Linear maps preserving products equal to vertices of a path algebra

Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal p-sylow subgroup

In this paper, we characterize linear bijective maps $\varphi$ on the space of all $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ having the property that the spectrum of $\varphi (A)$ and $\varphi (B)$ have at least one common eigenvalue for each similar matrices $A$ and $B$. Using this result, we characterize linear bijective maps having the property that the spectrum of $\varphi (A)$ and $\varphi (B)$ have common elements for each matrices $A$ and $B$ having the same spectrum. As a corollary, we also characterize linear bijective maps preserving the equality of the spectrum.

In this paper we define the notion of ampleness for two-sided tilting complexes over finite dimensional algebras and prove its basic properties. We call a finite dimensional k-algebra A of finite global dimension Fano if (A∗[−d])−1 is ample for some d ≥ 0. For example geometric algebras in the sense of Bondal-Polishchuk are Fano. We give a characterization of representation type of a quiver from a noncommutative algebro-geometric view point, that is, a finite acyclic quiver has finite representation type if and only if its path algebra is fractional Calabi-Yau, and a finite acyclic quiver has infinite representation type if and only if its path algebra is Fano. 0 Introduction Let X be a nonsingular projective variety over a field k and let ωX be its canonical bundle. Then the functor SX := −⊗X ωX [dimX] : D(cohX) −→ D(cohX) is the Serre functor ,i.e., HomX(G ·,F ·)∗ is functorially isomorphic to HomX(F ·, SX(G ·)) for F ·,G · ∈ D(cohX). By this fact, from a noncommutative (or categorical) algebro-geometric view point, one thinks of a triangulated category T as the derived category of coherent sheaves of some ”space” X and of the Serre functor ST of T (if exists) as the derived tensor product of ” dimX”-shifted ”canonical bundle” ωX . From this view point, the notion of Calabi-Yau algebra ( and Calabi-Yau category ) is defined and studied extensively by many researchers. In this paper we introduce the notion of ampleness for two-sided tilting complexes over finite dimensional k-algebras. Let A be a finite dimensional k-algebra of finite global dimension. Definition 0.1 (Definition 2.6). A two-sided tilting complex σ over A is called very ample if H(σ) = 0 for i ≥ 1 and σ is pure for n A 0. σ is called ample if σ is pure for n A 0. In Section 2, we justify this definition by using the theory of noncommutative projective schemes due to Artin-Zhang [AZ] and Polishchuk [Po]. In the theory of noncommutative projective schemes , for a graded coherent ring R over k we attach an imaginary geometric object projR = ( cohprojR, R, (1) ) . An abelian category cohprojR is considered as the category of coherent sheaves on projR. (See Section 1.) In Section 2 we show that the following facts hold. If σ is a very ample tilting complex over A, then the tensor algebra T := TA(H (σ)) of H(σ) over A is a graded connected coherent ring over A and there is a t-structure D defined by σ in Perf A and its heart H is equivalent to cohprojT . Moreover the following Theorem holds. Theorem 0.2 (Theorem 2.8). Let A be a finite dimensional k-algebra of finite global dimension and let σ be a very ample two-sided tilting complex. Then there is a natural equivalence of triangulated categories D (mod-A) ∼ −→ D (cohprojT ) . where T := TA(H (σ)) is the tensor algebra of H(σ) over A.

ABSTRACTLet R be a ring and 𝒬 be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category Rep(𝒬,R) of representations of 𝒬 by left R-modules. We also extend our formula to all terms of the minimal injective resolution of R𝒬. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra R𝒬 is k-Gorenstein if and only if and R is a k-Gorenstein ring, where n is the number of vertices of 𝒬.

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

In 2003, Araujo and Jarosz showed that every bijective linear map θ : A → B between unital standard operator algebras preserving zero products in two ways is a scalar multiple of an inner automorphism. Later in 2007, Zhao and Hou showed that similar results hold if both A,B are unital standard algebras on Hilbert spaces and θ preserves range or domain orthogonality. In particular, such maps are automatically bounded. In this paper, we will study linear orthogonality preservers in a unified way. We will show that every surjective linear map between standard operator algebras preserving range/domain orthogonality carries a standard form, and is thus automatically bounded.

