Congruence Kernels Research Articles

On symmetric representations of 𝑆𝐿₂(ℤ)

We introduce the notions of symmetric and symmetrizable representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} . The linear representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} . By investigating a Z / 2 Z \mathbb {Z}/2\mathbb {Z} -symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} that are subrepresentations of a symmetric one.

The stable representations of GL N over finite local principal ideal rings

Let be a discrete valuation ring with maximal ideal and with finite residue field , the field with q elements where q is a power of a prime p. For , we write for the reduction of modulo the ideal . An irreducible ordinary representation of the finite group is called stable if its restriction to the principal congruence kernel , where , consists of irreducible representations whose stabilizers modulo , where , are centralizers of certain matrices in , called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group for . Communicated by Scott Chapman

Fuzzy ideals and fuzzy congruences of distributive demi-p-algebras

In this article, we introduce *-fuzzy ideals of a distributive demi-p-algebra and its properties. We show that every *-fuzzy ideals of distributive demi-p-algebra L are precisely the kernels of the fuzzy congruences Θ on L if and only if its quotient is a Boolean algebra. Also we prove that the set of *-fuzzy ideals forms a complete Heyting algebra.

Congruence relations on pseudocomplemented almost distributive lattices

The concept of congruence kernels and congruence cokernels are studied in pseudocomplemented almost distributive lattices. A set of equivalent conditions are derived for a given subset J of a pseudocomplemented ADL L to be a congruence kernel (resp. a congruence cokernel). Moreover, for a given congruence kernel J of L the smallest and the largest *- congruences on L having J as a kernel are given and characterized in various forms.

L‐fuzzy congruence classes in universal algebras

In this paper, we study fuzzy congruence relations and their classes; so-called fuzzy congruence classes in universal algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law. Fuzzy congruence relations generated by a fuzzy relation are fully characterized in different ways. The main result in this paper is that, we give several Mal'cev-type characterizations for a fuzzy subset of an algebra A in a given variety to be a class of some fuzzy congruence on A. Some equivalent conditions are also given for a variety of algebras to possess fuzzy congruence classes which are also fuzzy subuniverses. Special fuzzy congruence classes called fuzzy congruence kernels are characterized in a more general context in universal algebras.

On GE-algebras

Hilbert algebras are important tools for certain investigations in intuitionistic logic and other non-classical logic and as a generalization of Hilbert algebra a new algebraic structure, called a GE-algebra (generalized exchange algebra), is introduced and studied its properties. We consider filters, upper sets and congruence kernels in a GE-algebra. We also characterize congruence kernels of transitive GE-algebras.

Normal elements in the Iwasawa algebras of Chevalley groups

For a prime $$p>2$$, let G be a semi-simple, simply connected, split Chevalley group over $${\mathbb {Z}}_p$$, G(1) be the first congruence kernel of G and $$\Omega _{G(1)}$$ be the mod-p Iwasawa algebra defined over the finite field $${\mathbb {F}}_p$$. Ardakov et al. (Adv Math 218: 865–901, 2008) have shown that if p is a “nice prime ” ($$p \ge 5$$ and $$p \not \mid (n+1)$$ if the Lie algebra of G(1) is of type $$A_n$$), then every non-zero normal element in $$\Omega _{G(1)}$$ is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof of Ardakov et al. result using explicit presentation of Iwasawa algebra developed by the second author of this article and thus eliminating the nice prime condition, therefore proving their conjecture. We also propose some potential topics regarding to the normal elements and ideals in the Iwasawa algebras of the pro-p Iwahori subgroups of general linear group $${\hbox {GL}}_n(\mathbb {Z}_p)$$ and discuss how to extend our current techniques and methods to the case of the pro-p Iwahori subgroups of $${\hbox {GL}}_n(\mathbb {Z}_p)$$.

