Simple Algebra Research Articles

Simple vertex algebras arising from congruence subgroups

For any congruence subgroup Γ, we consider the vertex algebra of Γ-invariant global sections of chiral de Rham complex on the upper half plane that are meromorphic at the cusps. We give a description of the linear structure of the Γ-invariant vertex algebra by exhibiting a linear basis determined by meromorphic modular forms, and generalize the Rankin-Cohen bracket of modular forms to meromorphic modular forms. We also show that the Γ-invariant vertex algebra is simple.

An isomorphism between unitals and between related classical groups

Abstract An isomorphism between two hermitian unitals is provided, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

Canonical structure constants for simple Lie algebras

AbstractLet $$\mathfrak {g}$$ g be a finite-dimensional simple Lie algebra over $$\mathbb {C}$$ C . In the 1950s Chevalley showed that $$\mathfrak {g}$$ g admits particular bases, now called “Chevalley bases”, for which the corresponding structure constants are integers. Such bases are not unique but, using Lusztig’s theory of canonical bases, one can single out a “canonical” Chevalley basis which is unique up to a global sign. In this paper, we give explicit formulae for the structure constants with respect to such a basis.

Narrow and wide regular subalgebras of semisimple Lie algebras

A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is regular extreme if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. We determine necessary and sufficient conditions for a simple module of a semisimple Lie algebra to remain indecomposable when restricted to a regular subalgebra. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras [10]. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Then, we show that all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme.

Exceptional Periodicity and Magic Star algebras: Generalized roots, gradings, Hermitian Vinberg T-algebras and their derivations

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

On the full Kostant–Toda hierarchy and its ℓ-banded reductions for the Lie algebras of type A,B and G

Abstract This paper is concerned with the solutions of the full Kostant–Toda (f-KT) hierarchy in the Hessenberg form and their reductions to the ℓ -banded Kostant–Toda ( ℓ -KT) hierarchies for 1 ⩽ ℓ ⩽ n , where the case with ℓ = 1 is the classical tridiagonal Toda hierarchy and the case with ℓ = n is the f-KT hierarchy. Based on the Hessenberg form of the f-KT hierarchy, we study the f-KT hierarchy and the corresponding ℓ -KT hierarchies on simple Lie algebras of type A , B and G as the root space reductions with proper choices of Chevalley systems. Explicit formulas of the polynomial solutions for the τ-functions are also given in terms of the Schur functions and Schur’s Q-functions.

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

Modular tensor categories arising from central extensions and related applications

A modular tensor category is a non-degenerate ribbon finite tensor category and a ribbon factorizable Hopf algebra is a Hopf algebra whose finite-dimensional representations form a modular tensor category. In this paper, we provide a method of constructing ribbon factorizable Hopf algebras using central extensions. We then apply this method to n-rank Taft algebras, which are considered finite-dimensional quantum groups associated with abelian Lie algebras (see Section 2 for the definition), and obtain a family of non-semisimple ribbon factorizable Hopf algebras Eq, thus producing non-semisimple modular tensor categories using their representation categories. And we provide a prime decomposition of Rep(Eq) (the representation category of Eq). By further studying the simplicity of Eq (whether it is a simple Hopf algebra or not), we conclude that(1)there exists a twist J of uq(sl2⊕3) such that uq(sl2⊕3)J is a simple Hopf algebra,(2)there is no relation between the simplicity of a Hopf algebra H and the primality of Rep(H),(3)there are many ribbon factorizable Hopf algebras that are distinct from some known ones, i.e., not isomorphic to any tensor products of trivial Hopf algebras (group algebras or their dual), Drinfeld doubles, and small quantum groups.

Quantum Sugawara operators in type A

The quantum Sugawara operators associated with a simple Lie algebra g are elements of the center of a completion of the quantum affine algebra Uq(gˆ) at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series ℓV(z) associated with every finite-dimensional representation V of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over Uq(gˆ) at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the q-characters and q-deformed W-algebras via the Harish-Chandra homomorphism.We produce explicit quantum Sugawara operators for the quantum affine algebra of type A which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the q-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.

