Underlying Vector Space Research Articles

Doubly alternating words in the positive part of [formula omitted]

This paper is about the positive part Uq+ of the q-deformed enveloping algebra Uq(slˆ2). The algebra Uq+ admits an embedding, due to Rosso, into a q-shuffle algebra ▪. The underlying vector space of ▪ is the free algebra on two generators x,y. Therefore, the algebra ▪ has a basis consisting of the words in x,y. Let U denote the image of Uq+ under the Rosso embedding. In our first main result, we find all the words in x,y that are contained in U. One type of solution is called alternating. The alternating words have been studied by Terwilliger. There is another type of solution, which we call doubly alternating. In our second main result, we display many commutator relations involving the doubly alternating words. In our third main result, we describe how the doubly alternating words are related to the alternating words.

DT INVARIANTS FROM VERTEX ALGEBRAS

AbstractWe obtain a new interpretation of the cohomological Hall algebra $\mathcal {H}_Q$ of a symmetric quiver Q in the context of the theory of vertex algebras. Namely, we show that the graded dual of $\mathcal {H}_Q$ is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of Q. Properties of that vertex algebra are shown to account for the key results about $\mathcal {H}_Q$ . In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of $\mathcal {H}_Q$ . Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson–Thomas invariants of Q (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

Non-Hermitian quantum walks and non-Markovianity: the coin-position interaction

A PT -symmetric, non-Hermitian Hamiltonian in the PT -unbroken regime can lead to unitary dynamics under the appropriate choice of the Hilbert space. The Hilbert space is determined by a Hamiltonian-compatible inner product map on the underlying vector space, facilitated by a ‘metric operator’. A more traditional method, however, involves treating the evolution as open system dynamics, and the state is constructed through normalization at each time step. In this work, we present a comparative study of the two methods of constructing the reduced dynamics of a system evolving under a PT -symmetric Hamiltonian. Our system is a one-dimensional quantum walk with the spin and position degrees of freedom forming its two subsystems. We compare the information flow between the subsystems under the two methods. We find that under the metric formalism, a power law decay of the information backflow to the subsystem gives a clear indication of the transition from PT -unbroken to the broken phase. This is unlike the information backflow under the normalized state method. We also note that even though non-Hermiticity models open system dynamics, pseudo-Hermiticity can increase entanglement between the subsystem in the metric Hilbert space, thus indicating that pseudo-Hermiticity cases can be seen as a resource in quantum mechanics.

An algebraic approach to circulant column parity mixers

Circulant Column Parity Mixers (CCPMs) are a particular type of linear maps, used as the mixing layer in permutation-based cryptographic primitives like Keccak-f (SHA3) and Xoodoo. Although being successfully applied, not much is known regarding their algebraic properties. They are limited to invertibility of CCPMs, and that the set of invertible CCPMs forms a group. A possible explanation is due to the complexity of describing CCPMs in terms of linear algebra. In this paper, we introduce a new approach to studying CCPMs using module theory from commutative algebra. We show that many interesting algebraic properties can be deduced using this approach, and that known results regarding CCPMs resurface as trivial consequences of module theoretic concepts. We also show how this approach can be used to study the linear layer of Xoodoo, and other linear maps with a similar structure which we call DCD-compositions. Using this approach, we prove that every DCD-composition where the underlying vector space with the same dimension as that of Xoodoo has a low order. This provides a solid mathematical explanation for the low order of the linear layer of Xoodoo, which equals 32. We design a DCD-composition using this module-theoretic approach, but with a higher order using a different dimension.

