Algebra Of Triangular Matrices Research Articles

Gradings on the algebra of triangular matrices as a Lie algebra: Revisited

We investigate the group gradings on the algebras of upper triangular matrices over an arbitrary field, viewed as Lie algebras. Classification results were obtained in 2017 by the same authors when the base field has characteristic different from 2. In this paper we provide streamlined proofs of these results. Moreover we present a complete classification of isomorphism classes of the group gradings on these algebras over an arbitrary field. Recall that two graded Lie algebras L1 and L2 are practically-isomorphic if there exists an (ungraded) algebra isomorphism L1→L2 that induces a graded-algebra isomorphism L1/z(L1)→L2/z(L2). We provide a classification of the practically-isomorphism classes of the group gradings on the Lie algebra of upper triangular matrices. The latter classification is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is 2, a topic that was neglected in previous works.

Products of infinite upper triangular quadratic matrices

Let F be a field and q(x) a quadratic polynomial in F[x] with q(0)≠0. We denote by T∞(F) the algebra of all infinite upper triangular matrices over the field F. A matrix A∈T∞(F) is called a quadratic matrix with respect to q(x) if q(A)=0. In this paper, we first investigate the subgroup in T∞(F) generated by all quadratic matrices with respect to q(x) and then present some applications.

A Novel Lie Hypergraph Based Lifetime Enhancement Routing Protocol for Environmental Monitoring in Wireless Sensor Networks

Wireless sensor network (WSN) is a rapid surging technology promising many fruitful innovations for the user community. The use of WSNs for continuous monitoring of environmental factors in severe situations, such as detection of volcanoes, forest fires, floods, and so on. Despite WSN’s various potential applications, energy and deployment are two major concerns influencing the network life span. The primary requirement is to monitor node energy consumption in order to improve network performance. Many approaches are proposed in recent times for energy efficiency, this article introduces an associated hypergraph with Lie algebra of upper triangular matrices for energy efficient clustering and routing. Initially, the sensor nodes are formulated as a hypergraph, and each hyperedge is treated as a cluster, from which cluster-head (CH) selection is exploited by minimum hypergraph transversal. The routing decision is employed by constructing the upper triangular matrix (UTM), and Lie commutators of the UTM Lie algebra identifies the best relay nodes for data forwarding. The performance of the proposed work is evaluated using simulation results which shows the effectiveness of the scheme over compared protocols in terms of, number of alive nodes, energy consumption and number of packets received in BS.

Lie triple centralizers of the algebra of dominant block upper triangular matrices

Lie triple centralizers of the algebra of dominant block upper triangular matrices

Graded identities with involution for the algebra of upper triangular matrices

Let F be a field of characteristic zero and let m≥2 be an integer. In this paper, we prove that if a group grading on UTm(F) admits a graded involution then this grading is a coarsening of a Z⌊m2⌋-grading on UTm(F) and the graded involution is equivalent to the reflection or symplectic involution on UTm(F), this grading is called the finest grading on UTm(F). Furthermore, if m≤4 the algebra UTm(F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z⌊m2⌋,⁎)-identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z⌊m2⌋,⁎)-codimensions is determined. As a consequence, we prove that for any G-grading on UTm(F) and any graded involution the (G,⁎)-exponent is m. Finally, we study the algebra UT3(F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z-grading and the Z2-grading induced by (0,1,0). We determine a basis for the (Z2,⁎)-identities and we compute the codimension sequence for the (Z2,⁎)-graded identities for UT3(F).

Transposed Poisson structures on the Lie algebra of upper triangular matrices

We describe transposed Poisson structures on the upper triangular matrix Lie algebra T_{n}(F) , n>1 , over a field F of characteristic zero. We prove that, for n>2 , any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for n=2 , there is one more class of transposed Poisson structures on T_{n}(F) . We also show that, up to isomorphism, the full matrix Lie algebra M_{n}(F) admits only one non-trivial transposed Poisson structure, and it is of Poisson type.

Lie hypergraph and chaos‐based privacy preserving protocol for wireless sensor networks in IoT environment

SummaryWireless sensor networks (WSN) are innately resource restrained and beneficial in a wide range of applications, including smart homes, e‐health care, law, military systems, disaster management, and emergency reprieve. These applications are linked to various devices that may communicate with one another through the internet, typically known as Internet of Things (IoT). The application of WSN plays an integral role in the IoT infrastructure. In a WSN, sensors are haphazardly placed in environments where the data transmission is challenged by privacy concerns. This paper proposes a methodology termed Lie hypergraph and chaos‐based secure routing (LH‐CSR) to perform an energy‐efficient routing with secure data transmission in WSN. In the first phase, the deployed sensor nodes are transformed into a hypergraph from which cluster head (CH) is elected by hypergraph transversal property, and for secure routing, the route is formed by the Lie commutators of the Lie algebra of upper triangular matrices. The second phase emphasizes privacy preservation by introducing the novel chaotic map formulation to process the key generation. The encryption and decryption processes are maintained by key generation to prevent data loss during retrieval. In this way, the data are retained confidentially with minimal computational overhead. The performance of the LH‐CSR is evaluated through a simulation, which shows that it outperforms over compared protocols in terms of cryptographic time, network lifetime, packet delivery ratio, end‐to‐end delay, and throughput.

