Algebras Of Rank Research Articles

The modern theory of Cuntz semigroups of $C^{\ast}$-algebras

We give a detailed introduction to the theory of Cuntz semigroups for C^{\ast} -algebras. Beginning with the most basic definitions and technical lemmas, we present several results of historical importance, such as Cuntz’s theorem on the existence of quasitraces, Rørdam’s proof that \mathcal{Z} -stability implies strict comparison, and Toms’ example of a non \mathcal{Z} -stable simple, nuclear C^{\ast} -algebra. We also give the reader an extensive overview of the state of the art and the modern approach to the theory, including the recent results for C^{\ast} -algebras of stable rank one (for example, the Blackadar–Handelman conjecture and the realization of ranks), as well as the abstract study of the Cuntz category Cu .

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

Biderivations, commuting linear maps, post-Lie algebra structure on solvable Lie algebras of maximal rank

In this paper, all biderivations of solvable Lie algebras of maximal rank g = Q ⊕ N are characterized. Namely, we consider a solvable Lie algebra of the form g = Q ⊕ N , where Q is the maximal torus subalgebra of g , N is the nilradical of g and dim Q = dim N / N 2 .In the case dim Q = N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given.

The Racah Algebra of Rank 2: Properties, Symmetries and Representation

The goals of this paper are threefold. First, we provide a new ''universal'' definition for the Racah algebra of rank 2 as an extension of the rank-1 Racah algebra where the generators are indexed by subsets and any three disjoint indexing sets define a subalgebra isomorphic to the rank-1 case. With this definition, we explore some of the properties of the algebra including verifying that these natural assumptions are equivalent to other defining relations in the literature. Second, we look at the symmetries of the generators of the rank-2 Racah algebra. Those symmetries allows us to partially make abstraction of the choice of the generators and write relations and properties in a different format. Last, we provide a novel representation of the Racah algebra. This new representation requires only one generator to be diagonal and is based on an expansion of the split basis representation from the rank-1 Racah algebra.

Representations of shifted quantum affine algebras and cluster algebras I: The simply laced case

AbstractWe introduce a family of cluster algebras of infinite rank associated with root systems of type , , . We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called ‐system. We conjecture that all cluster monomials are classes of simple objects of . In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply connected algebraic group. This explains the similarity between ‐system relations and certain generalized minor identities discovered by Fomin and Zelevinsky.

Axial view on pseudo-composition algebras and train algebras of rank 3

We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The corresponding axial algebras are called [Formula: see text]-axial algebras, where [Formula: see text] is an element of the ground field. As a first step toward their classification, we describe the [Formula: see text]- and [Formula: see text]-generated subalgebras of such algebras.

Functional Train Algebras of Rank ≤ 3

Functional Train Algebras of Rank ≤ 3

Mutation invariants of cluster algebras of rank 2

We consider the mutation invariants of cluster algebras of rank 2. We characterize the mutation invariants of finite type. Two examples are provided for the affine type and we prove the non-existence of Laurent mutation invariants of non-affine type. As an application, a class of Diophantine equations encoded with cluster algebras are studied.

On unital absorbing extensions of C*-algebras of stable rank one and real rank zero

Suppose that B B is a separable stable C ∗ C^* -algebra with real rank zero, stable rank one and ( K 0 ( B ) , K 0 + ( B ) ) (\mathrm {K}_0(B), \mathrm {K}_0^+(B)) is weakly unperforated in the sense of Elliott [Internat. J. Math. 1 (1990), no. 4, pp. 361–380]. Let A A be a unital simple separable nuclear C ∗ \mathrm {C}^* -algebra. We show that B B has the corona factorization property and any unital extension of A A by B B is absorbing.

Free bicommutative superalgebras

We introduce the variety Bsup of bicommutative superalgebras over an arbitrary field of characteristic different from 2. The variety consists of all nonassociative Z2-graded algebras satisfying the polynomial super-identities of super- left- and right-commutativityx(yz)=(−1)x‾y‾y(xz) and (xy)z=(−1)y‾z‾(xz)y, where u‾∈{0,1} is the parity of the homogeneous element u.We present an explicit construction of the free bicommutative superalgebras, find their bases as vector spaces and show that they share many properties typical for ordinary bicommutative algebras and super-commutative associative superalgebras. In particular, in the case of free algebras of finite rank we compute the Hilbert series and find explicitly its coefficients. As a consequence we give a formula for the codimension sequence. We establish an analogue of the classical Hilbert Basissatz for two-sided ideals. We see that the Gröbner-Shirshov bases of these ideals are finite, the Gelfand-Kirillov dimensions of finitely generated bicommutative superalgebras are nonnegative integers and the Hilbert series of finitely generated graded bicommutative superalgebras are rational functions. Concerning problems studied in the theory of varieties of algebraic systems, we prove that the variety of bicommutative superalgebras satisfies the Specht property. In the case of characteristic 0 we compute the sequence of cocharacters.

