Algebras Of Rank Research Articles

Proper identities of finitely generated commutative alternative algebras

We prove that every proper polynomial of degree at least 2n−2 is an identity of commutative alternative algebra of rank n⩾3. Using this we deduce that every commutative alternative algebra of rank n with the identity x3=0 is nilpotent of index at most 4n−2. We also prove that the index of nilpotency of the associator ideal in the free commutative alternative algebra of rank n⩾3 is equal to [2n3].

Primitively generated Hopf orders in characteristic p

ABSTRACTLet R be a characteristic p discrete valuation ring with field of fractions K. Let H be a commutative, cocommutative K-Hopf algebra of p-power rank which is generated as a K-algebra by primitive elements. We construct all of the R-Hopf orders of H in K; each Hopf order corresponds to a solution to a single matrix equation. For R complete, we greatly simplify the matrix equation and give explicit examples of Hopf orders in some rank p2 K-Hopf algebras.

A conditional determination of the average rank of elliptic curves

Under a hypothesis which is slightly stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both the average analytic rank and the average algebraic rank of elliptic curves in families of quadratic twists are exactly $\frac 12$. As a corollary we obtain that under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves in our family, and that asymptotically one half of these curves have algebraic rank $0$, and the remaining half $1$. We also prove an analogous result in the family of all elliptic curves. A way to interpret our results is to say that nonreal zeros of elliptic curve $L$-functions in a family have a direct influence on the average rank in this family. Results of Katz-Sarnak and of Young constitute a major ingredient in the proofs.

Extensions and automorphisms of Lie algebras

Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].

Flipping surfaces

We study semistable extremal threefold neighborhoods following earlier work of Mori, Koll\'ar, and Prokhorov. We classify possible flips and extend Mori's algorithm for computing flips of extremal neighborhoods of type k2A to more general neighborhoods of type k1A. In fact we show that they belong to the same deformation family as k2A, and we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which we interpret as a sequence of mutations in the cluster algebra of rank 2 with general coefficients. We identify, in the versal deformation space of a cyclic quotient singularity, the locus of deformations such that the total space admits a (terminal) antiflip. We show that these deformations come from at most two irreducible components of the versal deformation space. As an application, we give an algorithm for computing stable one-parameter degenerations of smooth projective surfaces (under some conditions) and describe several components of the Koll\'ar-Shepherd-Barron boundary of the moduli space of smooth canonically polarized surfaces of geometric genus zero.

A Presentation of the Free Lie Algebra M2,m,3

Let M2,m,3 be a free solvable nilpotent Lie algebra of rank 2 and nilpotency class m - 1. We show that M2,m,3 admits a minimal presentation whose set of defining relators consists of certain types of basic commutators using techniques in Grobner-Shirshov basis theory.

K-Primitivity and images of primitive elements

Let [Formula: see text] be a free Lie algebra of finite rank [Formula: see text] [Formula: see text]. We give another proof of the following criterion which is proven by Mikhalev and Zolotykh, using the idea of [Formula: see text]-primitivity: An endomorphism of [Formula: see text] preserving primitivity of elements is an automorphism.

Unistructurality of cluster algebras of type [formula omitted

It is conjectured by Assem, Schiffler and Shramchenko in [3] that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the cluster algebra structure. In other words, there exists a unique decomposition of the set of cluster variables into clusters. This conjecture has been proven to hold true for algebras of Dynkin type or rank 2, see [3]. The aim of this paper is to prove it for algebras of type A˜. We use triangulations of annuli and algebraic independence of clusters to prove unistructurality for algebras arising from annuli, which are of type A˜. We also prove the automorphism conjecture from [3] for algebras of type A˜ as a direct consequence.

Primitive cohomology of Hopf algebras

Primitive cohomology of a Hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra. Among many of its applications, two classifications are presented. Firstly we classify all non-locally PI, pointed Hopf algebra domains of Gelfand–Kirillov dimension two; and secondly we classify all pointed Hopf algebras of rank one. The first classification extends some results of Brown, Goodearl and others in an ongoing project to understand all Hopf algebras of low Gelfand–Kirillov dimension. The second generalizes results of Krop–Radford and Wang–You–Chen which classified Hopf algebras of rank one under extra hypothesis. Properties and algebraic structures of the primitive cohomology are discussed.

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ℚ-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.

Ideals and primitive elements of some relatively free Lie algebras.