ω-Circulant Matrices: A Selection of Modern Applications from Preconditioning of Approximated PDEs to Subdivision Schemes

It is well known that ω-circulant matrices with ω≠0 can be simultaneously diagonalized by a transform matrix, which can be factored as the product of a diagonal matrix, depending on ω, and of the unitary matrix Fn associated to the Fast Fourier Transform. Hence, all the sets of ω-circulants form algebras whose computational power, in terms of complexity, is the same as the classical circulants with ω=1. However, stability is a delicate issue, since the condition number of the transform is equal to that of the diagonal part, tending to max{|ω|,|ω|−1}. For ω=0, the set of related matrices is still an algebra, which is the algebra of lower triangular matrices, but they do not admit a common transform since most of them (all except the multiples of the identity) are non-diagonalizable. In the present work, we review two modern applications, ranging from parallel computing in preconditioning of PDE approximations to algorithms for subdivision schemes, and we emphasize the role of such algebra. For the two problems, few numerical tests are conducted and critically discussed and the related conclusions are drawn.

Upper triangular matrices of graded division algebras and their identities

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.

Graded polynomial identities for the Lie algebra of upper triangular matrices of order 3

We compute the graded polynomial identities and its graded codimension sequence for the elementary gradings of the Lie algebra of upper triangular matrices of order 3.

Y-proper graded cocharacters of the algebra UTm(F) of m × m upper triangular matrices over F

Let F be a field of characteristic 0 and let UTm(F) be the algebra of upper triangular matrices of order m with entries from F. In this paper we give a description of the Y-proper graded cocharacters of UTm(F) equipped with any elementary G-grading, where G is a finite group.

Images of multilinear graded polynomials on upper triangular matrix algebras

AbstractIn this paper, we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices $UT_n$ . For positive integers $q\leq n$ , we classify these images on $UT_{n}$ endowed with a particular elementary ${\mathbb {Z}}_{q}$ -grading. As a consequence, we obtain the images of multilinear graded polynomials on $UT_{n}$ with the natural ${\mathbb {Z}}_{n}$ -grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras $UT_{2}$ and $UT_{3}$ , for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra $UJ_{2}$ , and also for $UJ_{3}$ endowed with the natural elementary ${\mathbb {Z}}_{3}$ -grading.

Specht property for the algebra of upper triangular matrices of size two with a Taft’s algebra action

AbstractLet F be a field of characteristic zero, and let $UT_2$ be the algebra of $2 \times 2$ upper triangular matrices over F. In a previous paper by Centrone and Yasumura, the authors give a description of the action of Taft’s algebras $H_m$ on $UT_2$ and its $H_m$ -identities. In this paper, we give a complete description of the space of multilinear $H_m$ -identities in the language of Young diagrams through the representation theory of the hyperoctahedral group. We finally prove that the variety of $H_m$ -module algebras generated by $UT_2$ has the Specht property, i.e., every $T^{H_m}$ -ideal containing the $H_m$ -identities of $UT_2$ is finitely based.

Strongly Ad‐Nilpotent Elements of the Lie Algebra of Upper Triangular Matrices

In this paper, the strongly ad‐nilpotent elements of the Lie algebra of upper triangular complex matrices are studied. We prove that all the nilpotent matrices in are strongly ad‐nilpotent if and only if n ≤ 6. Additionally, we prove that all the elements exp(ad x), x strongly ad‐nilpotent generate the inner automorphism group Int t(n, ℂ).

τ-tilting finite triangular matrix algebras

First, we give a new example of silting-discrete algebras . Second, one explores when the algebra of triangular matrices over a finite dimensional algebra is τ -tilting finite. In particular, we classify algebras over which triangular matrix algebras are τ -tilting finite. Finally, we investigate when a triangular matrix algebra is silting-discrete.

Linear Lie centralizers of the algebra of dominant block upper triangular matrices

Let be the algebra of all dominant block upper triangular matrices over a field. In this paper, we explicitly describe all linear Lie centralizers of . We also describe linear Lie centralizers of the algebra of block upper triangular matrices over a field.

Local centrally essential subalgebras of triangular algebras

ABSTRACT We study local centrally essential subalgebras in the algebra of all upper triangular matrices over a field of characteristic . It is proved that the algebras of upper triangular or matrices have only commutative local centrally essential subalgebras. Every algebra of upper triangular matrices of order exceeding 6 contains a non-commutative local centrally essential subalgebra.

Central polynomials with involution for the algebra of 2 × 2 upper triangular matrices

Let F be a field of characteristic different from 2, and let be the algebra of upper triangular matrices over F. For every involution of the first kind on , we describe the set of all -central polynomials for this algebra.

Lie Triple Derivations of the Lie Algebra of Dominant Block Upper Triangular Matrices

Let [Formula: see text] be the Lie algebra of all n×n dominant block upper triangular matrices over a field 𝔽. In this paper, we explicitly describe all Lie triple derivations of [Formula: see text] when char(𝔽) ≠ 2. As an application, we characterize Lie derivations of [Formula: see text] when char(𝔽) ≠ 2.

Modular Hecke algebras over Möbius categories

We extend the modular Hecke operators of Connes and Moscovici by taking values in the modular incidence algebra M [ C ] over a Möbius category C . Here M is the ‘modular tower’ consisting of all modular forms of all weights across all levels. We construct multiple product structures on the collection A ( Γ ) [ C ] of modular Hecke operators of level Γ over C , where Γ is a principal congruence subgroup. These product structures are then shown to be well behaved with respect to Hopf actions on A ( Γ ) [ C ] . While A ( Γ ) [ C ] already carries an action of the Hopf algebra H 1 of codimension 1-foliations, the noncommutativity of the modular incidence algebra M [ C ] allows us to construct additional operators on A ( Γ ) [ C ] . The use of Möbius categories provides a single framework for describing modular Hecke operators taking values in various rings: from formal power series rings over M to arithmetic functions over M and algebras of upper triangular matrices with entries in M . Moreover, we use functors between Möbius categories to study relations between various modular Hecke algebras.