Unit group of semisimple group algebra of groups of order 36

Introduction. In this paper, we extend some of the classical theory of semi-simple algebraic groups and Lie algebras over the real numbers to an arbitrary real closed field. The existence of a Cartan decomposition for a semi-simple Lie algebra over a real closed field k is shown in ? 2. Such a decomposition is unique up to an inner automorphism of the Lie algebra. In ? 3, the theory of k-compact algebraic groups is developed. For such groups, any two maximal tori defined over k are conjugate by a k-rational element in the group . It is this fact that is utilized in ? 4 to show that the classification theory of connected semi-simple algebraic groups over k is the same as over the field of real numbers. A consequence of the existence of a Cartan decomposition for semi-simple Lie algebras over k is the existence of an Iwasawa decomposition. The corresponding decomposition for connected semi-simple algebraic groups defined over k also exists. These proofs are given in ? 5 and several application in ? 6. 1. Cartan decomposition for elements. In this section, we shall let k be a real closed field and C kV- 1. The non-trivial automorphism of C over k will be denoted by z zo. Also, G shall denote an algebraic group

The purpose of this article is to study in detail the actions of a semisimple Lie or algebraic group on its Lie algebra by the adjoint representation and on itself by the adjoint action. We will focus primarily on orbits through nilpotent elements in the Lie algebra; these are called nilpotent orbits for short. Many deep results about such orbits have been obtained in the last thirty-five years; we will collect some of the most significant of these that have found wide application to representation theory. We will primarily work in the setting of a semisimple Lie algebra and its adjoint group over an algebraically closed field of characteristic zero, but we will extend much of what we do to semisimple Lie algebras over the reals or an algebraically closed field of prime characteristic, and to conjugacy classes in semisimple algebraic groups. We will give detailed proofs of many results, including some which are difficult to ferret out of the literature. Other results will be summarized with reasonably complete references. The treatment is a more comprehensive version of that in [CM93]; there is also some overlap with Humphreys’s book [Hu95]. In the last chapter we summarize some of the most recent work being done in this topic and indicate some directions of current research.

On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior

Let \(K\) be an arbitrary field, whose characteristic does not divide the order of the dihedral group \(D_{2 m}\) of order \(2 m\), where \(m\) is odd, and \(K D_{2 m}\) be the group algebra of \(D_{2 m}\) over the field \(K\). The structure of the semisimple dihedral group algebra \(K D_{2 m}\) is examined in this work. We find a complete system of minimal central orthogonal idempotents of the group algebra for this purpose. We define the simple components of \(K D_{2 m}\) and its Wedderburn decomposition through it. The results are as general as possible, i.e. they do not require a finite field.

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.

Twisted group ring isomorphism problem and infinite cohomology groups

Let K be an arbitrary field, whose characteristic does not divide the order of the dihedral group D2m of order 2m, where m is odd. In this paper we examine the structure of the semisimple dihedral group algebra KD2m . For this purpose, we find a complete system of minimal central orthogonal idempotents of the group algebra. Through it we define the minimal components of KD2m and its Wedderburn decomposition. The results we get are as general as possible, i.e. without requiring the field to be finite.

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 $p > 3$ 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.

In this paper, we study the normal complement problem on semisimple group algebras and modular group algebras [Formula: see text] over a field [Formula: see text] of positive characteristic. We provide an infinite class of abelian groups [Formula: see text] and Galois fields [Formula: see text] that have normal complement in the unit group [Formula: see text] for semisimple group algebras [Formula: see text]. For metacyclic group [Formula: see text] of order [Formula: see text], where [Formula: see text] are distinct primes, we prove that [Formula: see text] does not have normal complement in [Formula: see text] for finite semisimple group algebra [Formula: see text]. Finally, we study the normal complement problem for modular group algebras over field of characteristic 2.

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 $$Q_{m,n}.$$

The topics of this conference all in some way evolved from the classical theory of real and complex Lie groups. Indeed, one of the important mathematical goals during the 1950's was to find analogs of the semisimple Lie groups of exceptional type over arbitrary fields. Chevalley completed the first crucial step by producing his famous basis theorem for simple complex Lie algebras, and later Steinberg succeeded in describing these analogs group-theoretically. An important development due to Tits was the theory of groups with a BN -pair and invented buildings; these buildings belong to arbitrary Chevalley groups as naturally as the projective spaces belong to the special linear groups. Since then the theories of algebraic groups and of buildings developed into various directions. However, due to their common origin both theories often lead naturally to similar questions which were attacked by completely different means. In the context of this conference, the PhD thesis by Bernhard Mühlherr and the work by Aloysius Helminck and coauthors on involutions of algebraic groups illustrate this in a quite remarkable way. Both projects contributed strongly to the understanding of the geometry of involutions of algebraic groups, but surprisingly each one has gone unnoticed by the researchers of the other until recently. One of the main objectives of this conference was to bring these two theories closer to each other. The first two lectures on Monday morning familiarised all participants with the concept of buildings; Pierre-Emmanuel Caprace explained the foundations of buildings from the simplicial point of view, while Bernhard Mühlherr introduced the chamber system approach to buildings and explained the power of filtrations when studying sub-geometries of (twin) buildings that arise from the action of certain subgroups of the isometry group of the (twin) building. As these two lectures were of an introductory nature and since their content is already well documented (we refer to the recently published book by Abramenko and Brown for the theory of buildings and to the contributions of Alice Devillers and of Hendrik Van Maldeghem to the Oberwolfach report 20/2008 for an account on filtrations and their powerful applications), we do not include acstracts of these lectures. On Monday afternoon and on Friday morning Aloysius Helminck presented the theory of involutions of algebraic groups, while in Monday's final lecture Max Horn showed how to combine Helminck's theory with M"uhlherr's PhD thesis in order to obtain general and powerful results on the geometry of involutions of groups with a root group datum, a class of groups that contains the semisimple algebraic groups, the split Kac–Moody groups, and the split finite groups of Lie type. Most of Tuesday and part of Wednesday were focussed on the Tits centre conjecture. In a series of two lectures Gerhard Röhrle and Michael Bate presented an algebraic-group approach towards proving the conjecture, while on Tuesday afternoon Katrin Tent presented a combinatorial approach and on Wednesday morning Linus Kramer reported on metric considerations in the context of the Tits centre conjecture. It is our impression that these four lectures have triggered additional activity towards proving the centre conjecture, and that one or more of these approaches will be successful in the near future. The fourth talk on Tuesday afternoon was given by Yiannis Sakellaridis on spherical varieties and automorphic forms, while the second talk on Wednesday by Lizhen Ji presented compactifications of locally symmetric spaces. Thursday's talks by Sergey Shpectorov and by Paul Levy concentrated on involutions of affine buildings, respectively automorphisms of finite order of semisimple Lie algebras. The remaining three talks were more topologically in nature. Guy Rousseau presented his theory of microaffine buildings, hovels, and Kac–Moody groups over ultrametric fields on Thursday. Thursday's fourth talk was by Bertrand Rémy on Satake compactifications of buildings via Berkovich theory. The conference was concluded by Pierre-Emmanuel Caprace's report on aspects of the structure of locally compact groups. We are particularly pleased by the lively interaction between the participants during the long afternoon breaks (each morning's lectures finished at 11.30 a.m. while the afternoon sessions only started at 4.20 p.m.) and during the evenings.