On the congruence kernel of isotropic groups over rings

Let R R be a connected noetherian commutative ring, and let G G be a simply connected reductive group over R R of isotropic rank ≥ 2 \ge 2 . The elementary subgroup E ( R ) E(R) of G ( R ) G(R) is the subgroup generated by U P + ( R ) U_{P^+}(R) and U P − ( R ) U_{P^-}(R) , where U P ± U_{P^\pm } are the unipotent radicals of two opposite parabolic subgroups P ± P^\pm of G G . Assume that 2 ∈ R × 2\in R^\times if G G is of type B n , C n , F 4 , G 2 B_n,C_n,F_4,G_2 and 3 ∈ R × 3\in R^\times if G G is of type G 2 G_2 . We prove that the congruence kernel of E ( R ) E(R) , defined as the kernel of the natural homomorphism E ( R ) ^ → E ( R ) ¯ \widehat {E(R)}\to \overline {E(R)} between the profinite completion of E ( R ) E(R) and the congruence completion of E ( R ) E(R) with respect to congruence subgroups of finite index, is central in E ( R ) ^ \widehat {E(R)} . In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of E ( R ) E(R) if R R is a local ring.

Directly decomposable ideals and congruence kernels of commutative semirings

As pointed out in the monographs by J. S. Golan and by W. Kuich and A. Salomaa on semirings, ideals play an important role despite the fact that they need not be congruence kernels as in the case of rings. Hence, having two commutative semirings S1 and S2, one can ask whether an ideal I of their direct product S = S1 x S2 can be expressed in the form I1 x I2 where Ij is an ideal of Sj for j=1,2. Of course, the converse is elementary, namely if Ij is an ideal of Sj for j=1,2 then I1 x I2 is an ideal of S1 x S2. Having a congruence on a commutative semiring S, its 0-class is an ideal of S, but not every ideal is of this form. Hence, the lattice Id S of all ideals of S and the lattice Ker S of all congruence kernels (i.e. 0-classes of congruences) of S need not be equal. Furthermore, we show that the mapping which assigns to every congruence its kernel need not be a homomorphism from Con S onto Ker S. Moreover, the question arises when a congruence kernel of the direct product S1 x S2 of two commutative semirings can be expressed as a direct product of the corresponding kernels on the factors. In the paper we present necessary and sufficient conditions for such direct decompositions both for ideals and for congruence kernels of commutative semirings. We also provide sufficient conditions for varieties of commutative semirings to have directly decomposable kernels.

Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SL n (ℤ p )

Abstract Iwasawa algebras of compact p-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p-adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over ℤ p {\mathbb{Z}_{p}} which were uniform pro-p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime p > n + 1 {p>n+1} , we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-p Iwahori subgroup of GL n ⁢ ( ℤ p ) {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro-p group.

Characterization of Minimal distributive semi lattice.

In this paper we give some characterization of minimal distributive semi lattice and proper filter in directed above semi lattice. Also this manuscript illustrates several principal results concerning congruence kernels of minimal distributive semi lattice S and provides a necessary and sufficient condition for a semi lattice congruence ≡ on Sis a * congruence. And also established a necessary and sufficient condition of an ideal of minimal distributive semi lattice S is a kernel ideal of semi lattice S.

Ideals and their complements in commutative semirings

We study conditions under which the lattice {{mathrm{mathbf {Id}}}}mathbf R of ideals of a given a commutative semiring {mathbf {R}} is complemented. At first we check when the annihilator I^* of a given ideal I of {mathbf {R}} is a complement of I. Further, we study complements of annihilator ideals. Next we investigate so-called Łukasiewicz semirings. These form a counterpart to MV-algebras which are used in quantum structures as they form an algebraic semantic of many-valued logics as well as of the logic of quantum mechanics. We describe ideals and congruence kernels of these semirings with involution. Finally, using finite unitary Boolean rings, a construction of commutative semirings with complemented lattice of ideals is presented.

Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over [formula omitted

It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple Lie algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over Zp, extending the proof given by Clozel for the group Γ1(SL2(Zp)), the first congruence kernel of SL2(Zp) for primes p>2.

Congruence kernels in Ockham algebras

We consider, in the context of an Ockham algebra \({{\mathcal{L} = (L; f)}}\), the ideals I of L that are kernels of congruences on \({\mathcal{L}}\). We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel \({I \neq L}\) is the intersection of the prime ideals P such that \({I \subseteq P}\), \({P \cap f(I) = \emptyset}\), and \({f^{2}(I) \subseteq P}\). The congruence kernels form a complete lattice which in general is not modular. For every non-empty subset X of L, we also describe the smallest congruence kernel to contain X, in terms of which we obtain necessary and sufficient conditions for modularity and distributivity. The case where L is of finite length is highlighted.