Complete descriptions of pseudo-Euclidean left-symmetric L-algebras and their pseudo-Euclidean modules

A pseudo-Euclidean left-symmetric algebra (A,.,〈,〉) is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form 〈,〉 such that left multiplications by any element of A are skew-symmetric with respect to 〈,〉. We recall that a pseudo-Euclidean Lie algebra (g,[,],〈,〉) is flat if and only if (g,.,〈,〉), its underlying vector space endowed with the Levi-Civita product associated with 〈,〉, is a pseudo-Euclidean left-symmetric algebra. In this paper, we study pseudo-Euclidean left-symmetric algebras (A,.,〈,〉) such that commutators of all elements of A are contained in the left annihilator of (A,.), these algebras will be called pseudo-Euclidean left-symmetric L-algebras. Next, we introduce and study pseudo-Euclidean modules of pseudo-Euclidean left-symmetric L-algebras. We develop double extension processes that allow us to have inductive descriptions of all pseudo-Euclidean left-symmetric L-algebras and of all its pseudo-Euclidean modules of any signature. This, in particular, improves some of the results obtained in [11].

Probabilistic bounds on best rank-1 approximation ratio

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors, our result recovers a known upper bound. For symmetric tensors, our upper bound unveils that the ratio of norms has the same order of magnitude as the trivial lower bound 1 / n d − 1 2 , when the order of a tensor d is fixed and the dimension of the underlying vector space n tends to infinity. However, when n is fixed and d tends to infinity, our lower bound is better than 1 / n d − 1 2 .

Higher Polynomial Identities for Mutations of Associative Algebras

We study polynomial identities satisfied by the mutation product xpy - yqx on the underlying vector space of an associative algebra A, where p, q are fixed elements of A. We simplify known results for identities in degree 4, proving that only two identities are necessary and sufficient to generate them all; in degree 5, we show that adding one new identity suffices; in degree 6, we demonstrate the existence of a significant number of new identities, which induce us to conjecture that the variety generated by mutation algebras of associative algebras is not finitely based.

On some Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation

There exists a multiplicative homomorphism from the braid group on k + 1 strands to the Temperley–Lieb algebra TLk . Moreover, the homomorphic images in TLk of the simple elements form a basis for the vector space underlying TLk . In analogy with the case of Bk , there exists a multiplicative homomorphism from the structure group G(X, r) of a non-degenerate, involutive set-theoretic solution (X, r), with , to an algebra, which extends to a homomorphism of algebras. We construct a finite basis of the underlying vector space of the image of G(X, r) using the Garsideness properties of G(X, r).

Автоморфные алгебры динамических систем и обобщенные контракции Иненю-Вигнера

Lie algebras a with a complex underlying vector space V are
 studied that are automorphic with respect to a given linear
 dynamical system on V , i.e., a 1-parameter subgroup Gt ⊂
 Aut(a) ⊂ GL(V ). Each automorphic algebra imparts a
 Lie algebraic structure to the vector space of trajectories of
 the group Gt. The basics of the general structure of automorphic
 algebras a are described in terms of the eigenspace
 decomposition of the operatorM ∈ der(a) that determines
 the dynamics. Symmetries encoded by the presence of nonabelian
 automorphic algebras are pointed out connected to
 conservation laws, spectral relations and root systems. It is
 shown that, for a given dynamics Gt, automorphic algebras
 can be found via a limit transition in the space of Lie algebras
 on V along the trajectories of the group Gt itself. This procedure
 generalises the well-known Inönü-Wigner contraction
 and links adjoint representations of automorphic algebras to
 isospectral Lax representations on gl(V ). These results can
 be applied to physically important symmetry groups and their
 representations, including classical and relativistic mechanics,
 open quantum dynamics and nonlinear evolution equations.
 Simple examples are given.

Circular Hessenberg pairs

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let M denote a Hessenberg matrix. Then M is called circular whenever the upper-right corner entry of M is nonzero and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. Let A,A⁎ denote a circular Hessenberg pair. We investigate six bases for the underlying vector space that we find attractive. We display the transition matrices between certain pairs of bases among the six. We also display the matrices that represent A and A⁎ with respect to the six bases. We introduce a special type of circular Hessenberg pair, said to be recurrent. We show that a circular Hessenberg pair A,A⁎ is recurrent if and only if A,A⁎ satisfy the tridiagonal relations. For a circular Hessenberg pair, there is a related object called a circular Hessenberg system. We classify up to isomorphism the recurrent circular Hessenberg systems. To this end, we construct four families of recurrent circular Hessenberg systems. We show that every recurrent circular Hessenberg system is isomorphic to a member of one of the four families.

