Arbitrary Algebra Research Articles

Uniqueness up to inner automorphism of regular exact Borel subalgebras

In [18], Külshammer, König and Ovsienko proved that for any quasi-hereditary algebra (A,≤A) there exists a Morita equivalent quasi-hereditary algebra (R,≤R) containing a basic exact Borel subalgebra B. The Borel subalgebra B constructed in [18] is in fact a regular exact Borel subalgebra as defined in [7]. Later, Conde [9] showed that given a quasi-hereditary algebra (R,≤R) with a basic regular exact Borel subalgebra B and a Morita equivalent quasi-hereditary algebra (R′,≤R′) with a basic regular exact Borel subalgebra B′, the algebras R and R′ are isomorphic, and Külshammer and Miemietz [20] showed that there is even an isomorphism φ:R→R′ such that φ(B)=B′.In this article, we show that if R=R′, then φ can be chosen to be an inner automorphism. Moreover, instead of just proving this for regular exact Borel subalgebras of quasi-hereditary algebras, we generalize this to an appropriate class of subalgebras of arbitrary finite-dimensional algebras. As an application, we show that if (A,≤A) is a finite-dimensional algebra and G is a finite group acting on A via automorphisms, then under some natural compatibility conditions, there is a Morita equivalent quasi-hereditary algebra (R,≤R) with a basic regular exact Borel subalgebra B such that g(B)=B for every g∈G.

Integral Classification of Endomorphisms of an Arbitrary Algebra with Finitary Operations

Integral Classification of Endomorphisms of an Arbitrary Algebra with Finitary Operations

Varieties of four-dimensional gauge theories

We use algebraic geometry to study the anomaly-free representations of an arbitrary gauge Lie algebra for 4-dimensional spacetime fermions. For irreducible representations, the problem reduces to studying the Lie algebras \U0001d530\U0001d532n for n ≥ 3. We show that there exist equivalence classes of such representations that are in bijection with the rational points on a projective variety that are dense in a region on the underlying real variety diffeomorphic to ℝn−3. It follows that the chiral ones overwhelm the non-chiral ones for n ≥ 5. We present an efficient algorithm to find explicit anomaly-free irreducible representations and discuss the generalization to reducible representations.

On Kursov's theorem for matrices over division rings

Let D be a division ring with center F and multiplicative group D×, where each element of the commutator subgroup of D× can be expressed as a product of at most s commutators. A known theorem of Kursov states that if D is finite-dimensional over F, then every element of the commutator subgroup of the general linear group over D can be expressed as a product of at most s+1 commutators. We show that this result remains valid when F has a sufficiently large number of elements, without requiring D to be finite-dimensional. Our approach not only improves upon recent results on matrix decompositions over division rings but also provides a look at the Engel word map for matrices over arbitrary algebras.

Subalgebras of octonion algebras

For an arbitrary unitary octonion algebra, we determine all subalgebras. It turns out that every subalgebra of dimension less than four is associative, while every subalgebra of dimension greater than four is not associative. In any split octonion algebra, there are both associative and non-associative subalgebras of dimension four. Except for one-dimensional subalgebras spanned by idempotents, any two isomorphic subalgebras are in the same orbit under automorphisms.

Commutative Poisson algebras from deformations of noncommutative algebras

It is well-known that a formal deformation of a commutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} leads to a Poisson bracket on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} and that the classical limit of a derivation on the deformation leads to a derivation on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The deformation leads in this case to a Poisson algebra structure on Π(A):=Z(A)×(A/Z(A))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A}){:}{=}Z(\\mathcal {A})\ imes (\\mathcal {A}/Z(\\mathcal {A}))$$\\end{document} and to the structure of a Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}-Poisson module on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The limiting derivations are then still derivations of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, but with the Hamiltonian belong to Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}, rather than to A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.

Realization of arbitrary Lie algebras by automorphisms of $\mathrm{CR}$ manifolds and symmetries of differential equations

For any finite-dimensional real Lie algebra $\mathfrak{h}$, we construct a germ of a real analytic hypersurface in complex space such that its Lie algebra of infinitesimal holomorphic automorphisms is isomorphic to $\mathfrak{h}$. For any $\mathfrak{h}$, we also construct a system of partial differential equations whose Lie algebra of symmetries is isomorphic to the complexification of the algebra $\mathfrak{h}$.

