Algebra Of Upper 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.

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.

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.

Involutions in Algebras of Upper-Triangular Matrices

Involutions in Algebras of Upper-Triangular Matrices

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).

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.

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.

Maximal and Minimal Triangular Matrices

We call a nonscalar matrix maximal (or minimal) if its centralizer is maximal (respectively minimal) in the poset of all centralizers of matrices. We discuss the form of maximal and minimal matrices in the algebra of upper triangular matrices.

Group gradings on the Lie and Jordan algebras of upper triangular matrices

Let K be a field and let $$UT_n=UT_n(K)$$ denote the associative algebra of upper triangular $$n\times n$$ matrices over K. The vector space of $$UT_n$$ can be given the structure of a Lie and of a Jordan algebra, respectively, by means of the new products: $$[a,b]=ab-ba$$ , and $$a\circ b= ab+ba$$ . We denote the corresponding Lie and Jordan algebra by $$UT_n^-$$ and by $$UT_n^+$$ , respectively. If G is a group, the G-gradings on $$UT_n$$ were described by Valenti and Zaicev (Arch Math 89(1):33–40, 2007); they proved that each grading on $$UT_n$$ is isomorphic to an elementary grading (that is every matrix unit is homogeneous). Also Di Vincenzo et al. (J Algebra 275(2):550–566, 2004) classified all elementary gradings on $$UT_n$$ . Here we study the gradings and the graded identities on $$UT_n^-$$ and on $$UT_n^+$$ , based on Koshlukov and Yukihide (J Algebra 473:66–79, 2017, J Algebra 477:294–311, 2017, Linear Algebra Appl, 2017). It turns out that the Lie and the Jordan cases are similar (though the methods used bear not much resemblance), and in turn, quite different from the associative case. We prove that, up to isomorphism, there are two kinds of gradings on $$UT_n^-$$ and on $$UT_n^+$$ : the elementary ones, and the so-called mirror type gradings. We classify all these gradings in the Lie and in the Jordan cases. Moreover we show that the gradings are completely determined by the graded identities they satisfy.

Group gradings on the Jordan algebra of upper triangular matrices

Let G be an arbitrary group and let K be a field of characteristic different from 2. We classify the G-gradings on the Jordan algebra UJn of upper triangular matrices of order n over K. It turns out that there are, up to a graded isomorphism, two families of gradings: the elementary gradings (analogous to the ones in the associative case), and the so called mirror type (MT) gradings. Moreover we prove that the G-gradings on UJn are uniquely determined, up to a graded isomorphism, by the graded identities they satisfy.

Degree-inverting involutions on matrix algebras

Let F be an algebraically closed field of characteristic zero, and G be a finite abelian group. If is a G-graded algebra, we study degree-inverting involutions on A, i.e. involutions on A satisfying , for all . We describe such involutions for the full matrix algebra over F and for the algebra of upper triangular matrices.

Group gradings on the Lie algebra of upper triangular matrices

The algebras UTn of the n×n upper triangular matrices over a field K are of significant importance in the theory of algebras with polynomial identities. Group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question. We classify the group gradings on the Lie algebra UTn(−). It was proved by Valenti and Zaicev in 2007 that every group grading on the associative algebra UTn is isomorphic to an elementary grading. The elementary gradings on UTn are also well understood, see [6]. It follows from our results that there are nonelementary gradings on UTn(−). Thus the gradings on the Lie algebra UTn(−) are much more intricate than those in the associative case.

Orthogonality graph of the algebra of upper triangular matrices

We study the connectedness and the diameter of orthogonality graphs of upper triangular matrix algebras over arbitrary fields.