Algebras Of Rank Research Articles

Integral structures in extended affine Lie algebras

We construct certain integral structures for the cores of reduced tame extended affine Lie algebras of rank at least 2. One of the main tools to achieve this is a generalization of Chevalley automorphisms in the context of extended affine Lie algebras. As an application, groups of extended affine Lie type associated to the adjoint representation are defined over arbitrary fields.

C*-algebras of stable rank one and their Cuntz semigroups

The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: we answer affirmatively, for the class of stable rank-one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the global Glimm halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one.

R-train algebras of rank 3, type (4, 2) with dim U2 = dim UV = 1 and U3 = 0

R-train algebras of rank 3, type (4, 2) with dim U2 = dim UV = 1 and U3 = 0

Partial Menger algebras and their weakly isomorphic representation

As generalization of semigroups, Karl Menger introduced in the 1940th algebras of multiplace operations. Such algebras satisfy the superassociative law, a generalization of the associative law. Menger algebras are defined as models of this superassociative law. Cayley’s theorem for semigroups says that any model of the associative law is isomorphic to a transformation semigroup. R. M. Dicker proved in 1963 that every Menger algebra of rank [Formula: see text] is isomorphic to a Menger algebra of [Formula: see text]-ary operations on some set. The composition of terms in which each variable occurs at most [Formula: see text]-times, so-called [Formula: see text]-terms, leads to partial algebras where the superassociative law is satisfied as a weak identity. In this paper, we introduce the concepts of a partial Menger algebra, a unitary partial Menger algebra and of a generalized partial Menger algebra. We prove that [Formula: see text]-terms of some type form a generalized partial Menger algebra with infinitely many nullary operations. Using weak identities and weak isomorphisms, Dicker’s result will be extended to partial Menger algebras and to unitary partial Menger algebras.

Symmetric polynomials in the free metabelian associative algebra of rank 2

Let $F$ be the free metabelian associative algebra generated by $x$ and $y$ over a field of characteristic zero. We call a polynomial $f\in F$ symmetric, if $f(x,y)=f(y,x)$. The set of all symmetric polynomials coincides with the algebra $F^{S_2}$ of invariants of the symmetric group $S_2$. In this paper, we give the full description of the algebra $F^{S_2}$.

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

AbstractWe discuss the$\ell $-adic case of Mazur’s ‘Program B’ over$\mathbb {Q}$: the problem of classifying the possible images of$\ell $-adic Galois representations attached to elliptic curvesEover$\mathbb {Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes$\ell =2$and$\ell \ge 13$are addressed by prior work, so we focus on the remaining primes$\ell = 3, 5, 7, 11$. For each of these$\ell $, we compute the directed graph of arithmetically maximal$\ell $-power level modular curves$X_H$, compute explicit equations for all but three of them and classify the rational points on all of them except$X_{\mathrm {ns}}^{+}(N)$, for$N = 27, 25, 49, 121$and two-level$49$curves of genus$9$whose Jacobians have analytic rank$9$.Aside from the$\ell $-adic images that are known to arise for infinitely many${\overline {\mathbb {Q}}}$-isomorphism classes of elliptic curves$E/\mathbb {Q}$, we find only 22 exceptional images that arise for any prime$\ell $and any$E/\mathbb {Q}$without complex multiplication; these exceptional images are realised by 20 non-CM rationalj-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on$X_{\mathrm {ns}}^+(\ell )$with$\ell \ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the$\ell $-adic images of Galois for any elliptic curve over$\mathbb {Q}$.In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on$\Gamma _1(N)$are of$\operatorname {GL}_2$-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of$X_H$.

Deformed Double Current Algebras via Deligne Categories

Abstract In this paper, we give an alternative construction of a certain class of deformed double current algebras. These algebras are deformations of $ U(\textrm {End}(\Bbbk ^r)[x,y]) $ and they were initially defined and studied by N. Guay in his papers. Here, we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank.

Control sets for bilinear and affine systems

For homogeneous bilinear control systems, the control sets are characterized using a Lie algebra rank condition for the induced systems on projective space. This is based on a classical Diophantine approximation result. For affine control systems, the control sets around the equilibria for constant controls are characterized with particular attention to the question when the control sets are unbounded.

Structure of finite dimensional exact estimation algebra on state dimension 3 and linear rank 2*

The estimation algebra plays an important role in classification of finite dimensional filters. When finite dimensional estimation algebra has maximal rank, Yau et al. [Yau (2003). Complete classification of finite-dimensional estimation algebras of maximal rank. International Journal of Control, 76(7), 657–677; Yau & Hu (2005). Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state-space dimension and Mitter conjecture. International Journal of Control, 78(10), 689–705.] have proved that η must be a degree 2 polynomial. In this paper, we study the structure of finite dimensional exact estimation algebra with state dimension 3 and rank 2. We establish a sufficient and necessary condition for estimation algebra with nonmaximal rank to be finite dimensional. Importantly, in the new filtering system, η needs not to be a degree 2 polynomial and can be of any degree . It is the first time to systematically analyse nonmaximal rank exact estimation algebra in which η is a polynomial of any degree . For Riccati-type equation, estimates have been done from the viewpoints of both classical solution and weak solution respectively. Finally, finite dimensional filters of Benés type are constructed successfully.