We study singularity properties of word maps on semisimple Lie algebras, semisimple algebraic groups and matrix algebras and obtain various applications to random walks induced by word measures on compact p p -adic groups. Given a word w w in a free Lie algebra L r \mathcal {L}_{r} , it induces a word map φ w : g r → g \varphi _{w}:\mathfrak {g}^{r}\rightarrow \mathfrak {g} for every semisimple Lie algebra g \mathfrak {g} . Given two words w 1 ∈ L r 1 w_{1}\in \mathcal {L}_{r_{1}} and w 2 ∈ L r 2 w_{2}\in \mathcal {L}_{r_{2}} , we define and study the convolution of the corresponding word maps φ w 1 ∗ φ w 2 ≔ φ w 1 + φ w 2 : g r 1 + r 2 → g \varphi _{w_{1}}*\varphi _{w_{2}}≔\varphi _{w_{1}}+\varphi _{w_{2}}:\mathfrak {g}^{r_{1}+r_{2}}\rightarrow \mathfrak {g} . By introducing new degeneration techniques, we show that for any word w ∈ L r w\in \mathcal {L}_{r} of degree d d , and any simple Lie algebra g \mathfrak {g} with φ w ( g r ) ≠ 0 \varphi _{w}(\mathfrak {g}^{r})\neq 0 , one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O ( d 4 ) O(d^{4}) self-convolutions of φ w \varphi _{w} . Similar results are obtained for matrix word maps. We deduce that a group word map of length ℓ \ell becomes (FRS), locally around identity, after O ( ℓ 4 ) O(\ell ^{4}) self-convolutions, for every semisimple algebraic group G _ \underline {G} . We furthermore provide uniform lower bounds on the log canonical threshold of the fibers of Lie algebra, matrix and group word maps. For the commutator word w 0 = [ X , Y ] w_{0}=[X,Y] , we show that φ w 0 ∗ 4 \varphi _{w_{0}}^{*4} is (FRS) for any semisimple Lie algebra, improving a result of Aizenbud-Avni, and obtaining applications in representation growth of compact p p -adic and arithmetic groups. The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form Z / p k Z \mathbb {Z}/p^{k}\mathbb {Z} . This allows us to relate them to properties of some natural families of random walks on finite and compact p p -adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to p p -adic probabilistic Waring type problems.

In this paper, we discuss the structure of the unit groups of the semisimple group algebras of 10 non-metabelian group having orders 150, 160, 168 and 180, respectively. In particular, we consider the only one non-metabelian group of orders 150 and 180, two nonmetabelian group of order 160 and six non-metabelian group of order 168 and study the unit groups of their corresponding semisimple group algebra. With this paper, the study of the unit groups of semisimple group algebra of all the groups up to order 180 is completed.

We study abelian codes in principal ideal group algebras (PIGAs). We first give an algebraic characterization of abelian codes in any group algebra and provide some general results. For abelian codes in a PIGA, which can be viewed as cyclic codes over a semisimple group algebra, it is shown that every abelian code in a PIGA admits generator and check elements. These are analogous to the generator and parity-check polynomials of cyclic codes. A characterization and an enumeration of Euclidean self-dual and Euclidean self-orthogonal abelian codes in a PIGA are given, which generalize recent analogous results for self-dual cyclic codes. In addition, the structures of reversible and complementary dual abelian codes in a PIGA are established, again extending results on reversible and complementary dual cyclic codes. Finally, asymptotic properties of abelian codes in a PIGA are studied. An upper bound for the minimum distance of abelian codes in a non-semisimple PIGA is given in terms of the minimum distance of abelian codes in semisimple group algebras. Abelian codes in a non-semisimple PIGA are then shown to be asymptotically bad, similar to the case of repeated-root cyclic codes.

In this paper, we give the characterization of the unit groups of semisimple group algebras of some non-metabelian groups of order [Formula: see text]. This study completes the study of unit groups of semisimple group algebras of all groups up to order [Formula: see text], except that of the symmetric group [Formula: see text] and groups of order [Formula: see text].