Let F be a free Lie algebra of finite rank over a field K. We prove that if an ideal leftlangle tilde{v}rightrangle of the algebra F/gamma _{m+1}left( F^{prime }right) contains a primitive element tilde{u} then the element tilde{v} is primitive. We also show that, in the Lie algebra F/gamma _{3}left( Fright) ^{prime } there exists an element bar{v} such that the ideal leftlangle bar{v}rightrangle contains a primitive element bar{u} but, bar{u} and bar{v} are not conjugate by means of an inner automorphism.

A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank

We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using $R$-matrices for $U_q(\mathfrak{sl}_N)$. Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.

Irreducible representations of the plactic algebra of rank four

Irreducible representations of the plactic monoid M of rank four are studied. Certain concrete families of simple modules over the plactic algebra K[M] over a field K are constructed. It is shown that the Jacobson radical J(K[M]) of K[M] is nilpotent. Moreover, the congruence ρ on M determined by J(K[M]) coincides with the intersection of the congruences determined by the primitive ideals of K[M] corresponding to the constructed simple modules. In particular, M/ρ is a subdirect product of the images of M in the corresponding endomorphism algebras.

Algebraic rank on hyperelliptic graphs and graphs of genus 3

Let G¯=(G,ω) be a vertex-weighted graph, and let δ be a divisor class on G. Let rG¯(δ) denote the (combinatorial) rank of δ. Caporaso has introduced the algebraic rank rG¯alg(δ) of δ by using nodal curves with dual graph G¯. In this paper, when G¯ is hyperelliptic or of genus 3, we show that rG¯alg(δ)≥rG¯(δ) holds, generalizing our previous result. We also show that, with respect to the specialization map from a nonhyperelliptic curve of genus 3 to its reduction graph, any divisor on the graph lifts to a divisor on the curve of the same rank.

Exchange relation planar algebras of small rank

The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras: subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author.

On higher real and stable ranks for $CCR$ $C^*-$algebras

We calculate the real rank and stable rank of CCR algebras which either have only finite dimensional irreducible representations or have finite topological dimension. We show that either rank of A is determined in a good way by the ranks of an ideal I and the quotient A/I in four cases: When A is CCR; when I has only finite dimensional irreducible representations; when I is separable, of generalized continuous trace and finite topological dimension, and all irreducible representations of I are infinite dimensional; or when I is separable, stable, has an approximate identity consisting of projections, and has the corona factorization property. We also present a counterexample on higher ranks of M(A), A subhomogeneous, and a theorem of P. Green on generalized continuous trace algebras.

Class polynomials for some affine Hecke algebras

Class polynomials attached to affine Hecke algebras were first introduced by He in [13]. They play an important role in the study of affine Deligne–Lusztig varieties. Motivated by [14], we compute the class polynomials attached to an affine Hecke algebra of type (twisted) A˜2. Using these class polynomials, we prove a conjecture of Görtz–Haines–Kottwitz–Reuman for the general linear group, unitary group, and division algebra of semisimple rank 2. Furthermore, we discuss some interesting patterns of affine Deligne–Lusztig varieties.

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$, ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$, ordered by height, in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.

Controllability Analysis of Spacecraft with Only Attitude Actuation Near Small Solar System Bodies

Abstract: This paper investigates the nonlinear controllability of the translational and rotational motion of an under actuated spacecraft near small Solar System bodies (asteroids and comets) using attitude actuation only The coupled dynamics of the spacecraft and small body is analyzed in the framework of the full two body problem, with the coupling provided by their mutual gravity Due to the symmetry in their mutual gravity the problem of analyzing the dynamics of the spacecraft is obtained by reducing it to that of a single body (the spacecraft) under the gravitational interaction provided by the spacecraft-small body pair. It has been previously shown that the rotational (attitude) dynamics of the spacecraft is strongly coupled with its translational (orbital) dynamics in this weak-gravity environment. For spacecraft missions to small bodies, this coupling can be used to obtain controlled trajectories within a bounded neighborhood of the small body using only attitude control, as the analysis here shows. Prior to designing guidance and control schemes for such spacecraft missions, it is important to show the controllability of the spacecraft dynamics on the 12-dimensional state space, i.e., the tangent bundle TSE(3), with only attitude control. Sufficient conditions for controllability are shown using the Lie Algebra Rank condition (LARC), and the weakly positively Poisson stable (WPPS) property of the drift vector field under certain initial conditions.

2-cyclic permutations of lattice vertex operator algebras

The irreducible modules of the 2-cyclic permutation orbifold models of lattice vertex operator algebras of rank one are classified, and the quantum dimensions of irreducible modules and the fusion rules are determined.