On the Gabriel quiver of extensions of Leibniz algebras

We compute the Gabriel quiver of simple objects in the category of bimodules over a simple Leibniz algebra and over the trivial 1-dimensional Leibniz algebra. Vertices of the quiver are the classes of simple objects, arrows are given by the dimensions of Ext 1 -groups.

Shapovalov elements for U q ( s l ( N + 1 ) )

For a simple Lie algebra, Shapovalov elements give rise to highest weight vectors in Verma modules. The usual construction of these elements uses induction on the length of a certain Weyl group element. If g = s l ( N + 1 ) explicit expressions for Shapovalov elements were given in I. M. Musson [Explicit expressions for Shapovalov elements in type A, J. Algebra 623 (2023), 358–394]. Here we adapt the argument to the quantized enveloping algebra of g .

Mathematical Modeling of Tumor Growth in Preclinical Mouse Models with Applications in Biomarker Discovery and Drug Mechanism Studies.

Oncology drug efficacy is evaluated in mouse models by continuously monitoring tumor volumes, which can be mathematically described by growth kinetic models. While past studies have investigated various growth models, their reliance on small datasets raises concerns about whether their findings are truly representative of tumor growth in diverse mouse models under different vehicle or drug treatments. Here, we systematically evaluated six parametric models (exponential, exponential quadratic, monomolecular, logistic, Gompertz, and von Bertalanffy) and the semi-parametric generalized additive model (GAM) on fitting tumor volume data from over 30,000 mice in 930 experiments conducted in patient-derived xenografts, cell line-derived xenografts, and syngeneic models. We found that the exponential quadratic model is the best parametric model and can adequately model 87% studies, higher than other models including von Bertalanffy (82%) and Gompertz (80%) models, the latter is often considered the standard growth model. On the mouse group level, 7.5% of growth data could not be fit by any parametric model and were fitted by GAM. We show that eGaIT, a GAM derived efficacy metric, is equivalent to eGR, a metric we previously proposed and conveniently calculated by simple algebra. Using five studies on Paclitaxel, anti-PD-1 antibody, Cetuximab, Irinotecan, and Sorafenib, we show that exponential and exponential quadratic models achieve similar performance in uncovering drug mechanism and biomarkers. We also compared eGR-based association analysis and exponential modeling approach in biomarker discovery and found they complement each other. Modeling methods herein are implemented in an open-source R package freely available at https://github.com/hjzhou988/TuGroMix.

A simple linear algebra identity to optimize large-scale neural network quantum states

Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with 3 × 105 parameters, achieving state-of-the-art ground-state energy in the J1–J2 Heisenberg model at J2/J1 = 0.5 on the 10 × 10 square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for neural-network quantum states, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle.

Rationality of holomorphic vertex operator algebras

We prove that if V is a unitary simple holomorphic vertex operator algebra of CFT-type, then V is rational, that is, all N-gradable V-modules are direct sums of copies of V.

The Automorphisms of Differential Extensions of Characteristic p

Nonassociative differential extensions are generalizations of associative differential extensions, either of a purely inseparable field extension K of exponent one of a field F, F of characteristic p, or of a central division algebra over a purely inseparable field extension of F. Associative differential extensions are well known central simple algebras first defined by Amitsur and Jacobson. We explicitly compute the automorphisms of nonassociative differential extensions. These are canonically obtained by restricting automorphisms of the differential polynomial ring used in the construction of the algebra. In particular, we obtain descriptions for the automorphisms of associative differential extensions of D and K, which are known to be inner.

EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS AND VIA CONTACTIFICATIONS

Abstract In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$ . Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$ . In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$ . The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$ th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$ . Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$ .

