Wedderburn Decomposition Research Articles

RATIONAL REPRESENTATIONS AND RATIONAL GROUP ALGEBRA OF VZ $\boldsymbol {p}$ -GROUPS

Abstract In this article, we study rational matrix representations of VZ p-groups (p is any prime). Using our findings on VZ p-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all p-groups of order $\leq p^4$ . Furthermore, we establish combinatorial formulae to determine the Wedderburn decompositions of rational group algebras for VZ p-groups and all p-groups of order $\leq p^4$ , ensuring simplicity in the process.

A Combinatorial Formula for the Wedderburn Decomposition of Rational Group Algebras of Split Metacyclic p-groups

A Combinatorial Formula for the Wedderburn Decomposition of Rational Group Algebras of Split Metacyclic <i>p</i>-groups

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

A short note on the Wedderburn components of a semisimple finite group algebra

One of the classical problems in the subject of group algebras is that of deducing Wedderburn decomposition of a finite semisimple group algebra. In this short note, we discuss how to check whether a matrix ring over a finite field is a Wedderburn component of the Wedderburn decomposition of a group algebra or not. Finally, we formulate an open problem in this direction.

Primitive decompositions of idempotents of the group algebras of dihedral groups and generalized quaternion groups

&lt;p&gt;In this paper, we introduce a method for computing the primitive decomposition of idempotents in any semisimple finite group algebra, utilizing 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}[Q_{8}] $ is described, under which the two complete sets of primitive orthogonal idempotents of these group algebras correspond bijectively.&lt;/p&gt;

A short note on Wedderburn decomposition of a group algebra

In this paper, we extend the result of Mittal and Sharma (Bull. Korean Math. Soc. 2022) on Wedderburn decomposition (WD) of a finite semisimple group algebra. It is known that, under certain conditions, WD of a finite semisimple group algebra FqG can be computed from WD of its subalgebra Fq(G/H), where H is a normal subgroup of G of prime order and q=pk for some prime p and positive integer k. We extend this result to any normal subgroup H of G of order n.

On the finite loop algebra F[M(Cm p x C2, 2)

Let G = Cm p o C2 be a generalized dihedral group for an odd prime p and a natural number m, L = M(G; 2) be the RA2 loop obtained from G and F be a finite field of characteristic 2. For the loop algebra F[L], we determine the Jacobson radical J(F[L]) of F[L] and the Wedderburn decomposition of F[L]=J(F[L]). The structure of 1 + J(F[L]) is also determined.

Computation of the Wedderburn decomposition of semisimple group algebras of groups up to order 120

Computation of the Wedderburn decomposition of semisimple group algebras of groups up to order 120

A Note on the Wedderburn Decomposition and Unit Group of the Semisimple Group Algebra FpSn

Abstract: In this short note, we give an algorithm to compute the unit group of a semisimple group algebra for any . To show the practicality of the algorithm, we explicitly find the unit groups of the semisimple group algebras

Polyadization of Algebraic Structures

A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented as block diagonal matrices (resulting in the Wedderburn decomposition), general forms of polyadic structures are given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures (“double” decomposition of two kinds). We then introduce the polyadization concept (a “polyadic constructor”), according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The “deformation” by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.

Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ from a subalgebra of factor group of $G$

In this paper, we derive a condition under which the Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ can be deduced from the Wedderburn decomposition of $\mathbb{F}_q(G/H)$, where $H$ is a normal subgroup of $G$ having two elements and $q=p^k$ for some prime $p$ and $k\in \mathbb{Z}^+$. In order to complement the abstract theory of the paper, we deduce the Wedderburn decomposition and hence the unit group of semisimple group algebra $\mathbb{F}_q(A_5\rtimes C_4)$, where $A_5\rtimes C_4$ is a non-metabelian group and $C_4$ is a cyclic group of order $4$.

Hilbertian Frobenius algebras

Commutative Hilbertian Frobenius algebras are those commutative semigroup objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this adjoint “comultiplication” is a bimodule map. In this note we show that the Frobenius relation forces the multiplication operators to be normal. We then prove that these algebras have a strong Wedderburn decomposition where the (ortho)complement of the Jacobson radical or equivalently of the annihilator, is the closure of the linear span of elements which essentially are the non-trivial characters. As a consequence such an algebra is semisimple if, and only if, its multiplication has a dense range. In particular every commutative special Hilbertian Frobenius algebra, that is, with a coisometric multiplication, is semisimple. Moreover we characterize from a setting a priori free of an involution, Ambrose’s commutative -algebras as the underlying algebras of Hilbertian Frobenius algebras. Extending a known result in the finite-dimensional situation, we prove that the structures of such Frobenius algebras on a given Hilbert space are in one-to-one correspondence with its bounded above orthogonal sets. We show, moreover, that the category of commutative Hilbertian Frobenius algebras is dually equivalent to a category of pointed sets.

