This is a survey paper on algorithms that have been developed during the last 25 years for the explicit computation of the structure of an associative algebra of finite dimension over either a finite field or an algebraic number field. This constructive approach was initiated in 1985 by Friedl and Ronyai and has since been developed by Cohen, de Graaf, Eberly, Giesbrecht, Ivanyos, Kuronya and Wales. I illustrate these algorithms with the case n = 2 of the rational semigroup algebra of the partial transformation semigroup PT_n on n elements; this generalizes the full transformation semigroup and the symmetric inverse semigroup, and these generalize the symmetric group S_n.
We investigate an analogue of the Wedderburn principal theorem for associative conformal algebras with finite faithful representations. It is shown that the radical splitting property for an algebra of this kind holds if the maximal semisimple factor of this algebra is unital, but does not hold in general.
Let $A$ 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 $I=I_{S}\oplus I_{R}$ where $I_{S}=I\cap S$. and $I_{R}=I\cap R$.
The paper is devoted to the study of evolution algebras that are power-associative algebras. We give the Wedderburn decomposition of evolution algebras that are power-associative algebras and we prove that these algebras are Jordan algebras. Finally, we use this decomposition to classify these algebras up to dimension six.
We construct universal associative envelopes for the nonassociative triple systems arising from the trilinear operations of Bremner and Peresi applied to the 2-dimensional simple associative triple system. We use noncommutative Gröbner bases to determine monomial bases, structure constants, and centers of the universal envelopes. We show that the infinite dimensional envelopes are closely related to the down-up algebras of Benkart and Roby. For the finite dimensional envelopes, we determine the Wedderburn decompositions and classify the irreducible representations.
When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ the ring of integers of an abelian number field $F$ and a finite group $G$) certain components in the Wedderburn decomposition of $FG$ cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 are division rings, type 2 are $2 \times 2$-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (w.r.t. quotients) having those division rings in …
Suppose a finite dimensional semisimple Lie algebra $\mathfrak g$ acts by derivations on a finite dimensional associative or Lie algebra $A$ over a field of characteristic $0$. We prove the $\mathfrak g$-invariant analogs of Wedderburn - Mal'cev and Levi theorems, and the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of $A$. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur's conjecture for finite dimensional associative algebras with an …
We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota--Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular we prove that there exists no post-Lie algebra structure on a pair $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is a simple Lie algebra and $\mathfrak{g}$ is a reductive Lie algebra, which is not isomorphic to $\mathfrak{n}$. We also show that there is no post-associative algebra structure on a pair $(A,B)$ arising from a …
In this paper, the complete algebraic structure of finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.
In this paper, we introduce a method computing the primitive decomposition of idempotents of any semisimple finite group algebra based on its matrix representations and Wedderburn decomposition. Particularly, we use this method to calculate the examples of the dihedral group algebras $\mathbb{C}[D_{2n}]$ and generalized quaternion group algebras $\mathbb{C}[Q_{4m}]$. Inspired by the orthogonality relations of the character tables of these two families of groups, we obtain two sets of trigonometric identities. Furthermore, a group algebra isomorphism between $\mathbb{C}[D_{8}]$ and $\mathbb{C}[…
Recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool. In this article we show how to generalise the QR decomposition and SVD to a wide class of real algebras, including all finite-dimensional semi-simple algebras, (twisted) group algebras and Clifford algebras. Two approaches are described for computing the QRD/SVD: one Jacobi method with a generalised Givens rotation, and one based on the Artin-Wedderburn theorem.
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition 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.
In this paper we study the moduli space of 4-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. Because the space of 4-dimensional algebras is large, we only classify the non-nilpotent algebras in this paper.
Matrices as graded BiHom-algebras and decompositions
School of Mathematics, Southeast University（东南大学数学学院）；Shing-Tung Yau Centre, School of Mathematics, Southeast University（东南大学数学中心（杨振宁中心））；School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院）
This paper concerns the associative lower central series ideals $M_i$ of the free algebra $A_n$ on $n$ generators. Namely, we study the successive quotients $N_i=M_i/M_{i+1}$, which admit an action of the Lie algebra $W_n$ of vector fields on $\Bbb C^n$. We bound the degree $|λ|$ of tensor field modules $F_λ$ appearing in the Jordan-Hölder series of each $N_i$, confirming a recent conjecture of Arbesfeld and Jordan. As an application, we compute these decompositions for small $n$ and $i$.
We consider two families of finite-dimensional simple Lie superalgebras of Cartan type, denoted by HO and KO, over an algebraically closed field of characteristic p>3. Using the weight space decompositions and the principal gradings we first show that neither HO nor KO possesses a nondegenerate associative form. Then, by means of computing the superderivations from the Lie superalgebras in consideration into their dual modules, the second cohomology groups with coefficients in the trivial modules are proved to be vanishing.
In this paper we study the structure of a class of algebras satisfying a polynomial identity of degree 6. We show, assuming the existence of a non-zero idempotent, that if an algebra satisfies such an identity, it admits a Peirce decomposition related to this idempotent. We studied the algebraic structure and highlighted the connections of the algebras of this class with Bernstein algebras, train algebras, Jordan algebras and power associative algebras. Keywords: Peirce decomposition, Bernstein algebra, Jordan algebra, Power associative algebra, train algebra,polynomial identity, idempotent.