On Singularity Properties of Word Maps and Applications to Probabilistic Waring Type Problems

We study singularity properties of word maps on semisimple Lie algebras, semisimple algebraic groups and matrix algebras and obtain various applications to random walks induced by word measures on compact p p -adic groups. Given a word w w in a free Lie algebra L r \mathcal {L}_{r} , it induces a word map φ w : g r → g \varphi _{w}:\mathfrak {g}^{r}\rightarrow \mathfrak {g} for every semisimple Lie algebra g \mathfrak {g} . Given two words w 1 ∈ L r 1 w_{1}\in \mathcal {L}_{r_{1}} and w 2 ∈ L r 2 w_{2}\in \mathcal {L}_{r_{2}} , we define and study the convolution of the corresponding word maps φ w 1 ∗ φ w 2 ≔ φ w 1 + φ w 2 : g r 1 + r 2 → g \varphi _{w_{1}}*\varphi _{w_{2}}≔\varphi _{w_{1}}+\varphi _{w_{2}}:\mathfrak {g}^{r_{1}+r_{2}}\rightarrow \mathfrak {g} . By introducing new degeneration techniques, we show that for any word w ∈ L r w\in \mathcal {L}_{r} of degree d d , and any simple Lie algebra g \mathfrak {g} with φ w ( g r ) ≠ 0 \varphi _{w}(\mathfrak {g}^{r})\neq 0 , one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O ( d 4 ) O(d^{4}) self-convolutions of φ w \varphi _{w} . Similar results are obtained for matrix word maps. We deduce that a group word map of length ℓ \ell becomes (FRS), locally around identity, after O ( ℓ 4 ) O(\ell ^{4}) self-convolutions, for every semisimple algebraic group G _ \underline {G} . We furthermore provide uniform lower bounds on the log canonical threshold of the fibers of Lie algebra, matrix and group word maps. For the commutator word w 0 = [ X , Y ] w_{0}=[X,Y] , we show that φ w 0 ∗ 4 \varphi _{w_{0}}^{*4} is (FRS) for any semisimple Lie algebra, improving a result of Aizenbud-Avni, and obtaining applications in representation growth of compact p p -adic and arithmetic groups. The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form Z / p k Z \mathbb {Z}/p^{k}\mathbb {Z} . This allows us to relate them to properties of some natural families of random walks on finite and compact p p -adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to p p -adic probabilistic Waring type problems.

A Study on the Classification of All Simple Commutative Near-Rings upto Isomorphism

This research paper aims to classify all simple commutative Near-Rings up to isomorphism, which are rings with no zero divisors and no ideals containing a nonzero element that is not prime ideal. The authors use various techniques from algebraic number theory and representation theory of semi simple algebras to construct representations of these rings over the integers or rational numbers. They also provide examples showing how these classes can be distinguished by their algebraic properties, such as whether they are PID (prime idempotent domain), zero-divisorless, or have ideals containing elements with prime index. The study contributes new results and insights into the classification of near-rings, which has implications for other areas of abstract algebra and number theory.

Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_2$ and Witt Algebra

We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to the modular group and $q$-deformed rational numbers defined by Morier-Genoud and Ovsienko and lead to $q$-deformed Möbius transformations acting on the hyperbolic plane.

Compiling with Abstract Interpretation

Rewriting and static analyses are mutually beneficial techniques: program transformations change the intensional aspects of the program, and can thus improve analysis precision, while some efficient transformations are enabled by specific knowledge of some program invariants. Despite the strong interaction between these techniques, they are usually considered distinct. In this paper, we demonstrate that we can turn abstract interpreters into compilers, using a simple free algebra over the standard signature of abstract domains. Functor domains correspond to compiler passes, for which soundness is translated to a proof of forward simulation, and completeness to backward simulation. We achieve translation to SSA using an abstract domain with a non-standard SSA signature. Incorporating such an SSA translation to an abstract interpreter improves its precision; in particular we show that an SSA-based non-relational domain is always more precise than a standard non-relational domain for similar time and memory complexity. Moreover, such a domain allows recovering from precision losses that occur when analyzing low-level machine code instead of source code. These results help implement analyses or compilation passes where symbolic and semantic methods simultaneously refine each other, and improves precision when compared to doing the passes in sequence.