Primitive elements of free non-associative algebras over finite fields

The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field.

Total Cuntz semigroup, extension, and Elliott Conjecture with real rank zero

AbstractIn this paper, we exhibit two unital, separable, nuclear ‐algebras of stable rank one and real rank zero with the same ordered scaled total K‐theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K‐theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of ‐algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the ‐algebras of stable rank one and real rank zero.

Ideal categories of the normed algebra of finite rank bounded operators on a Hilbert space

In this paper, we introduce the normal category [Formula: see text] [[Formula: see text]] of principal left [right] ideals of the normed algebra [Formula: see text] of all finite rank bounded operators on a Hilbert space [Formula: see text] and show that [Formula: see text] and [Formula: see text] are isomorphic using Hilbert space duality. We also prove that the semigroup of all normal cones in [Formula: see text] is isomorphic to the semigroup of all finite rank operators on [Formula: see text]. Further, we construct bounded normal cones in [Formula: see text] such that the set of all bounded normal cones in [Formula: see text] forms a normed algebra isomorphic to the normed algebra [Formula: see text].

Relative controllability of linear state-delay fractional systems

AbstractIn this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new $$\alpha $$ α -Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples.

On automorphisms of certain free nilpotent-by-abelian Lie algebras

For a positive integer [Formula: see text], let [Formula: see text] be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank n. We show that the subgroup of [Formula: see text] generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of [Formula: see text] is dense in [Formula: see text] with respect to the formal power series topology.

Annihilator Ideals of Indecomposable Modules of Finite-Dimensional Pointed Hopf Algebras of Rank One

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. We describe all annihilator ideals of indecomposable [Formula: see text]-modules by generators. In particular, we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.

On the accessibility and controllability of statistical linearization for stochastic control: Algebraic rank conditions and their genericity

On the accessibility and controllability of statistical linearization for stochastic control: Algebraic rank conditions and their genericity

Symmetric polynomials in free center-by-metabelian Lie algebras of rank 2

Let [Formula: see text] be the free center-by-metabelian Lie algebra of rank 2 freely generated by the set [Formula: see text]. Let [Formula: see text] be the Lie subalgebra of symmetric polynomials in [Formula: see text], that is, [Formula: see text] consists of all the elements [Formula: see text] in [Formula: see text] such that [Formula: see text]. We give a linear basis and a minimal infinite generating set for [Formula: see text], thus extending results of Ş. Findik and N. Ögüşlü [Palindromes in the free metabelian Lie algebras, Internat. J. Algebra Comput. 29(5) (2019) 885–891].

A Classification of 2-Dimensional Endo-Commutative Straight Algebras of Rank 1 over a non-Trivial Field

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of 2-dimensional endo-commutative straight algebras of rank one over an arbitrary non-trivial field, where a straight algebra of dimension 2 satisfies the condition that there exists an element x such that x and x2 are linearly independent. We list all multiplication tables of the algebras up to isomorphism.

DFT studies of 2D materials inspired by Lie algebras

Inspired by the root systems of Lie algebras of rank 2, we propose a mathematical method to engineer new 2D materials with double periodic structures tessellating the plane. Concretely, we investigate two geometries relaying on squares and hexagons exhibiting the D 4 × D 4 and D 6 × D 6 dihedral group invariances, respectively. Due to lack of empirical verifications of such double configurations, we provide a numerical investigation by help of the open source quantum espresso. Motivated by hybrid structures of the graphene, the silicene, and the germanene, we investigate two models involving the D 4 × D 4 and D 6 × D 6 dihedral symmetries which we refer to as Si4Ge4 and Si6C6 compounds, respectively. For simplicities, we study only the opto-electronic physical properties by applying an electromagnetic source propagating in linear and isotropic mediums. Among others, we find that such 2D materials exhibit metallic behaviors with certain optical features. Precisely, we compute and discuss the relavant optical quantities including the dielectric function, the absorption spectra, the refractive index, and the reflectivity. We believe that the Lie algebra inspiration of such 2D material studies, via density functional theory techniques, could open new roads to think about higher dimensional cases.