We show that the universal associative enveloping algebra of the simple anti-Jordan triple system of all $n \times n$ matrices $(n \ge 2)$ over an algebraically closed field of characteristic 0 is finite dimensional. We investigate the structure of the universal envelope and focus on the monomial basis, the structure constants, and the center. We explicitly determine the decomposition of the universal envelope into matrix algebras. We classify all finite dimensional irreducible representations of the simple anti-Jordan triple system, and show that the universal envelope is semisimple. We also …
We study a structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semi direct product of abelian groups. Using this decomposition we describe a structure of the lower central series and the series of derived subgroups for the group of unitriangular automorphisms and prove that every element from the derived subgroup is a commutator. In addition we prove that the group of unitriangular automorphisms of a free associative algebra of rang more than 2 is not linear and describe some two-generated subgroups from th…
We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras $A$ with a generalized Hopf action of an associative algebra $H$ with $1$ over an algebraically closed field of characteristic $0$ assuming only the invariance of the Jacobson radical $J(A)$ under the $H$-action and the existence of the decomposition of $A/J(A)$ into the sum of $H$-simple algebras. As a consequence, we show that the analog of Amitsur's conjecture holds for $G$-codimensions of finite dimensional associative algebras over a field of characteristic $0$ with an action of an arbitrary group $…
Suppose the ground field $\mathbb{F}$ is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras.
Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. In a previous article, we gave a necessary and sufficient condition for X to be free of given rank d over A. In the case that (i) the Wedderburn decomposition of E[G] is explicitly computable and (ii) each component is in fact a matrix ring over a field, this led to an algorithm that either gives elements that either gives an A-basis for X or determines that no such basis exists. In the present article, we generalise the algorithm by weakening condition (i…
The second cohomology group (SCG) of the Jordan superalgebra $\mathcal{D}_{t}$, $t\neq 0$, is calculated by using the coefficients which appear in the regular superbimodule $\mathrm{Reg}\mathcal{D}_t$. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem \cite{Faber1}. First, to calculate the SCG of a Jordan superalgebra we use split-null extension of the Jordan superalgebra and the Jordan superalgebra representation. We prove conditions that satisfy the bilinear forms $h$ that determine the SCG in Jordan superalg…
We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring $\Z G$ of a finite nilpotent group $G$ has a subgroup of finite index that is generated by three nilpotent groups for which we have an ex…
For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigid presentation and study the number of nonisomorphic direct summands and different complements. Based on that, we construct a simplicial complex governing the canonical decompositions of rigid presentations and provide some examples.
A Mathieu-Zhao subspace is a generalization of an ideal of an associative algebra $\mathcal A$ over a unital ring $R$ first formalized in 2010. A vertex algebra is an algebraic structure first developed in conjunction with string theory in the 1960s and later axiomatized by mathematicians in the 1980s. We formally introduce the definition of a Mathieu-Zhao subspace $M$ of a vertex algebra $V$. Building on natural connections to associative algebras, we classify an infinite set of non-trivial, non-ideal Mathieu-Zhao subspaces for simple and general vertex algebras by group action eigenspace dec…
We introduce a decomposition of associative algebras into a tensor product of cyclic modules. This produces a means to encode a basis with logarithmic information and thus extends the reach of calculation with large algebras. Our technique is an analogue to the Schreier-Sims algorithm for permutation groups and is a by-product of Frobenius reciprocity.
Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and simplicity; and (2) sum decompositions.
Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition of E[G] is explicitly computable and each component is in fact a matrix ring over a field, this leads to an algorithm that either gives an A-basis for X or determines that no such basis exists. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d=[K:E]. The algorith…
We construct a just infinite fractal 3-generated Lie superalgebra $\mathbf Q$ over arbitrary field, which gives rise to an associative hull $\mathbf A$, a Poisson superalgebra $\mathbf P$, and two Jordan superalgebras $\mathbf J$, $\mathbf K$. One has a natural filtration for $\mathbf A$ which associated graded algebra has a structure of a Poisson superalgebra and $\mathrm{gr} \mathbf A\cong\mathbf P$, also $\mathbf P$ admits an algebraic quantization. The Lie superalgebra $\mathbf Q$ is finely $\mathbb Z^3$-graded by multidegree in the generators, $\mathbf A$, $\mathbf P$ are $\mathbb Z^3$-gr…
We consider associative algebras L over a field provided with a direct sum decomposition of a two-sided ideal M and a sub-algebra A - examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of L. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.
Let $\mathfrak{g}$ be a finite dimensional complex Lie algebra and let $A$ be a finite dimensional complex, associative and commutative algebra with unit. We describe the structure of the derivation Lie algebra of the current Lie algebra $\mathfrak{g}_A= \mathfrak{g} \otimes A$, denoted by $\operatorname{Der}(\mathfrak{g}_A)$. Furthermore, we obtain the Levi decomposition of $\operatorname{Der}(\mathfrak{g}_A)$. As a consequence of the last result, if $\mathfrak{h}_m$ is the Heisenberg Lie algebra of dimension $2 m + 1$, we obtain a faithful representation of $\operatorname{Der}(\mathfrak{h}_{…
