. 327–332 this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev Arch. Math. (Basel) 89 (2007), pp. 33–40.
In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some ...multilinear polynomial is a central polynomial, or a trace zero polynomial, and we use this approach to present an equivalent statement to the Lvov-Kaplansky conjecture.
The aim of the paper is to give a description of multiplicative Lie n-derivations for a certain class of generalized matrix algebras. As a consequence multiplicative Lie n-derivations of full matrix ...algebras are determined. This solves a conjecture due to Benkovič and Eremita in 2012.
In the present paper we study UT(D1,…,Dn), a G-graded algebra of block triangular matrices where G is a group and the diagonal blocks D1,…,Dn are graded division algebras. We prove that any two such ...algebras are G-isomorphic if and only if they satisfy the same graded polynomial identities. We also discuss the number of different isomorphism classes obtained by varying the grading and we exhibit its connection with the factorability of the T-ideal of graded identities. Moreover we give some results about the generators of the graded polynomial identities for these algebras. In particular we generalize the results about the graded identities of UTn to the case in which the diagonal blocks D1,…,Dn are all isomorphic.
Let F be an algebraically closed field of characteristic zero, and G be a finite abelian group. If Formula omitted. is a G-graded algebra, we study degree-inverting involutions on A, i.e. involutions ...Formula omitted. on A satisfying Formula omitted. , for all Formula omitted. . We describe such involutions for the full Formula omitted. matrix algebra over F and for the algebra of Formula omitted. upper triangular matrices.
Let F be a field of characteristic zero and p a prime. In the present paper it is proved that a variety of Zp-graded associative PI F-algebras of finite basic rank is minimal of fixed Zp-exponent d ...if, and only if, it is generated by an upper block triangular matrix algebra UTZp(A1,…,Am) equipped with a suitable elementary Zp-grading, whose diagonal blocks are isomorphic to Zp-graded simple algebras A1,…,Am satisfying dimF(A1⊕⋯⊕Am)=d.
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