Realization of arbitrary Lie algebras by automorphisms of $\mathrm{CR}$ manifolds and symmetries of differential equations

Для произвольной конечномерной вещественной алгебры Ли $\mathfrak{h}$ выписан росток вещественно аналитической гиперповерхности комплексного пространства, такой что его алгебра Ли инфинитезимальных голоморфных автоморфизмов изоморфна $\mathfrak{h}$. Также для произвольной $\mathfrak{h}$ построена система уравнений в частных производных, алгебра Ли симметрий которой изоморфна комплексификации алгебры $\mathfrak{h}$. Библиография: 13 наименований.

Conservative algebras of 2-dimensional algebras, IV

The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra [Formula: see text] of all algebras (i.e. bilinear maps) on the [Formula: see text]-dimensional vector space. If [Formula: see text], then the algebra [Formula: see text] does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like [Formula: see text] in the theory of conservative algebras plays a similar role to the role of [Formula: see text] in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra [Formula: see text] for some [Formula: see text] The present paper is part of a series of papers, which is dedicated to the study of the algebra [Formula: see text] and its principal subalgebras.

Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States.

The Jordan-Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan-Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.

On categorical structures arising from implicative algebras: From topology to assemblies

Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.

Gleason’s theorem for composite systems

Gleason’s theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.

Jordan 3-graded Lie algebras with polynomial identities

We study Jordan 3-graded Lie algebras satisfying 3-graded polynomial identities. Taking advantage of the Tits-Kantor-Koecher construction, we interpret the PI-condition in terms of their associated Jordan pairs, which allows us to formulate an analogue of Posner-Rowen Theorem for strongly prime PI Jordan 3-graded Lie algebras. Arbitrary PI Jordan 3-graded Lie algebras are also described by introducing the Kostrikin radical of the Lie algebras.

A functor for constructing R$R$‐matrices in the category O$\mathcal {O}$ of Borel quantum loop algebras

AbstractWe tackle the problem of constructing ‐matrices for the category associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebra . For this, we define an invertible exact functor from the category linked to to the one linked to . This functor is compatible with tensor products, preserves irreducibility, and interchanges the subcategories and of Hernandez and Leclerc (Algebra Number Theory 10 (2016) 2015–2052). We construct ‐matrices for by applying on the braidings already found for by Hernandez (Represent. Theory 26 (2022) 179–210). We also use the factorization of the latter intertwiners in terms of stable maps to deduce an analogous factorization for our new braidings. We finally obtain as byproducts new relations for the Grothendieck ring as well as a functorial interpretation of a remarkable ring isomorphism of Hernandez–Leclerc.

Horizontally Affine Functions on Step-2 Carnot Algebras

In this paper, we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-n Carnot algebra is isomorphic to the exterior algebra of {mathbb {R}}^n. Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.

-PERPENDICULAR WIDE SUBCATEGORIES

Abstract Let $\Lambda $ be a finite-dimensional algebra. A wide subcategory of $\mathsf {mod}\Lambda $ is called left finite if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of $\mathsf {mod}\Lambda $ arising from $\tau $ -tilting reduction are precisely the Serre subcategories of left-finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $\tau $ -tilting reduction. This leads to a natural way to extend the definition of the “ $\tau $ -cluster morphism category” of $\Lambda $ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the $\tau $ -tilting finite case and by Igusa–Todorov in the hereditary case.

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

Idempotent pre-endomorphisms of algebras

In the study of pre-Lie algebras, the concept of pre-morphism arises naturally as a generalization of the standard notion of morphism. Pre-morphisms can be defined for arbitrary (not-necessarily associative) algebras over any commutative ring k with identity, and can be dualized in various ways to generalized morphisms (related to pre-Jordan algebras) and anti-pre-morphisms (related to anti-pre-Lie algebras). We consider idempotent pre-endomorphisms (generalized endomorphisms, anti-pre-endomorphisms). Idempotent pre-endomorphisms are related to semidirect-product decompositions of the sub-adjacent anticommutative algebra.

Discovering sparse representations of Lie groups with machine learning

Recent work has used deep learning to derive symmetry transformations, which preserve conserved quantities, and to obtain the corresponding algebras of generators. In this letter, we extend this technique to derive sparse representations of arbitrary Lie algebras. We show that our method reproduces the canonical (sparse) representations of the generators of the Lorentz group, as well as the U(n) and SU(n) families of Lie groups. This approach is completely general and can be used to find the infinitesimal generators for any Lie group.

The characteristic polynomial in calculation of exponential and elementary functions in Clifford algebras

Formulas to calculate multivector exponentials in a basis‐free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalized to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems ofanalysis of evolution of the entangled quantum states.