arXiv subjects

Agata Smoktunowicz

Publications and source records attributed to Agata Smoktunowicz.

At least 19 recordsLinked to original sources

arXiv2026

On a conjecture by Michael Wemyss regarding the calculation of GV invariants

Contraction algebras are noncommutative algebras introduced by Donovan and Wemyss to classify of 3-dimensional flops. Wemyss conjectures that contraction algebras can be deformed to a single semisimple algebra. This gives an intrinsic method of calculating Gopakumar-Vafa invariants of the flop. The main result is a proof of Wemyss' conjecture for types A and D. In the course of the proof, we recall and introduce new techniques for constructing flat deformations of associative algebras and compare various notions of deformations. We also put forward two conjectures which hint towards a deeper theory.

math.RA↗

arXiv2025

On flat deformations and their applications

We say that a formal deformation from an algebra $N$ to algebra $A$ is strongly flat if for every real number $e $ there is a real number $0 0$ gives an algebra isomorphic to $A$. It is shown that all semisimple algebras which can be obtained as a specialisation of such a deformation are isomorphic. We also show that every strongly flat deformation $\mathcal N=N\{t\}$ from a finite-dimensional $\mathbb C$-algebra $N$ to a semisimple $\mathbb C$-algebra $A$ specialised at $t=s$ for all sufficiently small real numbers $s>0$ gives an algebra isomorphic to $A$. A remark by Joachim Jelisiejew is also included which allows us to obtain this result as an application of Gabriel's theorem [6]. We also give a characterisation of semisimple algebras $A$ to which a given algebra $N$ cannot be deformed to. This gives a partial answer to a question of Michael Wemyss on Acons [26]. We also give a partial answer to question 6.5 from [1].

math.RA↗

arXiv2025

On constructing deformations of noncommutative algebras

We show that if there is a flat deformation from a finite-dimensional algebra N to an algebra A, then such a deformation can be obtained by a slight generalization of the construction from [2]. The generalization allows negative powers of t to occur in the formula for f. The proof uses a result from [19]. We also prove that this modified construction produces well-defined formal deformations that are flat deformations from N to A.

math.RA↗

arXiv2024

On some connections between braces and pre-Lie rings outside of the context of Lazard's correspondence

Let $p>3$ be a prime number and let $A$ be a brace whose additive group is a direct sum of cyclic groups of cardinalities larger than $p^{\alpha }$ for some $\alpha $. Suppose that either (i) $A^{\lfloor{\frac {p-1}4}\rfloor}\subseteq pA$ or that (ii) the additive group of brace $A$ has rank smaller than ${\lfloor{\frac {p-1}4}\rfloor}$. It is shown that for every natural number $i\leq \alpha- {\frac {4\alpha }{p-1}}$ the factor brace $A/p^{i}A$ is obtained by a formula similar to the group of flows from a left nilpotent pre-Lie ring.

math.GR↗

arXiv2024

On pre-Lie rings related to some non-Lazard braces

Let A be a brace of cardinality $p^{n}$ for some prime number $p$. Suppose that either (i) the additive group of brace $A$ has rank smaller than $p-3$, or (ii) $A^{\frac {p-1}2}\subseteq pA$ or (iii) $p^{i}A$ is an ideal in in $A$ for each $i$. It is shown that there is a pre-Lie ring associated to brace $A$. The left nilpotency index of this pre-Lie ring can be arbitrarily large. Let $A$ be a brace of cardinality $p^{n}$ for some prime number $p$. Denote $ann(p^{i})=\{a\in A: p^{i}a=0\}$. Suppose that for $i=1,2,\ldots $ and all $a,b\in A$ we have \[a*(a*(\cdots *a*b))\in pA, a*(a*(\cdots *a*ann(p^{i})))\in ann(p^{i-1})\] where $a$ appears less than $\frac {p-1}4$ times in this expression. Let $k$ be such that $p^{k(p-1)}A=0$. It is shown that the brace $A/ann(p^{4k})$ is obtained from a left nilpotent pre-Lie ring by a formula which depends only on the additive group of brace $A$. We also obtain some applications of this result.

math.GR↗

arXiv2023

Numerical stability of the symplectic $LL^T$ factorization

In this paper we give the detailed error analysis of two algorithms $W_1$ and $W_2$ for computing the symplectic factorization of a symmetric positive definite and symplectic matrix $A \in \mathbb R^{2n \times 2n}$ in the form $A=LL^T$, where $L \in \mathbb R^{2n \times 2n}$ is a symplectic block lower triangular matrix. We prove that Algorithm $W_2$ is numerically stable for a broader class of symmetric positive definite matrices $A \in \mathbb R^{2n \times 2n}$. It means that Algorithm $W_2$ is producing the computed factors $\tilde L$ in floating-point arithmetic with machine precision $\mathcal{u}$ such that $||A-\tilde L {\tilde L}^T||_{2} = {\cal O}(\mathcal{u} ||{A}||_{2})$. On the other hand, Algorithm $W_1$ is unstable, in general, for symmetric positive definite and symplectic matrix $A$. In this paper we also give corresponding bounds for Algorithm $W_1$ that are weaker. We show that the factorization error depends on the condition number $\kappa_2(A_{11})$ of the principal submatrix $A_{11}$. Bounds for the loss of symplecticity of the lower block triangular matrices $L$ for both Algorithms $W_1$ and $W_2$ that hold in exact arithmetic for a broader class of symmetric positive definite matrices $A$ (but not necessarily symplectic) are also given. The tests performed in \textsl{MATLAB} illustrate that our error bounds for considered algorithms are reasonably sharp.

