Algebraic Group Research Articles

Some Isomorphisms between Incidence Algebras and Group Algebras

Some Isomorphisms between Incidence Algebras and Group Algebras

Recovering the Picard group of quadratic algebras from Wood’s binary quadratic forms

Let [Formula: see text] be a scheme such that [Formula: see text] is not a zero divisor. In this paper, we address the following question: given a quadratic algebra over [Formula: see text], how can we parametrize its Picard group in terms of quadratic forms? In 2011, Wood established a set-theoretical bijection between isomorphism classes of primary binary quadratic forms over [Formula: see text] and isomorphism classes of pairs [Formula: see text] where [Formula: see text] is a quadratic algebra over [Formula: see text] and [Formula: see text] is an invertible [Formula: see text]-module. Unexpectedly, examples suggest that a refinement of Wood’s bijection is needed in order to parametrize Picard groups. This is why we start by classifying quadratic algebras over [Formula: see text]; this is achieved by using two invariants, the discriminant and the parity. Extending the notion of orientation of quadratic algebras to the non-free case is another key step, eventually leading us to the desired parametrization. All along the paper, we illustrate various notions and obstructions with a wide range of examples.

Classifying group algebras in which the socle of the center is an ideal

Let F be a field. We study the structure of finite groups G for which the socle of the center of the group algebra FG is an ideal in FG. In previous work (Brenner-Külshammer, 2024), we studied which general implications this ring-theoretic property of FG has on the structure of G. In the present article, we now focus on a detailed structural analysis of the groups with this property. We showcase its strength by deriving a complete classification for a prototype class of groups.

Reversibility in matrix rings and group algebras

Reversibility in matrix rings and group algebras

The pointed p-groups on a block algebra

A pointed p-group is a pointed group Pγ such that P is a p-group. We parameterize the pointed p-groups on a group algebra or on a block algebra of a group algebra. This is equivalent to a parameterization of the isomorphism classes of indecomposable direct summands of the algebra as a bimodule with the group acting on the left and a p-subgroup acting on the right. The parameterization involves p-subgroups and irreducible characters of centralizers of p-subgroups.

Units of twisted group rings and their correlations to classical group rings

This paper is centred around the classical problem of extracting properties of a finite group G from the ring isomorphism class of its integral group ring ZG. This problem is considered via describing the unit group U(ZG) generically for a finite group. Since the ‘90s’ several well known generic constructions of units are known to generate a subgroup of finite index in U(ZG) if QG does not have so-called exceptional simple epimorphic images, e.g. M2(Q). However it remained a major open problem to find a generic construction under the presence of the latter type of simple images. In this article we obtain such generic construction of units. Moreover, this new construction also exhibits new properties, such as providing generically free subgroups of large rank. As an application we answer positively for several classes of groups recent conjectures on the rank and the periodic elements of the abelianisation U(ZG)ab. To obtain all this, we investigate the group ring RΓ of an extension Γ of some normal subgroup N by a group G, over a domain R. More precisely, we obtain a direct sum decomposition of the (twisted) group algebra of Γ over the fraction field F of R in terms of various twisted group rings of G over finite extensions of F. Furthermore, concrete information on the kernel and cokernel of the associated projections is obtained. Along the way we also launch the investigations of the unit group of twisted group rings and of U(RΓ) via twisted group rings.

Differential embeddings into algebras of topological stable rank 1

We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative C*-algebras whose openness of multiplication is completely determined by the topological stable rank of the target C*-algebra. We then show that group algebras of Abelian groups of unbounded exponent fail to have uniformly open convolution. Finally, we completely characterize in the complex case (uniform) openness of multiplication in algebras of continuous functions in terms of the covering dimension.

Operator Kantor pairs

Kantor pairs, (quadratic) Jordan pairs, and similar structures have been instrumental in the study of Z-graded Lie algebras and algebraic groups. We introduce the notion of an operator Kantor pair, a generalization of Kantor pairs to arbitrary (commutative, unital) rings, similar in spirit as to how quadratic Jordan pairs and algebras generalize linear Jordan pairs and algebras. Such an operator Kantor pair is formed by a pair of Φ-groups (G+,G−) of a specific kind, equipped with certain homogeneous operators. For each such a pair (G+,G−), we construct a 5-graded Lie algebra L together with actions of G± on L as automorphisms. Moreover, we can associate a group G(G+,G−)⊂Aut(L) to this pair generalizing the projective elementary group of Jordan pairs. If the non-0-graded part of L is projective, we can uniquely recover G+,G− from G(G+,G−) and the grading on L alone. We establish, over rings Φ with 1/30∈Φ, a one to one correspondence between Kantor pairs and operator Kantor pairs. Finally, we construct operator Kantor pairs for the different families of central simple structurable algebras.

