Universal enveloping algebra
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BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras Open
A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β:A→A such that α(a)(bc)=(ab)β(c), for all a,b,c∈A. This concept arose in the study of algebras in so-called group Hom-cate…
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Classification of nilpotent associative algebras of small dimension Open
We classify nilpotent associative algebras of dimensions up to [Formula: see text] over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods…
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Higher enveloping algebras Open
We provide spectral Lie algebras with enveloping algebras over the operad of little [math] –framed [math] –dimensional disks for any choice of dimension [math] and structure group [math] , and we describe these objects in two complementary…
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The Hopf Algebra of Fliess Operators and Its Dual Pre-lie Algebra Open
We study the Hopf algebra H of Fliess operators coming from Control Theory in\nthe one-dimensional case. We prove that it admits a graded, finte-dimensional,\nconnected gradation. Dually, the vector space IR is both a pre-Lie algebra for\n…
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Associative realizations of the extended Snyder model Open
The star product usually associated to the Snyder model of noncommutative\ngeometry is nonassociative, and this property prevents the construction of a\nproper Hopf algebra. It is however possible to introduce a well-defined Hopf\nalgebra …
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Pseudo-Euclidean Jordan Algebras Open
International audience
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Hom-structures on semi-simple Lie algebras Open
A Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is als…
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Squares and associative representations of two dimensional evolution algebras Open
We associate an square to any two dimensional evolution algebra. This geomet- ric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degr…
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Geometrical aspects of the Lie algebra S-expansion procedure Open
In this article it is shown that S-expansion procedure affects the geometry of a Lie group, changing it and leading us to the geometry of another Lie group with higher dimensionality. A method for determining the semigroup, which would pro…
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Remarks on quantum unipotent subgroups and the dual canonical basis Open
We prove the tensor product decomposition of the half of quantized universal enveloping algebra associated with a Weyl group element which was conjectured by Berenstein and Greenstein.
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Post-Lie algebras in Regularity Structures Open
In this work, we construct the deformed Butcher-Connes-Kreimer Hopf algebra coming from the theory of Regularity Structures as the universal envelope of a post-Lie algebra. We show that this can be done using either of the two combinatoria…
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Post-Hopf algebras, relative Rota–Baxter operators and solutions to the Yang–Baxter equation Open
In this paper, first, we introduce the notion of post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and the fact that there is naturally a post-Hopf algebra structure on the universal enveloping al…
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Algebras with representable representations Open
Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to …
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Associative algebras for (logarithmic) twisted modules for a vertex operator algebra Open
We construct two associative algebras from a vertex operator algebra $V$ and a general automorphism $g$ of $V$. The first, called a $g$-twisted zero-mode algebra, is a subquotient of what we call a $g$-twisted universal enveloping algebra …
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On the algebra of parallel endomorphisms of a pseudo-Riemannian metric Open
On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e.\nsections of End(TM) may be parallel for g. They form an associative algebra A,\nwhich is also the commutant of the holonomy group of g. As any associative\nalgebra…
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Correspondences among CFTs with different W-algebra symmetry Open
W-algebras are constructed via quantum Hamiltonian reduction associated with\na Lie algebra $\\mathfrak{g}$ and an $\\mathfrak{sl}(2)$-embedding into\n$\\mathfrak{g}$. We derive correspondences among correlation functions of\ntheories havi…
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<i>V</i>-universal Hopf algebras (co)acting on Ω-algebras Open
We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra struc…
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Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere Open
Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is …
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On q-deformed infinite-dimensional n-algebra Open
The q-deformation of the infinite-dimensional n-algebras is investigated. Based on the structure of the q-deformed Virasoro–Witt algebra, we derive a nontrivial q-deformed Virasoro–Witt n-algebra which is nothing but a sh-n-Lie algebra. Fu…
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Root vectors of the composition algebra of the Kronecker algebra Open
According to the canonical isomorphism between
\n the positive part Uq⁺(g) of the Drinfeld–Jimbo quantum group
\n Uq(g) and the generic composition algebra C(∆) of Λ, where the
\n Kac–Moody Lie algebra g and the finite dimensional heredita…
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Conservative algebras of 2-dimensional algebras, II Open
In 1990 Kantor defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does not belong to any well-known class of algebras (such as associative, Li…
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The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example Open
In this paper we study in detail algebraic properties of the algebra\n$\\mathcal D(W)$ of differential operators associated to a matrix weight of\nGegenbauer type. We prove that two second order operators generate the algebra,\nindeed $\\m…
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Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules Open
A classification of the simple weight modules is given for the (6-dimensional) Euclidean Lie algebra 𝔢(3) = 𝔰𝔩2⋉V3. As an intermediate step, a classification of all simple modules is given for the centralizer C of the Cartan element H (in …
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Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras Open
We introduce a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We show that the transposed Poisson algebra thus defined not only shares common proper…
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Lie, associative and commutative quasi-isomorphism Open
Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. This answers a folklore problem in rational homot…
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Universal Central Extension of the Lie Algebra of Hamiltonian Vector Fields: Table 1. Open
We determine the universal central extension of the Lie algebra of\nhamiltonian vector fields, thereby classifying its central extensions.\nFurthermore, we classify the central extensions of the Lie algebra of\nsymplectic vector fields, of…
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On associative representations of non-associative algebras Open
We define a notion of associative representation for algebras. We prove the existence of faithful associative representations for any alternative, Mal’cev, and Poisson algebra, and prove analogs of Ado-Iwasawa theorem for each of these cas…
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Hopf algebras of prime dimension in positive characteristic Open
We prove that a Hopf algebra of prime dimension $p$ over an algebraically\nclosed field, whose characteristic is equal to $p$, is either a group algebra\nor a restricted universal enveloping algebra. Moreover, we show that any Hopf\nalgebr…
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Envelopes and refinements in categories, with applications to functional analysis Open
An envelope in a category is a construction that generalizes the operations\nof "exterior completion", like completion of a locally convex space, or\nStone-\\v{C}ech compactification of a topological space, or universal enveloping\nalgebra…
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Universal property of skew PBW extensions Open
In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commuta…