Dimensional Associative Algebra Research Articles

On the order types of hammocks for domestic string algebras

In the representation-theoretic study of finite dimensional associative algebras over an algebraically closed field, Brenner introduced certain partially ordered sets known as hammocks to encode factorizations of maps between indecomposable finitely generated modules. In the context of domestic string algebras, Schröer introduced a simpler version of hammocks in his doctoral thesis that are bounded discrete linear orders. In this paper, we characterize the class of order types(=order isomorphism classes) of hammock linear orders for domestic string algebras as the bounded discrete ones amongst the class LOfp of finitely presented linear orders–the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphisms, order reversal, finite order sums and antilexicographic products.In fact, we provide a multi-step algorithm to compute the order type of any closed interval in the hammock, and prove the correctness of this algorithm. A major step of this algorithm is the construction of a variation, which we call the arch bridge quiver, of a finite combinatorial gadget called the bridge quiver introduced by Schröer. He utilised the graph-theoretic properties of the bridge quiver for the computation of some representation-theoretic numerical invariants of domestic string algebras. The vertices of the bridge quiver are (representatives of cyclic permutations of) bands and its arrows are certain band-free strings. There is a natural but ill-behaved partial binary operation, ∘, on a superset of the set of bridges consisting of weak bridges such that bridges are precisely the ∘-irreducibles. We equip an even larger yet finite set of weak arch bridges with another partial binary operation, ∘H, to obtain a finite category. The binary operation ∘H uses isomorphisms between hammocks and explicitly relies on the description of the domestic string algebra as a bound quiver algebra. Each weak arch bridge admits a unique ∘H-factorization into arch bridges, i.e., the ∘H-irreducibles.

Graded group actions and generalized H-actions compatible with gradings

We introduce the notion of a graded group action on a graded algebra or, which is the same, a group action by graded pseudoautomorphisms. An algebra with such an action is a natural generalization of an algebra with a super- or a pseudoinvolution. We study groups of graded pseudoautomorphisms, show that the Jacobson radical of a group graded finite dimensional associative algebra A over a field of characteristic 0 is stable under graded pseudoautomorphisms, prove the invariant version of the Wedderburn–Artin Theorem and the analog of Amitsur's conjecture for the codimension growth of graded polynomial G-identities in such algebras A with a graded action of a group G.

Differential codimensions and exponential growth

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let cnL(A), n≥1, be its differential codimension sequence. Such sequence is exponentially bounded and expL⁡(A)=limn→∞⁡cnL(A)n is an integer that can be computed, called differential PI-exponent of A.In this paper we prove that for any Lie algebra L, expL⁡(A) coincides with exp⁡(A), the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth.

The quantum detection of projectors in finite-dimensional algebras and holography

We define the computational task of detecting projectors in finite dimensional associative algebras with a combinatorial basis, labelled by representation theory data, using combinatorial central elements in the algebra. In the first example, the projectors belong to the centre of a symmetric group algebra and are labelled by Young diagrams with a fixed number of boxes n. We describe a quantum algorithm for the task based on quantum phase estimation (QPE) and obtain estimates of the complexity as a function of n. We compare to a classical algorithm related to the projector identification problem by the AdS/CFT correspondence. This gives a concrete proof of concept for classical/quantum comparisons of the complexity of a detection task, based in holographic correspondences. A second example involves projectors labelled by triples of Young diagrams, all having n boxes, with non-vanishing Kronecker coefficient. The task takes as input the projector, and consists of identifying the triple of Young diagrams. In both of the above cases the standard QPE complexities are polynomial in n. A third example of quantum projector detection involves projectors labelled by a triple of Young diagrams, with m, n and m + n boxes respectively, such that the associated Littlewood-Richardson coefficient is non-zero. The projector detection task is to identify the triple of Young diagrams associated with the projector which is given as input. This is motivated by a two-matrix model, related via the AdS/CFT correspondence, to systems of strings attached to giant gravitons. The QPE complexity in this case is polynomial in m and n.

