Family Of Algebras Research Articles

APPLICATION OF SYSTEMS ANALYSIS AND COMPUTER ALGORITHMS IN STUDYING ORBITS OF 7-DIMENSIONAL LIE ALGEBRAS

We discuss a systematic approach to the problem of describing holomorphically homogeneous real hypersurfaces in the space C4, each of which is an orbit of some real Lie algebra. When studying the family of 7-dimensional Lie algebras, which plays an important role in the problem at hand and contains more than a thousand different types of algebras, it is natural to use computer algorithms. With the participation of the authors of this article, classification results on the orbits of several large blocks of algebras from this family were previously obtained. Relations are established between the presence and dimensions of nilpotent and Abelian subalgebras of the original Lie algebras and such properties of their orbits in C4 as Levi degeneracy and tubularity. In this article, the above ideas and computer algorithms are applied to a family of 18 types of 7-dimensional Lie algebras that have a common 6-dimensional nil-radical. Holomorphic realizations in C4 of these algebras are constructed and by integrating them, all holomorphically homogeneous (in the local sense) Levi-nondegenerate 7-dimensional orbits of this family are obtained. Using a quadratic change of variables, it is shown that all these orbits are holomorphically equivalent to tubular hypersurfaces.

Remarks on generating families of matrix algebras of small orders

Remarks on generating families of matrix algebras of small orders

On types of elements, Gelfand and strongly harmonic rings of skew PBW extensions over weak compatible rings

We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, π-regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.

Operator Kantor pairs

Kantor pairs, (quadratic) Jordan pairs, and similar structures have been instrumental in the study of Z-graded Lie algebras and algebraic groups. We introduce the notion of an operator Kantor pair, a generalization of Kantor pairs to arbitrary (commutative, unital) rings, similar in spirit as to how quadratic Jordan pairs and algebras generalize linear Jordan pairs and algebras. Such an operator Kantor pair is formed by a pair of Φ-groups (G+,G−) of a specific kind, equipped with certain homogeneous operators. For each such a pair (G+,G−), we construct a 5-graded Lie algebra L together with actions of G± on L as automorphisms. Moreover, we can associate a group G(G+,G−)⊂Aut(L) to this pair generalizing the projective elementary group of Jordan pairs. If the non-0-graded part of L is projective, we can uniquely recover G+,G− from G(G+,G−) and the grading on L alone. We establish, over rings Φ with 1/30∈Φ, a one to one correspondence between Kantor pairs and operator Kantor pairs. Finally, we construct operator Kantor pairs for the different families of central simple structurable algebras.

3-Preprojective Algebras of Type D

AbstractWe present a family of selfinjective algebras of type D, which arise from the 3-preprojective algebras of type A by taking a $$\mathbb {Z}_3$$ Z 3 -quotient. We show that a subset of these are themselves 3-preprojective algebras, and that the associated 2-representation-finite algebras are fractional Calabi-Yau. In addition, we show our work is connected to modular invariants for SU(3).

Multiplicities of Representations in Algebraic Families

Multiplicities of Representations in Algebraic Families

Extended derivations of algebras

We study the concept of extended derivations of algebras which expands diverse definitions of generalized derivations given in the literature. We concentrate on the family of the anti-commutative algebras and classify such spaces of derivations by considering a suitable notion of equivalence when we restrict attention to sucssh algebras. Afterwards, we investigate a link between the well-known ( α , β , γ ) -derivations and some new extended derivations obtained in our classification. Finally, we present applications of extended derivations to study degenerations of algebras.

Classical Dynamical r-matrices for the Chern–Simons Formulation of Generalized 3d Gravity

AbstractClassical dynamical r-matrices arise naturally in the combinatorial description of the phase space of Chern–Simons theories, either through the inclusion of dynamical sources or through a gauge fixing procedure involving two punctures. Here we consider classical dynamical r-matrices for the family of Lie algebras which arise in the Chern–Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical r-matrices in this case and show that they can be viewed as generalized complexifications, in a sense which we define, of the equations governing dynamical r-matrices for $$\mathfrak {su}(2)$$ su ( 2 ) and $$\mathfrak {sl}(2,{\mathbb {R}})$$ sl ( 2 , R ) . We obtain explicit families of solutions and relate them, via Weierstrass factorization, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.

Representations of shifted quantum affine algebras and cluster algebras I: The simply laced case

AbstractWe introduce a family of cluster algebras of infinite rank associated with root systems of type , , . We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called ‐system. We conjecture that all cluster monomials are classes of simple objects of . In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply connected algebraic group. This explains the similarity between ‐system relations and certain generalized minor identities discovered by Fomin and Zelevinsky.

On the cohomology of restricted Heisenberg Lie algebras

We show that the Heisenberg Lie algebras over a field F of characteristic p>0 admit a family of restricted Lie algebras, and we classify all such non-isomorphic restricted Lie algebra structures. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the restricted 1-dimensional central extensions, including explicit formulas for the Lie brackets and ⋅[p]-operators.