The high-SNR capacity of a block-faded non-coherent channel can be achieved by a multi-dimensional Grassmannian modulation. This paper proposes an analytical quasi-Gray labelling for Reed-Muller Grassmannian constellations. The proposed labelling method enables to reduce the block error rate (BLER) of a system with error correction code, by minimizing the average Hamming distance between labels of neighboring modulation symbols. In particular, we first define an inherent generation label from which the Reed-Muller Grassmannian constellation is constructed, and show it fulfills a homogeneity property. Based on this, a bijective linear mapping between a quasi-Gray labelling and the former labelling is proposed, where the bijective linear mapping matrix is obtained by a low-complexity algorithm. Numerical results show that the proposed quasi-Gray labelling method can achieve 40% reduction of the average Hamming distance between neighboring modulation symbols compared to the generation label or random labelling. Finally, link-level simulation results further demonstrate that the proposed quasi-Gray labelling can effectively reduce the BLER.

We characterize bijective linear maps on [Formula: see text] that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions and a map preserving square-zero matrices. Next, we consider bijective linear maps that preserve the square roots of a rank-one nilpotent matrix. These maps do not have standard forms when compared to similar linear preserver problems.

Let $$ \mathbb{F} $$ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and for the whole matrix space M n ( $$ \mathbb{F} $$ ). It is known that for n = 2, there are bijective linear maps Φ on $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and M n ( $$ \mathbb{F} $$ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ϕ), where Φ is an arbitrary bijective map on matrices and ϕ : $$ \mathbb{F} $$ → $$ \mathbb{F} $$ is an arbitrary map such that per A = ϕ(det Φ(A)) for all matrices A from the spaces $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and M n ( $$ \mathbb{F} $$ ), respectively. Moreover, for the space M n ( $$ \mathbb{F} $$ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $$ \mathbb{F} $$ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.

We prove that a unital, bijective linear map between absolute order unit spaces is an isometry if and only if it is absolute value preserving. We deduce that, on (unital) JB-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that a unital, bijective $$*$$ -linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) $$\hbox {C}^*$$ -algebras such maps are precisely $$\hbox {C}^*$$ -algebra isomorphisms.

Let V {\mathcal {V}} be an n n -dimensional complex linear space and L ( V ) {\mathcal {L}}({\mathcal {V}}) the algebra of all linear transformations on V {\mathcal {V}} . We prove that every linear map on L ( V ) {\mathcal {L}}({\mathcal {V}}) , which maps every operator into an operator with isomorphic lattice of invariant subspaces, is an inner automorphism or an inner antiautomorphism multiplied by a nonzero constant and additively perturbed by a scalar type operator. The same result holds if we replace the lattice of invariant subspaces by the lattice of hyperinvariant subspaces or the set of reducing subspaces. Some of these results are extended to linear transformations of finite-dimensional linear spaces over fields other than the complex numbers. We also characterize linear bijective maps on the algebra of linear bounded operators on an infinite-dimensional complex Hilbert space which have similar properties with respect to the lattice of all invariant subpaces (not necessarily closed).

For a natural number n≥2, denote by Mn the space of all n×n matrices over the complex field C. Let x0∈Cn be a fixed nonzero vector, and fix also two nonempty subsets K1,K2⊆C, each having at most n distinct elements. Under the assumption that |K1|≤|K2|, we characterize linear bijective maps φ on Mn having the property that, for each matrix T, we have that K2 is a subset of the local spectrum of φ(T) at x0 whenever K1 is a subset of the local spectrum of T at x0. As a corollary, we also characterize linear maps φ on Mn having the property that, for each matrix T, we have that K1 is a subset of the local spectrum of T at x0 if and only if K2 is a subset of the local spectrum of φ(T) at x0, without the bijectivity assumption on the map φ and with no assumption made regarding the number of elements of K1 and K2.

Linear maps preserving inclusion of fixed subsets into the spectrum

Geometrically, curves and surfaces are usually considered to be sets of points with some special properties, living in a space consisting of “points.” Typically, one is also interested in geometric properties invariant under certain transformations, for example, translations, rotations, projections, etc. One could model the space of points as a vector space, but this is not very satisfactory for a number of reasons. One reason is that the point corresponding to the zero vector (0), called the origin, plays a special role, when there is really no reason to have a privileged origin. Another reason is that certain notions, such as parallelism, are handled in an awkward manner. But the deeper reason is that vector spaces and affine spaces really have different geometries. The geometric properties of a vector space are invariant under the group of bijective linear maps, whereas the geometric properties of an affine space are invariant under the group of bijective affine maps, and these two groups are not isomorphic. Roughly speaking, there are more affine maps than linear maps.

On linear preservers of permanental rank