Finite-dimensional Algebra Research Articles

HOPF CO-BRACE, BRAID EQUATION AND BICROSSED

In this paper, we mainly give some equivalent characterisations of Hopf cobraces, show that the full subcategory HCB(A) of Hopf co-braces is equivalent to the full subcategory C(A) of bijective 1-cocycles, and prove that the full subcategory HCB(A) is also equivalent to the category M(A) of Hopf matched pairs. Moreover, we construct many Hopf co-braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel’d doubles of finite dimensional Hopf algebras. And we also give a sufficient and necessary condition for a given bicrossed coproduct A ▷◁ H to be a Hopf co-brace if A or H is a Hopf co-brace

HOMOLOGY THEORY OF MULTIPLICATIVE HOM-LIE ALGEBRAS

In this article, we establish Serre-Hochschild spectral sequences of Hom-Lie algebras. By using these spectral sequences, we describe homology groups of finite dimensional multiplicative Hom-Lie algebras in terms of homology groups of Lie algebras and abe

Smooth DG algebras and twisted tensor product

The twisted tensor product of DG algebras is studied and sufficient conditions for smoothness of such a product are presented. It is shown that in the case of finite-dimensional DG algebras, applying this operation offers great possibilities for constructing new examples of smooth DG algebras and algebras. In particular, examples are given of families of algebras of finite global dimension with two simple modules that have non-trivial moduli spaces. Bibliography: 24 titles.

On a classification of faithful representations of the Galilean lie algebra on the polynomial ring in three variables

We show a complete classification of faithful representations of the 2 + 1 space-times Galilean Lie algebra on the polynomial ring in three variables, where actions of the Galilean Lie algebra are given by derivations with coefficients of degree at most one. In particular, all such representations of the Galilean Lie algebra are explicitly constructed and classified by one parameter. In a more general setting we show that, with respect to a nonzero abelian ideal of a finite-dimensional Lie algebra, there is at most one such representation up to graded-equivalence.

Smooth DG algebras and twisted tensor product

The twisted tensor product of DG algebras is studied and sufficient conditions for smoothness of such a product are presented. It is shown that in the case of finite-dimensional DG algebras, applying this operation offers great possibilities for constructing new examples of smooth DG algebras and algebras. In particular, examples are given of families of algebras of finite global dimension with two simple modules that have non-trivial moduli spaces. Bibliography: 24 titles.

On Simple Finite-Dimensional Algebras with Infinite Bases of Identities

В 1993 г. И. П. Шестаков поставил вопрос о существовании центральной простой конечномерной алгебры над полем нулевой характеристики, тождества которой не задаются конечным набором тождеств. В 2012 г. И. М. Исаевым и автором построен пример семимерной центральной простой алгебры над любым полем, не имеющей конечного базиса тождеств. В данной работе строится пример шестимерной центральной простой алгебры над полем нулевой характеристики, не имеющей конечного базиса тождеств. Библиография: 22 названия.

One-Sided Isotopes of Finite-Dimensional Algebras

It is shown that the right isotopes of finite-dimensional (−1, 1)-algebras cannot be reduced to left isotopes. It is proved that no unital isotope of the Mikheev algebra is a left alternative algebra. In particular, the opposite algebra, generally speaking, is not an isotope of the original algebra.

4–Manifold invariants from Hopf algebras

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.

Virtual mutations of weighted surface algebras

The finite-dimensional symmetric algebras over algebraically closed fields, based on surface triangulations, motivated by the theory of cluster algebras, have been extensively investigated and applied. In particular, the weighted surface algebras and their deformations were introduced and studied in [14–18], and it was shown that all these algebras, except a few singular cases, are symmetric tame periodic algebras of period 4. In this article, using the general form of a weighted surface algebra from [17], we introduce and study so called virtual mutations of weighted surface algebras, which constitute a new large class of symmetric tame periodic algebras of period 4. We prove that all these algebras are derived equivalent but not isomorphic to weighted surface algebras. The results of this paper form an essential step towards a classification of all tame symmetric periodic algebras.