V-Regular Ternary Menger Algebras and Left Translations of Ternary Menger Algebras

Let n be a fixed natural number. Ternary Menger algebras of rank n, which was established by the authors, can be regarded as a suitable generalization of ternary semigroups. In this article, we introduce the notion of v-regular ternary Menger algebras of rank n, which can be considered as a generalization of regular ternary semigroups. Moreover, we investigate some of its interesting properties. Based on the concept of n-place functions (n-ary operations), these lead us to construct ternary Menger algebras of rank n of all full n-place functions. Finally, we study a special class of full n-place functions, the so-called left translations. In particular, we investigate a relationship between the concept of full n-place functions and left translations.

The capability and certain functors of some nilpotent lie algebras of class two

Recently, the authors obtained the Schur multiplier, the non-abelian tensor square and the non-abelian exterior square of d-generator generalized Heisenberg Lie algebras of rank Here, we intend to obtain the same results for d-generator generalized Heisenberg Lie algebras of rank t when Then, as a result, we give similar consequences for a nilpotent Lie algebra L of class two when such that

Tame and Wild Automorphisms of Differential Polynomial Algebras of Rank 2

It is proved that the tame automorphism group of a differential polynomial algebra k{x, y} over a field k of characteristic 0 in two variables x, y with m commuting derivations δ1, . . . , δm is a free product with amalgamation. An example of a wild automorphism of the algebra k{x, y} in the case of m ≥ 2 derivations is constructed.

Finite quasi-quantum groups of rank two

This is a contribution to the structure theory of finite pointed quasi-quantum groups. We classify all finite-dimensional connected graded pointed Majid algebras of rank two which are not twist equivalent to ordinary pointed Hopf algebras.

A unitary Cuntz semigroup for C⁎-algebras of stable rank one

We introduce a new invariant for C⁎-algebras of stable rank one that merges the Cuntz semigroup information together with the K1-group information. This semigroup, termed the Cu1-semigroup, is constructed as equivalence classes of pairs consisting of a positive element in the stabilization of the given C⁎-algebra together with a unitary element of the unitization of the hereditary subalgebra generated by the given positive element. We show that the Cu1-semigroup is a well-defined continuous functor from the category of C⁎-algebras of stable rank one to a suitable codomain category that we write Cu∼. Furthermore, we compute the Cu1-semigroup of some specific classes of C⁎-algebras. Finally, in the course of our investigation, we show that we can recover functorially Cu, K1 and K⁎:=K0⊕K1 from Cu1.

Representations of power-associative train algebras of rank 4

This paper is devoted to the study of four-dimensional commutative power-associative non-Jordan train algebras. The existence of a unique orthogonal idempotent element in such kind of algebras enables us to define a non-zero algebra homomorphism over their ground field, which gives rise to train algebras of rank 4. We describe the irreducible representations over an algebraically closed field of characteristic prime to 30 by means of multilinear identities which define it. Furthermore, we provide sufficient conditions to guarantee whenever a representation is decomposable.

Relaxed highest-weight modules II: Classifications for affine vertex algebras

This is the second of a series of papers devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first [K. Kawasetsu and D. Ridout, Relaxed highest-weight modules I: Rank [Formula: see text] cases, Commun. Math. Phys. 368 (2019) 627–663, arXiv:1803.01989 [math.RT]] studied the simple “rank-[Formula: see text]” affine vertex superalgebras [Formula: see text] and [Formula: see text], with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones). Here, we turn to the question of classifying relaxed highest-weight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification of highest-weight modules by generalizing Olivier Mathieu’s coherent families [O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier [Formula: see text]Grenoble[Formula: see text] 50 (2000) 537–592]. We formulate this algorithmically and illustrate its practical implementation with several detailed examples. We also show how to use coherent family technology to establish the non-semisimplicity of category [Formula: see text] in one of these examples.

τ-exceptional sequences

We introduce the notions of τ-exceptional and signed τ-exceptional sequences for any finite dimensional algebra. We prove that for a fixed algebra of rank n, and for any positive integer t≤n, there is a bijection between the set of signed τ-exceptional sequences of length t, and (basic) ordered support τ-rigid objects with t indecomposable direct summands. If the algebra is hereditary, our notions coincide with exceptional and signed exceptional sequences. The latter were recently introduced by Igusa and Todorov, who constructed a similar bijection in the hereditary setting.

Exceptional sequences and idempotent functions

We prove that there is a one-to-one correspondence between the following three sets: idempotent functions on a set of size n, complete exceptional sequences of linear radical square zero Nakayama algebras of rank n and rooted labeled forests with n nodes and height of at most one. Therefore, the number of exceptional sequences is given by the sum ∑j=1nnjjn−j.

On the triple tensor product of nilpotent Lie algebras

In this paper, we give the explicit structure of and where H is a generalized Heisenberg Lie algebra of rank at most 2. Moreover, for a non-abelian nilpotent Lie algebra L, we obtain an upper bound for the dimension of

Ideals 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. In this paper we show that any finite-dimensional indecomposable [Formula: see text]-module is generated by one element. In particular, any indecomposable submodule of [Formula: see text] under the adjoint action is generated by a special element of [Formula: see text]. Using this result, we show that the Hopf algebra [Formula: see text] is a principal ideal ring, i.e., any two-sided ideal of [Formula: see text] is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras.