Block Triangular Matrix Algebras Research Articles

Counting gradings on block-triangular matrix algebras

Counting gradings on block-triangular matrix algebras

Gradings on block-triangular matrix algebras

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].

On the factorability of polynomial identities of upper block triangular matrix algebras graded by cyclic groups

Let F be an algebraically closed field of characteristic zero and G be an arbitrary finite cyclic group. In this paper, given an m-tuple (A1,…,Am) of finite dimensional G-simple algebras, we focus on the study of the factorability of the TG-ideals IdG((UT(A1,…,Am),α˜)) of the G-graded upper block triangular matrix algebras UT(A1,…,Am) endowed with elementary G-gradings induced by some maps α˜.When G is a cyclic p-group we prove that the factorability of the ideal IdG((UT(A1,…,Am),α˜) is equivalent to the G-regularity of all (except for at most one) the G-simple components A1,…,Am, as well to the existence of a unique isomorphism class of α˜-admissible elementary G-gradings for UT(A1,…,Am). Moreover, we present some necessary and sufficient conditions to the factorability of IdG((UT(A1,A2),α˜)), even in case G is not a p-group, with some stronger assumptions on the gradings of the algebras A1 and A2.

Graded involutions on block-triangular matrix algebras

In this paper we classify the ordinary and graded involutions on block-triangular matrix algebras over an algebraically closed field of characteristic ≠2.

Graded identities and isomorphisms on algebras of upper block-triangular matrices

Let G be an abelian group and K an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the G-gradings on upper block-triangular matrix algebras provided that G is finite. We prove that their result holds for any abelian group G: any grading is isomorphic to the tensor product A⊗B of an elementary grading A on an upper block-triangular matrix algebra and a division grading B on a matrix algebra. We then consider the question of whether graded identities A⊗B, where B is an algebra with a division grading, determine A⊗B up to graded isomorphism. In our main result, Theorem 3, we reduce this question to the case of elementary gradings on upper block-triangular matrix algebras which was previously studied by O.M. Di Vincenzo and E. Spinelli.

Graded isomorphisms on upper block triangular matrix algebras

We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence we describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of characteristic zero) graded by a finite abelian group.

Graded polynomial identities on upper block triangular matrix algebras

Let G be an arbitrary abelian group. In the present paper we study the question of whether G-graded upper block triangular matrix algebras over a field of characteristic zero are determined, up to G-graded isomorphism, by their G-graded polynomial identities. We obtain some results for an elementary grading and, as a consequence, for any grading over an algebraically closed field.

Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G n , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.

Abelian Gradings on Upper Block Triangular Matrices

AbstractLet G be an arbitrary finite abelian group. We describe all possible G-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

Rank-one non-increasing additive maps between block triangular matrix algebras

Let m and n be positive integers with n ≤ m. In this note, we characterize rank-one non-increasing additive maps ψ from n-square block triangular matrix algebras to m-square matrix algebras over an arbitrary field with the assumption that ψ(In ) is of rank n. We deduce from this result a complete classification of adjugate-commuting additive maps between block triangular matrix algebras. As a corollary, we characterize adjugate-commuting additive maps between square matrix algebras.

Additive rank-one preservers on block triangular matrix spaces

In this note, we address a description of the general form of additive rank-one preservers on block triangular matrix spaces over an arbitrary field. As an application, we characterize additive mappings preserving rank-additivity on block triangular matrix algebras.

Block-triangular matrix algebras and factorable ideals of graded polynomial identities

We study the graded polynomial identities of block-triangular matrix algebras with respect to the grading defined by an abelian group G. In particular, we describe conditions for the T G -ideal of a such algebra to be factorable as a product of T G -ideals corresponding to the algebras defining the diagonal blocks. Moreover, for the factorable T 2-ideal of a superalgebra we give a formula for computing its sequence of graded cocharacters once given the sequences of cocharacters of the T 2-ideals that factorize it. We finally apply these results to a specific example of block-triangular matrix superalgebra.

Asymptotics for the standard and the Capelli identities

Let {c n (St k )} and {c n (C k )} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold: $$\begin{gathered} c_n \left( {St_{2k} } \right) \simeq c_n \left( {C_{k^2 + 1} } \right) \simeq c_n \left( {M_k \left( F \right)} \right), \hfill \\ c_n \left( {St_{2k + 1} } \right) \simeq c_n \left( {M_{k \times 2k} \left( F \right) \oplus M_{2k \times k} \left( F \right)} \right), \hfill \\ \end{gathered} $$ whereM k (F) is the algebra ofk×k matrices andM k×l (F) is the algebra of (K+l)×(k+l) matrices having the lastl rows and the lastk columns equal to zero. The precise asymptotics ofc n (M k (F)) are known and those ofM k×2k (F) andM 2k×k (F) can be easily deduced. For Capelli polynomials we show that also upper block triangular matrix algebras come into play.

Isometries for Ky-Fan norms on block triangular matrix algebras

We characterize the linear isometries for Ky-Fan norms on the space of block triangular matrices.

Codimension growth and minimal superalgebras

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope G(A) of a finite dimensional superalgebra A. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: A is a minimal superalgebra if and only if the ideal of identities of G(A) is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties V such that exp(V) = d > 2 and exp(U) < d for all proper subvarieties U of V. This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003). As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.

Some linear preserver problems on block triangular matrix algebras

In this article we classify rank-one nonincreasing maps ψ on n-square block triangular matrix algebras with the assumption that ψ( I n ) is of rank n. As applications, we obtain complete classifications of adjugate-commuting maps, and compound-commuting maps on block triangular matrix algebras.