A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS

In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra \(\mathcal{D}\) and the seven-dimensional central simple commutative algebra \(\mathcal{C}\). We prove that every local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is a derivation, and every 2-local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also a derivation. We also prove that every local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is an automorphism, and every 2-local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also an automorphism.

Hopf Differential Graded Galois Extensions

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that D(R#H)≅D(RH) is equivalent to that R/RH is a Hopf differential graded Galois extension. We present a weaker version of Hopf differential graded Galois extensions and show the relationships between Hopf differential graded Galois extensions and Hopf Galois extensions.

The images of multilinear and semihomogeneous polynomials on the algebra of octonions

The generalized L'vov–Kaplansky conjecture states that for any finite-dimensional simple algebra A the image of a multilinear polynomial on A is a vector space. In this paper, we prove it for the algebra of octonions O over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field R ). In fact, letting V be the space of pure octonions in O , we prove that the image set must be either { 0 } , F, V or O . We discuss possible evaluations of semihomogeneous polynomials on O and of arbitrary polynomials on the corresponding Malcev algebra.

Finite generation for Hochschild cohomology of Gorenstein monomial algebras

We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall–Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann–Holloway–Snashall–Solberg–Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.

Finite-dimensional Nichols algebras over the Suzuki algebras I: simple Yetter-Drinfeld modules of $A_{N\,2n}^{\mu\lambda}$

The Suzuki algebra $A_{Nn}^{\mu \lambda}$, introduced by Suzuki Satoshi in 1998, is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{\mu\lambda}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;\I_2)$ and the Nichols algebra $\ufo(8)$. There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N\,2n}^{\mu \lambda}$. The $64$-dimensional Nichols algebras are of dihedral rack type $\Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $\mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{\mu \lambda}$. Using a result of Masuoka, we prove that $\dim\mathfrak{B}(V_{abe})=\infty$ under the condition $b^2=(ae)^{-1}$, $b\in\Bbb{G}_{m}$ for $m\geq 5$.

Poles of finite-dimensional representations of Yangians

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $Y_{\hbar}(\mathfrak{g})$ be the Yangian of $\mathfrak{g}$. In this paper, we study the sets of poles of the rational currents defining the action of $Y_{\hbar}(\mathfrak{g})$ on an arbitrary finite-dimensional vector space $V$. Using a weak, rational version of Frenkel and Hernandez' Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of $V$ and the inverse of the $q$-Cartan matrix of $\mathfrak{g}$. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double.

The Polynomial Modules over Quantum Group Uq(sl3)

Let [Formula: see text] be a finite dimensional complex simple Lie algebra with Cartan subalgebra [Formula: see text]. Then [Formula: see text] has a [Formula: see text]-module structure if and only if [Formula: see text] is of type [Formula: see text] or of type [Formula: see text]; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over [Formula: see text] have been classified. In this paper, we prove that the quantum group [Formula: see text] has no rank one polynomial module.

Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so that the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field , being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed.

Finite-dimensional Algebras are (m> 2)-Calabi-Yau-tilted

Finite-dimensional Algebras are (m> 2)-Calabi-Yau-tilted

On the nilpotency index of the radical of a module category

Let A be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to determine which vertices of QA are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of A. Precisely, we study the above mentioned problem whenever A is a monomial or a toupie algebra where their Auslander-Reiten quivers are component with length. We also study the above problem for string algebras. We give concrete formulas to compute such index for many classes of algebras such as toupie algebras, tree algebras and some classes of pullback algebras over the hereditary algebras of type An, Dn and E6. Furthermore, we determine the index taking into account the ordinary quiver of the given algebras.

Homological systems and bocses

We show that, up to Morita equivalence, any standardly stratified algebra, admits an exact Borel subalgebra. In fact, we show this in the more general case of finite-dimensional algebras possessing an admissible homological system. This generalizes a theorem by Koenig, Külshammer, and Ovsienko, which holds for quasi-hereditary algebras. Our proof follows the same general scheme proposed by these authors, in a more general context: we associate a differential graded tensor algebra with relations, using the structure of A∞-algebra of a suitable Yoneda algebra, and use its category of modules to describe the category of filtered modules associated to the given homological system.