Diagonal Reduction Research Articles

Ghost center and representations of the diagonal reduction algebra of osp(1|2)

Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models. In this paper we continue the study of the diagonal reduction superalgebra A of the orthosymplectic Lie superalgebra osp(1|2). We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of A is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for osp(1|2)). As another application, we classify all finite-dimensional irreducible representations of A. Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.

Illumination correction of dyed fabric based on extreme learning machine with improved ant lion optimizer

AbstractIn order to eliminate the influence of scene illumination on the evaluation of the color difference of dyed fabrics, this paper proposes a dyed fabric illumination correction algorithm based on the extreme learning machine (ELM) with grey wolf optimizer (GWO)‐optimized ant lion optimizer (ALO). Firstly, the Grey Edge framework is used to extract the features of the dyed fabric image as the input vector. Then, to improve the optimization ability of the ALO algorithm, the GWO algorithm is used to provide a set of optimized initial populations to the ALO algorithm, and then the improved ALO algorithm is used to optimize the parameters of the ELM. Finally, the proposed GWO‐ALO‐ELM algorithm is used to correct the illumination of the dyed fabric, and restore the graphics to the effect display under standard illumination through the diagonal reduction model. Compared with the experimental results of GWO‐ELM, ALO‐ELM, backpropagation (BP), ELM, random vector function link (RVFL), and other algorithms, it can be seen that the GWO‐ALO‐ELM algorithm proposed in this paper has good predictive value and quasi‐bias effect, and good stability.

Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $$( \mathfrak{G} , \mathfrak{g} )$$ has previously been considered by Khoroshkin and Ogievetsky in the case of the diagonal reduction algebra for $$\mathfrak{gl}(n)$$ . In this paper, we consider the diagonal reduction algebra of the pair of Lie superalgebras $$( \mathfrak{osp}(1|2) \times \mathfrak{osp}(1|2) , \mathfrak{osp}(1|2) )$$ as a double coset space having an associative $$ \mathrel{\scriptstyle\lozenge} $$ -product and give a complete presentation in terms of generators and relations. We also provide a PBW basis for this reduction algebra along with Casimir-like elements and a subgroup of automorphisms.

Reduction of matrices over simple Ore domains

We study the theory of diagonal reductions of matrices over simple Ore domains of finite stable range. We cover the cases of 2-simple rings of stable range 1, Ore domains and certain cases of Bezout domains.

A Noise Study of the PSW Signature Family: Patching DRS with Uniform Distribution †

At PKC 2008, Plantard et al. published a theoretical framework for a lattice-based signature scheme, namely Plantard–Susilo–Win (PSW). Recently, after ten years, a new signature scheme dubbed the Diagonal Reduction Signature (DRS) scheme was presented in the National Institute of Standards and Technology (NIST) PQC Standardization as a concrete instantiation of the initial work. Unfortunately, the initial submission was challenged by Yu and Ducas using the structure that is present on the secret key noise. In this paper, we are proposing a new method to generate random noise in the DRS scheme to eliminate the aforementioned attack, and all subsequent potential variants. This involves sampling vectors from the n-dimensional ball with uniform distribution. We also give insight on some underlying properties which affects both security and efficiency on the PSW type schemes and beyond, and hopefully increase the understanding on this family of lattices.

Elementary matrix reduction over Bézout domains

A ring [Formula: see text] is an elementary divisor ring if every matrix over [Formula: see text] admits a diagonal reduction. If [Formula: see text] is an elementary divisor domain, we prove that [Formula: see text] is a Bézout duo-domain if and only if for any [Formula: see text], [Formula: see text] such that [Formula: see text]. We explore certain stable-like conditions on a Bézout domain under which it is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

Diagonal reduction of matrices over commutative semihereditary Bezout rings

It is proven that every commutative semihereditary Bezout ring in which any regular element is Gelfand (adequate), is an elementary divisor ring.

LOCAL COMPARABILITY OF EXCHANGE IDEALS

An exchange ideal $I$ of a ring $R$ is locally comparable if for every regular $x\in I$ there exists a right or left invertible $u\in 1+I$ such that $x=xux$. We prove that every matrix extension of an exchange locally comparable ideal is locally comparable. We thereby prove that every square regular matrix over such ideal admits a diagonal reduction.

Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals

We investigate commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals. In particular, we have described the class of such rings, which are elementary divisor rings. A ring $R$ is called an elementary divisor ring if every matrix over $R$ has a canonical diagonal reduction (we say that a matrix $A$ over $R$ has a canonical diagonal reduction if for the matrix $A$ there exist invertible matrices $P$ and $Q$ of appropriate sizes and a diagonal matrix $D=\mathrm{diag}(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r,0,\dots,0)$ such that $PAQ=D$ and $R\varepsilon_i\subseteq R\varepsilon_{i+1}$ for every $1\le i\le r-1$). We proved that a commutative Bezout domain $R$ in which any nonze\-ro prime ideal is contained in a finite set of maximal ideals and for any nonzero element $a\in R$ the ideal $aR$ a decomposed into a product $aR = Q_1\ldots Q_n$, where $Q_i$ ($i=1,\ldots, n$) are pairwise comaximal ideals and $\mathrm{rad}\,Q_i\in\mathrm{spec}\, R$, is an elementary divisor ring.