Rational and quasi-permutation representations of holomorphs of cyclic $p$-groups

‎For a finite group $G$‎, ‎three of the positive integers governing its‎ ‎representation theory over $mathbb{C}$ and over $mathbb{Q}$ are‎ ‎$p(G),q(G),c(G)$‎. ‎Here‎, ‎$p(G)$ denotes the {it minimal degree} of a‎ ‎faithful permutation representation of $G$‎. ‎Also‎, ‎$c(G)$ and $q(G)$‎ ‎are‎, ‎respectively‎, ‎the minimal degrees of a faithful representation‎ ‎of $G$ by quasi-permutation matrices over the fields $mathbb{C}$‎ ‎and $mathbb{Q}$‎. ‎We have $c(G)leq q(G)leq p(G)$ and‎, ‎in general‎, ‎either inequality may be strict‎. ‎In this paper‎, ‎we study the‎ ‎representation theory of the group $G =$ Hol$(C_{p^{n}})$‎, ‎which is‎ ‎the {it holomorph} of a cyclic group of order $p^n$‎, ‎$p$ a prime‎. ‎This group is metacyclic when $p$ is odd and metabelian but not‎ ‎metacyclic when $p=2$ and $n geq 3$‎. ‎We explicitly describe the set‎ ‎of all {it isomorphism types} of irreducible representations of $G$‎ ‎over the field of complex numbers $mathbb{C}$ as well as the‎ ‎isomorphism types over the field of rational numbers $mathbb{Q}$‎. ‎We compute the {it Wedderburn decomposition} of the rational group‎ ‎algebra of $G$‎. ‎Using the descriptions of the irreducible‎ ‎representations of $G$ over $mathbb{C}$ and over $mathbb{Q}$‎, ‎we‎ ‎show that $c(G) = q(G) = p(G) = p^n$ for any prime $p$‎. ‎The proofs‎ ‎are often different for the case of $p$ odd and $p=2$‎.

On unit group of finite semisimple group algebras of non-metabelian groups of order 108

In this paper, we characterize the unit groups of semisimple group algebras $\mathbb{F}_qG$ of non-metabelian groups of order $108$, where $F_q$ is a field with $q=p^k$ elements for some prime $p &gt; 3$ and positive integer $k$. Up to isomorphism, there are $45$ groups of order $108$ but only $4$ of them are non-metabelian. We consider all the non-metabelian groups of order $108$ and find the Wedderburn decomposition of their semisimple group algebras. And as a by-product obtain the unit groups.

Second cohomology group of the finite-dimensional simple Jordan superalgebra 𝒟t, t≠0

The second cohomology group (SCG) of the Jordan superalgebra [Formula: see text], [Formula: see text], over an algebraically closed field [Formula: see text] of characteristic zero is calculated by using the coefficients which appear in the regular superbimodule [Formula: see text]. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem [F. A. Gómez-González, Wedderburn principal theorem for Jordan superalgebras I, J. Algebra 505 (2018) 1–32]. 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 [Formula: see text] that determine the SCG in Jordan superalgebras. We use these to calculate the SCG for the Jordan superalgebra [Formula: see text], [Formula: see text]. Finally, we prove that [Formula: see text], [Formula: see text].

Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

Abstract We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endo-morphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

English

We study the inverse problem of the determination of a group algebra from the knowledge of its Wedderburn decomposition. We show that a certain class of matrix rings always occur as summands of finite group algebras.

A cancellation theorem for modules over integral group rings

AbstractA long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups G for which the integral group ring ℤG has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that ℤG has SFC provided at most one copy of the quaternions ℍ occurs in the Wedderburn decomposition of the real group ring ℝG. This generalises the Eichler condition in the case of integral group rings.

Commutative power-associative algebras of small dimension

This paper is concerned with the algebraic and geometric classification of the commutative power-associative algebras of dimension over an algebraically closed field k with characteristic relative prime with 30. We prove that an algebra of this variety is Jordan if its dimension as vector space over k is not greater than 3. Albeit in dimension 4, non-Jordan commutative power-associative algebras exist, we can show that such algebras yet admits Wedderburn decomposition. We use the action of over the affine variety CPA4 of four-dimensional commutative power-associative algebras over to get 12 irreducible components that correspond to the Zariski closure of the orbit of rigid algebras, two of which are non-Jordan algebras.

Структура конечной групповой алгебры одного полупрямого произведения абелевых групп и её приложения

В 1978 году Р. Мак-Элисом построена первая асимметричная кодовая криптосистема, основанная на применении помехоустойчивых кодов Гоппы, при этом эффективные атаки на секретный ключ этой криптосистемы до сих пор не найдены. К настоящему времени известно много криптосистем, основанных на теории помехоустойчивого кодирования. Одним из способов построения таких криптосистем является модификация криптосистемы Мак-Элиса с помощью замены кодов Гоппы на другие классы кодов. Однако, известно что криптографическая стойкость многих таких модификаций уступает стойкости классической криптосистемы Мак-Элиса. В связи с развитием квантовых вычислений кодовые криптосистемы, наряду с криптосистемамми на решётках, рассматриваются как альтернатива теоретико-числовым. Поэтому актуальна задача поиска перспективных классов кодов, применимых в криптографии. Представляется, что для этого можно использовать некоммутативные групповые коды, т.е. левые идеалы в конечных некоммутативных групповых алгебрах.Для исследования некоммутативных групповых кодов полезной является теорема Веддерберна, доказывающая существование изоморфизма групповой алгебры на прямую сумму матричных алгебр. Однако конкретный вид слагаемых и конструкция изоморфизма этой теоремой не определены, и поэтому для каждой группы стоит задача конструктивного описания разложения Веддерберна. Это разложение позволяет легко получить все левые идеалы групповой алгебры, т.е. групповые коды. В работе рассматривается полупрямое произведение $$Q_{m,n} = (\mathbb{Z}_m \times \mathbb{Z}_n) \leftthreetimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$$ абелевых групп и конечная групповая алгебра $$\mathbb{F}_q Q_{m,n}$$ этой группы. Для этой алгебры при условиях $$n \mid q -1$$ и $$\text{НОД}(2mn, q) = 1$$ построено разложение Веддербёрна. В случае поля чётной характеристики, когда эта групповая алгебра не является полупростой, также получена сходная структурная теорема. Описаны все неразложимые центральные идемпотенты этой групповой алгебры. Полученные результаты используются для алгебраического описания всех групповых кодов над $$Q_{m,n}.$$