Graded contractions of the [formula omitted]-grading on [formula omitted]

Graded contractions of the Z23-grading on the complex exceptional Lie algebra g2 are classified up to equivalence and up to strong equivalence. The non-toral fine Z23-grading is highly symmetric, with all the homogeneous components Cartan subalgebras. This makes possible a combinatorial treatment based on certain nice subsets of the set of 21 edges of the Fano plane. There are 24 such nice sets up to collineation. Each of these is the support of an admissible graded contraction, one of which is present in every equivalence class of graded contractions. Each nice set gives rise to a single Lie algebra, except for three of the cases in which families depending on one or two parameters are found. In particular, a large family of 14-dimensional Lie algebras arise, most of which are solvable. The properties of each of these Lie algebras are studied.

A classification of automorphic Lie algebras on complex tori

Abstract We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$ . For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2({\mathbb{Z}})$ , apart from four cases, which are all isomorphic to Onsager’s algebra.

A Generalization of the First Tits Construction

Let F be a field of characteristic, not 2 or 3. The first Tits construction is a well-known tripling process to construct separable cubic Jordan algebras, especially Albert algebras. We generalize the first Tits construction by choosing the scalar employed in the tripling process outside of the base field. This yields a new family of non-associative unital algebras which carry a cubic map, and maps that can be viewed as generalized adjoint and generalized trace maps. These maps display properties often similar to the ones in the classical setup. In particular, the cubic norm map permits some kind of weak Jordan composition law.

On the isomorphism between four-dimensional Sklyanin and elliptic algebras

Let [Formula: see text] be an elliptic curve, and let [Formula: see text] be a point on [Formula: see text]. In his groundbreaking paper [E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representations of a quantum algebra, Funktsional. Anal. i Prilozhen. 17(4) (1983) 34–48], Sklyanin introduced a family of algebras [Formula: see text], consisting of four generators and six relations. These algebras hold profound connections with his work on the Quantum Scattering Method. Subsequently, Feigin and Odesskii defined algebras [Formula: see text] for any number of generators [Formula: see text] [Sklyanin’s algebras associated to elliptic curves, Inst. Theor. Phys., Akad. Nauk Ukrain. SSR, Kiev (1988)] and asserted that, for [Formula: see text], they coincide with Sklyanin’s algebras. In this paper, we provide an explicit isomorphism between the algebras [Formula: see text] and [Formula: see text]. As an application, we present a new proof of the Calabi–Yau property for [Formula: see text].

Hochschild homology of reductive $p$-adic groups

Consider a reductive p -adic group G , its (complex-valued) Hecke algebra \mathcal H (G) , and the Harish-Chandra–Schwartz algebra \mathcal S (G) . We compute the Hochschild homology groups of \mathcal H (G) and of \mathcal S (G) , and we describe the outcomes in several ways. Our main tools are algebraic families of smooth G -representations. With those we construct maps from HH_{n} (\mathcal H (G)) and HH_{n} (\mathcal S(G)) to modules of differential n -forms on affine varieties. For n = 0 , this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) G -representations. It is known from [J. Algebra 606 (2022), 371–470] that every Bernstein ideal \mathcal H (G)^{\mathfrak s} of \mathcal H (G) is closely related to a crossed product algebra of the form \mathcal O (T)\rtimes W . Here \mathcal O (T) denotes the regular functions on the variety T of unramified characters of a Levi subgroup L of G , and W is a finite group acting on T . We make this relation even stronger by establishing an isomorphism between HH_{*} (\mathcal H (G)^{\mathfrak s}) and HH_{*} (\mathcal O (T)\rtimes W) , although we have to say that in some cases it is necessary to twist \mathbb{C} [W] by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal \mathcal S (G)^{\mathfrak s} of \mathcal S (G) is isomorphic to HH_{*} (C^{\infty} (T_{u})\rtimes W) , where T_{u} denotes the Lie group of unitary unramified characters of L . In these pictures of HH_{*} (\mathcal H (G)) and HH_{*} (\mathcal S (G)) , we also show how the Bernstein centre of \mathcal H (G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of \mathcal H (G) and of \mathcal S (G) and we relate that to topological K-theory.

On Hopf algebras whose coradical is a cocentral abelian cleft extension

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn , n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K 3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216.

An inductive approach to representations of general linear groups over compact discrete valuation rings

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called PSH-algebras. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.

Chain algebras of finite distributive lattices

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h-vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension n>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n>2$$\\end{document}, we show that the defining ideal has minimal generators of degree at least n.

Equivariant Fusion Subcategories

AbstractWe provide a parameterization of all fusion subcategories of the equivariantization by a group action on a fusion category. As applications, we classify the Hopf subalgebras of a family of semisimple Hopf algebras of Kac-Paljutkin type and recover Naidu-Nikshych-Witherspoon classification of the fusion subcategories of the representation category of a twisted quantum double of a finite group.

A dichotomy between twisted tensor products of bialgebras and Frobenius algebras

We endow twisted tensor products with a natural notion of counit and comultiplication, and we provide sufficient and necessary conditions making the twisted tensor product a counital coassociative coalgebra. We then characterize when the twisted tensor product of bialgebras is a bialgebra, and when the twisted tensor product of Frobenius algebras is a Frobenius algebra. Our methods are purely diagrammatic, so these results hold for (braided) monoidal categories. As an application, we recover that some quantum complete intersections are Frobenius algebras, and we construct families of noncommutative symmetric Frobenius algebras. Along the way, we also characterize when twisted tensor products of separable algebras are separable, and we prove that twisted tensor products of special Frobenius algebras are special Frobenius.