Diagonal reduction algebra and the reflection equation

We describe the diagonal reduction algebra D(gl n ) of the Lie algebra gl n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl n ).

Diagonal matrix reduction over Hermite rings

ABSTRACTA commutative ring R is an elementary divisor ring if every matrix over R admits a diagonal reduction. In this paper, we define the term ‘Zabvasky subset’ of a ring to study diagonal matrix reduction. Let S be a Zabavsky subset of a Hermite ring R. We prove that R is an elementary divisor ring if and only if with implies that there exist such that . If with implies that there exists a such that , then R is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

Elementary matrix reduction over J-stable rings

ABSTRACTA commutative ring R is J-stable provided that R∕aR has stable range 1 for all a∉J(R). A commutative ring R in which every finitely generated ideal principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g. [3, Theorem 8], [4, Theorem 4.1], [7, Theorem 3.7], [8, Theorem], [9, Theorem 2.1], [14, Theorem 1] and [18, Theorem 7].

A Mathematical modeling and Simulation Analysis of Lattice reduction algorithms for large MIMO detections

Abstract – This paper analyses the performance of Multiple Input Multiple Output (MIMO) wireless communication system by combining it with an efficient algorithm Diagonal Reduction (DR). DR is a powerful tool to achieve more efficiently a high performance with less complexity. This paper proposes a DR algorithm in order to reduce its complexity named Greedy Diagonal Reduction (GDR) algorithm which gives reduced complexity with efficient performance at the receiver using the MATLAB communication tool box version 7.0 as a simulation tool. From the simulation results of various reduction algorithms it is observed that, DR can reduce the number of iterations using size reduction operations. Proposed DR algorithm gives identical Bit Error Rate (BER) performance with LLL algorithms when applied to Successive Interference cancellation (SIC) decoding. Greedy DR reduces the computational complexity in Multiple Input Multiple Output systems by improving the efficiency in terms of size reduction operations. Sphere Decoding (SD) technique gives better BER performance on comparing with Successive Interference cancellation decoding technique.

Rings of Fractions of Reduction Algebras

We establish the absence of zero divisors in the reduction algebra of a Lie algebra ${\mathfrak{g}}$ with respect to its reductive Lie subalgebra ${\mathfrak{k}}$ . We identify the field of fractions of the diagonal reduction algebra of ${\mathfrak{sl}}_2$ with the standard skew field; as a by-product we obtain a two-parametric family of realizations of this diagonal reduction algebra by differential operators. We also present a new proof of the Poincare–Birkhoff–Witt theorem for reduction algebras.

A Diagonal Lattice Reduction Algorithm for MIMO Detection

Recently, an efficient lattice reduction method, called the effective LLL (ELLL) algorithm, was presented for the detection of multiinput multioutput (MIMO) systems. In this letter, a novel lattice reduction criterion, called diagonal reduction, is proposed. The diagonal reduction is weaker than the ELLL reduction, however, like the ELLL reduction, it has identical performance with the LLL reduction when applied for the sphere decoding and successive interference cancelation (SIC) decoding. It improves the efficiency of the ELLL algorithm by significantly reducing the size-reduction operations. Furthermore, we present a greedy column traverse strategy, which reduces the column swap operations in addition to the size-reduction operations.

Generalized Stable Rings and Regularity

We prove in this article that the generalized stable property is invariant under Morita contexts. Further, we show that many classes of square matrices over generalized stable regular rings admit a diagonal reduction. Related examples are constructed as well.

Structure Constants of Diagonal Reduction Algebras of gl Type

We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra $\gl_n$ into $\gl_n\oplus\gl_n$. Its representation theory is related to the theory of decompositions of tensor products of $\gl_n$-modules.

On Generalized Stable Ideals

An ideal I of a ring R is generalized stable in case aR + bR = R with a ∈ I, b ∈ R implies that there exist s, t ∈ 1 + I such that s(a + by)t = 1 for a y ∈ R. We establish, in this article, necessary and sufficient conditions for an ideal of a regular ring to be generalized stable. It is shown that every regular square matrix over such ideals admits a diagonal reduction. These extend the corresponding results of generalized stable regular rings.

Diagonal reduction algebras of gl type

Several general questions concerning a reduction algebra, namely, rings of definition and the algorithmic efficiency of the set of ordering relations, are discussed. For reduction algebras related to the diagonal embedding of the Lie algebra gl n into gl n ⊕gl n , a stabilization phenomenon is established and a complete set of defining relations is given.

UNIT-REGULARITY AND STABLE RANGE ONE

Let R be a ring, and let <TEX>$\Psi$</TEX>(R) be the ideal generated by the set {x <TEX>$\in$</TEX>R | 1 + sxt <TEX>$\in$</TEX> R is unit-regular for all s, t <TEX>$\in$</TEX> R}. We show that <TEX>$\Psi$</TEX>(R) has "radical-like" property. It is proven that <TEX>$\Psi$</TEX>(R) has stable range one. Thus, diagonal reduction of matrices over such ideal is reduced.