math.NA↗

arXiv2023

From braces to pre-Lie rings

Let $A$ be a brace of cardinality $p^{n}$ where $p>n+1$ is prime, and let $ann (p^{2})$ be the set of elements of additive order at most $p^{2}$ in this brace. We construct a pre-Lie ring related to the brace $A/ann(p^{2})$. In the case of strongly nilpotent braces of nilpotency index $k<p$ the brace $A/ann(p^{2})$ can be recovered by applying the construction of the group of flows to the resulting pre-Lie ring. We don't know whether, when applied to braces which are not right nilpotent, our construction is related to the group of flows. We use powerful Lie rings associated with finite $p$-groups in the study of brace automorphisms with few fixed points. As an application we bound the number of elements which commute with a given element in a brace, as well as the number of elements which multiplied from left by a given element give zero. We also study various Lie rings associated to powerful groups and braces whose adjoint groups are powerful, and show that the obtained Lie and pre-Lie rings are also powerful. We also show that braces whose adjoint groups are powerful and powerful left nilpotent pre-Lie rings are in one-to-one correspondence and that they are left and right nilpotent under some cardinality assumptions.

math.RA↗

arXiv2023

A construction of deformations to general algebras

One of the questions investigated in deformation theory is to determine to which algebras can a given associative algebra be deformed. In this paper we investigate a different but related question, namely: for a given associative finite-dimensional C-algebra A, find algebras N which can be deformed to A. We develop a simple method which produces associative and flat deformations to investigate this question. As an application of this method we answer a question of Michael Wemyss about deformations of contraction algebras.

math.AG↗

arXiv2022

From pre-Lie rings back to braces

Let A be a brace of cardinality p^{n} where p>n+1 is prime, and ann(p^{i}) be the set of elements of additive order at most p^{i} in this brace. A pre-Lie ring related to the brace A/ann(p^{2}) was constructed in [8]. We show that there is a formula dependent only on the additive group of the brace A which reverses the construction from [8]. As an application example it is shown that the brace A/ann(p^{4}) is the group of flows of a left nilpotent pre-Lie algebra.

math.RA↗

arXiv2022

From Braces to Hecke algebras & Quantum Groups

We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a novel class of quantum discrete integrable systems and we derive symmetries for the corresponding periodic transfer matrices.

math-ph↗

arXiv2022

On the passage from finite braces to pre-Lie rings

Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality $p^{n}$, for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality $p^{n}$ for n+1<p. For the passage from pre-Lie rings to braces we use exactly the same method as suggested in [41].

math.RA↗

arXiv2021

A new formula for Lazard's correspondence for finite braces and pre-Lie algebras

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite Fp-braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent Fp-braces generated by one element of cardinality p^4 is obtained, answering a question posed by Leandro Vendramin. It is also shown that the sum of a finite number of left nilpotent ideals in a left brace is a left nilpotent ideal, therefore every finite brace contains the largest left nilpotent ideal.

math.RA↗

arXiv2021

Set theoretic Yang-Baxter & reflection equations and quantum group symmetries

Connections between set-theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic $R$-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for $R$-matrices being Baxterized solutions of the $A$-type Hecke algebra ${\cal H}_N(q=1)$. We show in the case of the reflection algebra that there exists a ``boundary'' finite sub-algebra for some special choice of ``boundary'' elements of the $B$-type Hecke algebra ${\cal B}_N(q=1, Q)$. We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the $B$-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the $B$-type Hecke algebra. These are universal statements that largely generalize previous relevant findings, and also allow the investigation of the symmetries of the double row transfer matrix.

math-ph↗

arXiv2020

Algebraic approach to Rump's results on relations between braces and pre-Lie algebras

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right R-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper we explain Rump's correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.

math.RA↗

arXiv2019

Combinatorial solutions to the reflection equation

We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions to the parameter-dependent reflection equation.

math.RA↗

arXiv2018

On Radicals of Ore Extensions and Related Questions

We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If $R$ is prime radical and $δ$ is a derivation of $R$, then the differential polynomial ring $R[X;δ]$ is locally nilpotent. This answers an open question posed in by Nielsen and Ziembowski. The nil radical of a differential polynomial ring $R[X;δ]$ takes the form $I[X;δ]$ for some ideal $I$ of $R$, provided that the base field is infinite. This answers an open question posed by Hong, Kim, Lee and Nielsen for algebras over infinite fields. If $R$ is a graded algebra generated in degree $1$ over a field of characteristic zero and $δ$ is a grading preserving derivation on $R$, then the Jacobson radical of $R$ is $δ$-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp. Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent. The power series ring $R[[X;σ,δ]]$ is well-defined whenever $δ$ is a locally nilpotent $σ$-derivation; this answers a conjecture by Bergen and Grzeszczuk and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on $δ$.

math.RA↗

arXiv2018

Skew left braces of nilpotent type

We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces.

math.RA↗

arXiv2017

A note on set-theoretic solutions of the Yang-Baxter equation

This paper shows that every finite non-degenerate involutive set theoretic solution (X,r) of the Yang-Baxter equation whose symmetric group has cardinality which a cube-free number is a multipermutation solution. Some properties of finite braces are also investigated (Theorems 3, 5 and 11). It is also shown that if A is a left brace whose cardinality is an odd number and (-a) b=-(ab) for all a, b A, then A is a two-sided brace and hence a Jacobson radical ring. It is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level.

math.RA↗