Linear degenerations of algebras and certain representations of the general linear group

Let where is a field with be the space of structure vectors of algebras having the n-dimensional -space V as the underlying vector space. Also let Regarding as a G-module via the ‘change of basis’ action of G on V, we determine the composition factors of various G-submodules of which correspond to certain important families of algebras. This is achieved by introducing the notion of linear degeneration which allows us to obtain analogues over of certain known results on degenerations of algebras. As a result, the GL(V)-structure of is determined.

Ergodic theorem in CAT(0) spaces in terms of inductive means

AbstractLet $(G,+)$ be a compact, abelian, and metrizable topological group. In this group we take $g\in G$ such that the corresponding automorphism $\tau _g$ is ergodic. The main result of this paper is a new ergodic theorem for functions in $L^1(G,M)$ , where M is a Hadamard space. The novelty of our result is that we use inductive means to average the elements of the orbit $\{\tau _g^n(h)\}_{n\in \mathbb {N}}$ . The advantage of inductive means is that they can be explicitly computed in many important examples. The proof of the ergodic theorem is done firstly for continuous functions, and then it is extended to $L^1$ functions. The extension is based on a new construction of mollifiers in Hadamard spaces. This construction has the advantage that it only uses the metric structure and the existence of barycenters, and does not require the existence of an underlying vector space. For this reason, it can be used in any Hadamard space, in contrast to those results that need to use the tangent space or some chart to define the mollifier.

A comparison of vector symbolic architectures

Vector Symbolic Architectures combine a high-dimensional vector space with a set of carefully designed operators in order to perform symbolic computations with large numerical vectors. Major goals are the exploitation of their representational power and ability to deal with fuzziness and ambiguity. Over the past years, several VSA implementations have been proposed. The available implementations differ in the underlying vector space and the particular implementations of the VSA operators. This paper provides an overview of eleven available VSA implementations and discusses their commonalities and differences in the underlying vector space and operators. We create a taxonomy of available binding operations and show an important ramification for non self-inverse binding operations using an example from analogical reasoning. A main contribution is the experimental comparison of the available implementations in order to evaluate (1) the capacity of bundles, (2) the approximation quality of non-exact unbinding operations, (3) the influence of combining binding and bundling operations on the query answering performance, and (4) the performance on two example applications: visual place- and language-recognition. We expect this comparison and systematization to be relevant for development of VSAs, and to support the selection of an appropriate VSA for a particular task. The implementations are available.

Classifying and completing word analogies by machine learning

Analogical proportions are statements of the form ‘a is to b as c is to d’, formally denoted a:b::c:d. They are the basis of analogical reasoning which is often considered as an essential ingredient of human intelligence. For this reason, recognizing analogies in natural language has long been a research focus within the Natural Language Processing (NLP) community. With the emergence of word embedding models, a lot of progress has been made in NLP, essentially assuming that a word analogy like man:king::woman:queen is an instance of a parallelogram within the underlying vector space. In this paper, we depart from this assumption to adopt a machine learning approach, i.e., learning a substitute of the parallelogram model. To achieve our goal, we first review the formal modeling of analogical proportions, highlighting the properties which are useful from a machine learning perspective. For instance, the postulates supposed to govern such proportions entail that when a:b::c:d holds, then seven permutations of a,b,c,d still constitute valid analogies. From a machine learning perspective, this provides guidelines to build training sets of positive and negative examples. Taking into account these properties for augmenting the set of positive and negative examples, we first implement word analogy classifiers using various machine learning techniques, then we approximate by regression an analogy completion function, i.e., a way to compute the missing word when we have the three other ones. Using a GloVe embedding, classifiers show very high accuracy when recognizing analogies, improving state of the art on word analogy classification. Also, the regression processes usually lead to much more successful analogy completion than the ones derived from the parallelogram assumption.