On the congruence subgroup property for GGS-groups

We show that all GGS-groups with a non-constant defining vector satisfy the congruence subgroup property. This provides, for every odd prime p p , many examples of finitely generated, residually finite, non-torsion groups whose profinite completion is a pro- p p group, and among them we find torsion-free groups. This answers a question of Barnea. On the other hand, we prove that the GGS-group with a constant defining vector has an infinite congruence kernel and is not a branch group.

Some properties of pseudo-BCK- and pseudo-BCI-algebras

Pseudo-BCI-algebras generalize both BCI-algebras and pseudo-BCK-algebras, which are a non-commutative generalization of BCK-algebras. In this paper, following [37], we show that pseudo-BCI-algebras are the residuation subreducts of semi-integral residuated po-monoids and characterize those pseudo-BCI-algebras which are direct products of pseudo-BCK-algebras and groups (regarded as pseudo-BCI-algebras). We also show that the quasivariety of pseudo-BCI-algebras is relatively congruence modular; in fact, we prove that this holds true for all relatively point regular quasivarieties which are relatively ideal determined, in the sense that the kernels of relative congruences can be described by means of ideal terms.

On the congruence kernel for simple algebraic groups

This paper contains several results about the structure of the congruence kernel C(S)(G) of an absolutely almost simple simply connected algebraic group G over a global field K with respect to a set of places S of K. In particular, we show that C(S)(G)) is always trivial if S contains a generalized arithmetic progression. We also give a criterion for the centrality of C(S)(G) in the general situation in terms of the existence of commuting lifts of the groups G(Kv) for v ∉ S in the S-arithmetic completion Ĝ(S). This result enables one to give simple proofs of the centrality in a number of cases. Finally, we show that if K is a number field and G is K-isotropic, then C(S)(G) as a normal subgroup of Ĝ(S) is almost generated by a single element.

On the Congruence Kernel for Simple Algebraic Groups

This paper contains several results about the structure of the congruence kernel C^(S)(G) of an absolutely almost simple simply connected algebraic group G over a global field K with respect to a set of places S of K. In particular, we show that C^(S)(G) is always trivial if S contains a generalized arithmetic progression. We also give a criterion for the centrality of C^(S)(G) in the general situation in terms of the existence of commuting lifts of the groups G(K_v) for v \notin S in the S-arithmetic completion \widehat{G}^(S). This result enables one to give simple proofs of the centrality in a number of cases. Finally, we show that if K is a number field and $G$ is K-isotropic then C^(S)(G) as a normal subgroup of \widehat{G}^(S) is almost generated by a single element.

On ideals and congruences of distributive demi-p-algebras

We identify the $${{}^\star}$$?-ideals of a distributive demi-pseudocomplemented algebra L as the kernels of the boolean congruences on L, and show that they form a complete Heyting algebra which is isomorphic to the interval $${[G,\iota]}$$[G,?] of the congruence lattice of L where G is the Glivenko congruence. We also show that the notions of maximal $${{}^\star}$$?-ideal, prime $${{}^\star}$$?-ideal, and falsity ideal coincide.

Congruence kernels around affine curves

Abstract Let S be a smooth affine algebraic curve, and let S ˚ ${\text{{$\mathring{S}$}}}$ be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of mapping class groups by showing that the congruence kernel ker Mod ( S ˚ ) ^ → Out ( π 1 ( S ˚ ) ^ ) $ \ker \Bigl (\widehat{\mathrm {Mod}(\mathring{S})} \rightarrow \mathrm {Out}(\widehat{\pi _1(\mathring{S})})\Bigr ) $ lies in the centralizer of every braid in Mod ( S ˚ ) ${{\mathrm {Mod}(\text{{$\mathring{S}$}})}}$ . As a corollary, we obtain a new proof of Asada's theorem that the congruence subgroup property holds in genus one. We also obtain simple-connectivity of Boggi's procongruence curve complex 𝒞 ˇ ( S ˚ ) ${\check{\mathcal {C}}(\text{{$\mathring{S}$}})}$ when S is affine, and a new proof of Matsumoto's theorem that the congruence kernel depends only on the genus in the affine case.