Algorithm method for generalized derivations of two dimensional associative algebras

تمت دراسة مفهوم الاشتقاقات المعممة للجبر الترابطي ثنائي الأبعاد وخصائصها. تم إعطاء خوارزمية لإيجاد الاشتقاقات المعممة ومسافاتها الفرعية للجبر الترابطي [1]. تحتوي الجبر الترابطية ثنائية الأبعاد على 5 فئات تماثلية للجبر. كل فئة من فئات التشابه لها جدول ضرب مختلف.

Generalization of Jordan-Lie inner ideals of finite dimensional associative algebras

We generalize Baranov and Shlaka’s results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let [Formula: see text] be a finite dimensional [Formula: see text]-perfect associative algebras [Formula: see text] over an algebraically closed field [Formula: see text] of arbitrary characteristic [Formula: see text] and let [Formula: see text] be a subalgebra of [Formula: see text]. We prove that for any bar-minimal Jordan-Lie inner ideal [Formula: see text] a [Formula: see text], there is a bar-minimal Jordan-Lie inner ideal of [Formula: see text] that contains [Formula: see text] and if [Formula: see text] is regular, then is a regular inner ideal of [Formula: see text] that contains [Formula: see text]. We also prove that for any strict orthogonal pair [Formula: see text] in [Formula: see text], there is a strict orthogonal idempotent pair [Formula: see text] in [Formula: see text] such that [Formula: see text].

The Automorphisms Group and the Classification of Gradings of Finite Dimensional Associative Algebras

A $4$-algebra is a commutative algebra $A$ over a field $k$ such that $(a^2)^2 = 0$, for all $a \in A$. We have proved recently \cite{Mil} that $4$-algebras play a prominent role in the classification of finite dimensional Bernstein algebras. Let $A$ be a $4$-algebra, $E$ a vector space and $\pi : E \to A$ a surjective linear map with $V = {\rm Ker} (\pi)$. All $4$-algebra structures on $E$ such that $\pi : E \to A$ is an algebra map are described and classified by a global cohomological object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$. Any such $4$-algebra is isomorphic to a crossed product $V \# A$ and ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is a coproduct, over all $4$-algebras structures $\cdot_V$ on $V$, of all non-abelian cohomologies ${\mathbb H}^{2}_{\rm nab} \, \bigl(A, \, (V, \, \cdot_{V} )\bigl)$, which are the classifying objects for all extensions of $A$ by $V$. Several applications and examples are provided: in particular, ${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$ and ${\mathbb G} {\mathbb H}^{2} \, (k, \, V)$ are explicitly computed and the Galois group ${\rm Gal} \, (V \# A/ V )$ of the extension $V \hookrightarrow V \# A$ is described.

Morita Invariance for Infinitesimal Deformations

Let A and B be two Morita equivalent finite dimensional associative algebras over a field 𝕜. It is well known that Hochschild cohomology is invariant under Morita equivalence. Since infinitesimal deformations are connected with the second Hochschild cohomology group, we explicitly describe the transfer map connecting HH2(A) with HH2(B). This allows us to transfer Morita equivalence between A and B to that between infinitesimal deformations of them. As an application, when 𝕜 is algebraically closed, we consider the quotient path algebra associated to A and describe the presentation by quiver and relations of the infinitesimal deformations of A.

Retraction article: Eight-dimensional octonion-like but associative normed division algebra

We present an eight-dimensional even sub-algebra of the -dimensional associative Clifford algebra and show that its eight-dimensional elements denoted as and respect the norm relation thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

Locally Finite Associative Algebras and Their Lie Subalgebras

An infinite dimensional associative algebra over a field is called locally finite associative algebra if every finite set of elements is contained in a finite dimensional subalgebra of . Given any associative algebra over field of any characteristic. Consider a new multiplication on called the Lie multiplication which defined by [a, b] = ab − ba for all a, b ∈ A, where ab is the associative multiplication in . Then L = (−) together with the Lie multiplication form a Lie subalgebra of . It is natural to expect that the structures of L and are connected closely. In this paper, we study and discuss the structure of infinite dimensional locally finite Lie and associative algebras. The relation between them, their ideals and their inner ideals is considered. A brief discussion of the simple associative algebras and simple Lie algebras is also be provided.