Recoding Lie algebraic subshifts

<p style='text-indent:20px;'>We study internal Lie algebras in the category of subshifts on a fixed group – or Lie algebraic subshifts for short. We show that if the acting group is virtually polycyclic and the underlying vector space has dense homoclinic points, such subshifts can be recoded to have a cellwise Lie bracket. On the other hand there exist Lie algebraic subshifts (on any finitely-generated non-torsion group) with cellwise vector space operations whose bracket cannot be recoded to be cellwise. We also show that one-dimensional full vector shifts with cellwise vector space operations can support infinitely many compatible Lie brackets even up to automorphisms of the underlying vector shift, and we state the classification problem of such brackets. <p style='text-indent:20px;'>From attempts to generalize these results to other acting groups, the following questions arise: Does every f.g. group admit a linear cellular automaton of infinite order? Which groups admit abelian group shifts whose homoclinic group is not generated by finitely many orbits? For the first question, we show that the Grigorchuk group admits such a CA, and for the second we show that the lamplighter group admits such group shifts.

Double Yangian and the universal R-matrix

We describe the double Yangian of the general linear Lie algebra $\mathfrak{gl}_N$ by following a general scheme of Drinfeld. This description is based on the construction of the universal $R$-matrix for the Yangian. To make the exposition self contained, we include the proofs of all necessary facts about the Yangian itself. In particular, we describe the centre of the Yangian by using its Hopf algebra structure, and provide a proof of the analogue of the Poincar\'e-Birkhoff-Witt theorem for the Yangian based on its representation theory. This proof extends to the double Yangian, thus giving a description of its underlying vector space.

A Multistage Method for SCMA Codebook Design Based on MDS Codes

Sparse Code Multiple Access (SCMA) has been recently proposed for the future generation of wireless communication standards. SCMA system design involves specifying several parameters. In order to simplify the procedure, most works consider a multistage design approach. Two main stages are usually emphasized in these methods: sparse signatures design (equivalently, resource allocation) and codebook design. In this paper, we present a novel SCMA codebook design method. The proposed method considers SCMA codebooks structured with an underlying vector space obtained from classical block codes. In particular, when using maximum distance separable (MDS) codes, our proposed design provides maximum signal-space diversity with a relatively small alphabet. The use of small alphabets also helps to maintain desired properties in the codebooks, such as low peak-to-average power ratio and low-complexity detection.

Quadratic D-forms with applications to hermitian forms

We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for these forms. We then associate to every (skew) hermitian form over a division algebra with involution of the first kind a quadratic form defined on its underlying vector space. It is shown that this quadratic form, with its generalized notions of isotropy and isometry, can be used to determine the isotropy behaviour and the isometry class of (skew) hermitian forms.

Polynomials and tensors of bounded strength

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan–Hochster in their proof of Stillman’s conjecture and generalized here to other tensors, is universal among these ranks in the following sense: any non-trivial Zariski-closed condition on tensors that is functorial in the underlying vector space implies bounded strength. This generalizes a theorem by Derksen–Eggermont–Snowden on cubic polynomials, as well as a theorem by Kazhdan–Ziegler which says that a polynomial all of whose directional derivatives have bounded strength must itself have bounded strength.

Double Constructions of Compatible Associative Algebras

A compatible associative algebra is a pair of associative algebras satisfying that any linear combination of the two associative products is still an associative product. We construct a compatible associative algebra with a decomposition into the direct sum of the underlying vector space of another compatible associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant. This compatible associative algebra is equivalent to a certain bialgebra structure of compatible associative algebras, which is an analogue of a Lie bialgebra. Many properties of the bialgebra are presented. In particular, the coboundary bialgebra theory leads to the system of associative Yang–Baxter equations in compatible associative algebras, which is an analogue of the classical Yang–Baxter equation in a Lie algebra. Furthermore, the bialgebra can also be regarded as a “compatible version” of antisymmetric infinitesimal bialgebras, that is, a pair of antisymmetric infinitesimal bialgebras satisfying any linear combination of them is still an antisymmetric infinitesimal bialgebra.