Non-associative Frobenius algebras of type [formula omitted] with trivial Tits algebras

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). In a previous paper, we gave an explicit description of these algebras for groups of type G2 and F4 in terms of the octonion algebras and the Albert algebras, respectively. In this paper, we attempt a similar approach for type E6.

Rubik’s cube and GeoGebra: a visual exploration of permutation groups

This article aims to use the construction of the Rubik’s Cube in GeoGebra as a primary tool for visualization and manipulation in the teaching of permutation groups in Abstract Algebra. We bring a brief discussion about the concept of group, aspects of the Rubik's cube, the Rubik’s group as a group of permutations and possibilities for its exploration in GeoGebra. Based on this study, we recognize the potential to delve into permutation groups in Abstract Algebra through a visual interface that associates their properties with a tangible and manipulable object. Additionally, there is the potential for simulating their movements using Dynamic Geometry software, such as GeoGebra.

Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

We define quantum automorphism groups of a wide range of discrete structures. The central tool for their construction is a generalisation of the Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras M, and any concrete unitary 2-category of finite type Hilbert-M-bimodules C, under reasonable conditions, we construct an algebraic quantum group G which acts on M by a action α, such that the category of α-equivariant representations of G on finite type Hilbert-M-bimodules is equivalent to C. Moreover, we explicitly describe how to get such categories from connected locally finite discrete structures. As an example, we define the quantum automorphism group of a quantum Cayley graph.

On the Structure of Automorphism Groups of Some Low-Dimensional Leibniz Algebras

On the Structure of Automorphism Groups of Some Low-Dimensional Leibniz Algebras

Equivariant algebraic concordance of strongly invertible knots

AbstractBy considering a particular type of invariant Seifert surfaces we define a homomorphism from the (topological) equivariant concordance group of directed strongly invertible knots to a new equivariant algebraic concordance group . We prove that lifts both Miller and Powell's equivariant algebraic concordance homomorphism (J. Lond. Math. Soc. (2023), no. 107, 2025‐2053) and Alfieri and Boyle's equivariant signature (Michigan Math. J. 1 (2023), no. 1, 1–17). Moreover, we provide a partial result on the isomorphism type of and obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox–Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that can obstruct equivariant sliceness for knots with Alexander polynomial one.

Category 𝒞_{𝓀} of multi-loop algebra representations versus modular representations: Questions of G. Lusztig

Lusztig defined an abelian category C k {\mathscr {C}}_{k} of a class of representations of a multi-loop algebra and asked various questions connecting it to the modular representation theory of simple algebraic groups in char. p > 0 p>0 . The aim of this paper is to show that some of these questions have negative answer.

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.

On J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras

AbstractWe introduce the combinatorial model of J-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam’s Plancherel Theorem.

Self-similarity of p-adic groups

We show that a compact open subgroup H of a simple algebraic p-adic group G is self-similar if and only if it is isotropic.

Visualization of finite groups: The case of the Rubik's cube and supporting properties in GeoGebra

In this text, we present a discussion about the Rubik's Cube and its relationship with Group Theory, particularly permutation groups, as well as possibilities for exploring it using the GeoGebra software interface. We bring a brief discussion about the concept of group, aspects of the Rubik's cube, the Rubik's group as a group of permutations and possibilities for its exploration in GeoGebra. Based on this study, we recognize the potential to delve into permutation groups in Abstract Algebra through a visual interface that associates their properties with a tangible and manipulable object. Additionally, there is the potential for simulating their movements using Dynamic Geometry software, such as GeoGebra. These findings highlight the relevance of GeoGebra as a useful tool for visualizing and understanding permutation groups, promoting a more intuitive and accessible approach to Group Theory in the educational context.

On pairs of root groups and reflections associated to the Chevalley group E 6(K), in characteristic two

This paper is concerned with computing the number of orbits under the action of the algebraic group E = E 6 ( K ) on the set of all pairs of root subgroups of itself. Furthermore, the number of orbits of the action of the group generated by a pair of certain reflections, on the 27 dimensional module A, has been computed. For the computations done in this paper, an explicit representation of E inside GL ( A ) has been fixed right at the onset.

Abstract irreducible representations of reductive algebraic groups with Borel-stable line

Abstract irreducible representations of reductive algebraic groups with Borel-stable line