Jordan-Lie inner ideals of finite dimensional associative algebras

We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and prove that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.

On geometric degenerations and Gerstenhaber formal deformations

We study the degeneration relations on the varieties of associative and Lie algebra structures on a fixed finite dimensional vector space and give a description of them in terms of Gerstenhaber formal deformations. We use this result to show how the orbit closure of the $3$-dimensional Lie algebras can be determined using homological algebra. For the case of finite dimensional associative algebras, we prove that the $N$-Koszul property is preserved under the degeneration relation for all $N\geq2$.

Non integral exponential growth of central polynomials

Let A be an algebra over a field F of characteristic zero. For every $$n\ge 1$$ , let $$\delta _n(A)$$ be the number of linearly independent multilinear proper central polynomials of A in n fixed variables. It was shown in [8] that if A is a finite dimensional associative algebra, the limit $$\delta (A)=\lim _{n\rightarrow \infty }\root n \of {\delta _n(A)}$$ always exists and is an integer. Here we show that such a result cannot be extended in general to non associative algebras. In fact we construct a five-dimensional non associative algebra such that the above limit exists and $$\delta (A)\approx 3.61$$ , a non integer.

Diagram calculus for a type affine C Temperley–Lieb algebra, II

In a previous paper, we presented an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley–Lieb algebra having a basis indexed by the fully commutative elements of the Coxeter group of type affine C. We also provided an explicit description of a basis for the diagram algebra. In this paper, we show that the diagrammatic representation is faithful and establish a correspondence between the basis diagrams and the so-called monomial basis of the Temperley–Lieb algebra of type affine C.

Wedderburn–Malcev decomposition of one-sided ideals of finite dimensional algebras

ABSTRACTLet A be a finite dimensional associative algebra over a perfect field and let R be the radical of A. We show that for every one-sided ideal I of A there exists a semisimple subalgebra S of A such that where IS = I∩S and IR = I∩R.

Algebras and semigroups of locally subexponential growth

We prove that a countable dimensional associative algebra (resp. a countable semigroup) of locally subexponential growth is M∞-embeddable as a left ideal in a finitely generated algebra (resp. semigroup) of subexponential growth. Moreover, we provide bounds for the growth of the finitely generated algebra (resp. semigroup). The proof is based on a new construction of matrix wreath product of algebras.

DERIVATIONS AND CENTROIDS OF FOUR-DIMENSIONAL ASSOCIATIVE ALGEBRAS

In this paper, we focus on derivations and centroids of four dimensional associative algebras. Using an existing classification result of low dimensional associative algebras, we describe the derivations and centroids of four dimensional associative algebras. We also identify algebra(s) that belong to the characteristically nilpotent class among the algebras of four dimensional associative algebras.

Factorization of Temperley–Lieb diagrams

The Temperley--Lieb algebra is a finite dimensional associative algebra that arose in the context of statistical mechanics and occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$. It is often realized in terms of a certain diagram algebra, where every diagram can be written as a product of "simple diagrams." These factorizations correspond precisely to factorizations of the so-called fully commutative elements of the Coxeter group that index a particular basis. Given a reduced factorization of a fully commutative element, it is straightforward to construct the corresponding diagram. On the other hand, it is generally difficult to reconstruct the factorization given an arbitrary diagram. We present an efficient algorithm for obtaining a reduced factorization for a given diagram.

Existence of Gradings on Associative Algebras

In this article, we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra A does not have a nontrivial grading if and only if A is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist nontrivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.

Semigroup graded algebras and codimension growth of graded polynomial identities

We show that if T is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative T-graded algebra over a field of characteristic 0 such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if T is a left or right zero band and the T-graded algebra is unital, or T is a cancellative semigroup, then the T-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.