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In this paper we study the Diophantine problem in Chevalley groups G π (Φ , R) , where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1. We establish a variant of double centralizer theorem for elementary unipotents x α (1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup X α , α ∈ Φ , is Diophantine in G. Then we prove that the Diophantine problem in G π (Φ , R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings. [ABSTRACT FROM AUTHOR]

We show that for Chevalley groups G(R) of rank at least 2 over an integral domain R each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that R and G(R) are bi-interpretable, yielding a new approach to biinterpretability for algebraic groups over a wide range of rings and fields. For such groups it then follows that the group G(R) is (finitely) axiomatizable in the appropriate class of groups provided R is (finitely) axiomatizable in the corresponding class of rings. [ABSTRACT FROM AUTHOR]

Given a Chevalley group G.q and a parabolic subgroup P G.q we prove that for any set A there is a certain growth of A relatively to P, namely, either AP or PA is much larger than A. Also, we study a question about the intersection of An with parabolic subgroups P for large n. We apply our method to obtain some results on a modular form of Zaremba's conjecture from the theory of continued fractions, and make the first step towards Hensley's conjecture about some Cantor sets with Hausdorff dimension greater than 1=2. [ABSTRACT FROM AUTHOR]

Let F be a subfield of the algebraic closure of a finite field F p of characteristic p ≠ 2 , and let R be any ring such that F [ t ] ⊆ R ⊊ F (t). Let G (R) be a classical Chevalley group of adjoint type defined over R. We prove that the group G (R) has the R ∞ -property. [ABSTRACT FROM AUTHOR]

We exhibit presentations of the Von Dyck groups D(2,3,m), m≥3, in terms of two generators of order m satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases m=3,4,5, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When m≥6 we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for m≥7 when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain three-relator presentations of PSL(2,m). We discover two general formulas presenting these as factors of D(2,3,m). The first one works for any odd m and is essentially equivalent to the shortest known presentation of Sunday [J. Sunday, Presentations of the groups SL(2,m) and PSL(2,m), Canadian J. Math., 24 (1972) 1129--1131]. The second applies to the cases m≡±2 (mod 3), m≡/ 11(mod 30), and is substantively shorter. Additionally, by random search, we find many efficient presentations of finite simple Chevalley groups PSL(2,q) as factors of D(2,3,m) where m divides the order of the group. The only other simple group that we found in this way is the sporadic Janko group J2. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let F be a finite field. Consider a direct sum V of an infinite number of copies of F , consider the dual space V ⋄ , i.e., the direct product of an infinite number of copies of F. Consider the direct sum V = V ⋄ ⊕ V. The object of the paper is the group GL ‾ of continuous linear operators in V. We reduce the theory of unitary representations of GL ‾ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over F. In fact we consider a certain family Q ‾ α of subgroups in GL ‾ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to Q ‾ α , and reduce this multiplication to products of linear relations. We show that this group has type I and obtain an 'upper estimate' of the set of all irreducible unitary representations of GL ‾. [ABSTRACT FROM AUTHOR]

This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory. [ABSTRACT FROM AUTHOR]

Abstract: We show that the third cohomology of the finite general linear group GL 6 (F 2) with trivial mod 2 coefficients is non-zero. The necessarily unique non-trivial element restricts to the third Milgram–Priddy class. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study certain geometric properties of generalized quadrangle (Ω , 𝔏) of type O 6 − (2). We define certain root elements which generate a Lie algebra of type E 6 (K) for fields K of characteristic two. The construction will be mainly based on the geometric properties of the generalized quadrangle (Ω , 𝔏). In fact, we will explicity construct a Chevalley base of this Lie algebra. [ABSTRACT FROM AUTHOR]

We generalize to the case of singular blocks the result in Bezrukavnikov and Lachowska [Quantum groups, Contemporary Mathematics 433 (American Mathematical Society, Providence, RI, 2007) 89–101] that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in Lachowska and Qi [Int. Math. Res. Not., Preprint, 2017, arXiv:1604.07380], we present a linear algebro‐geometric approach to compute the dimensions of the singular blocks and of the entire center of the small quantum group associated with a complex semisimple Lie algebra. A conjectural formula for the dimension of the center of the small quantum group at an lth root of unity is formulated in type A. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that Lie algebras of Chevalley type (An, Bn, Cn, Dn, E6, E7, E8, F4, and G2) over associative commutative rings with 1/2 (with 1/2 and 1/3 in the case of G2) have unique addition. As a corollary of this theorem, we note the uniqueness of addition in semisimple Lie algebras of Chevalley type over fields of characteristic ≠ 2 (≠ 2, 3 in the case of G2). [ABSTRACT FROM AUTHOR]

It is shown that each element of the elementary Chevalley group of rank greater than 2 over a ring of stable rank 1 can be expressed as a product of few commutators. [ABSTRACT FROM AUTHOR]

Let p be a prime integer and let ℤ p {\mathbb{Z}_{p}} be the ring of p-adic integers. By a purely computational approach we prove that each nonzero normal element of a noncommutative Iwasawa algebra over the special linear group SL 3 ⁢ (ℤ p) {\mathrm{SL}_{3}(\mathbb{Z}_{p})} is a unit. This gives a positive answer to an open question in [F. Wei and D. Bian, Erratum: Normal elements of completed group algebras over SL n ⁢ (ℤ p) \mathrm{SL}_{n}(\mathbb{Z}_{p}) [mr2747414], Internat. J. Algebra Comput. 23 2013, 1, 215] and makes up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over SL n ⁢ (ℤ p) \mathrm{SL}_{n}(\mathbb{Z}_{p}) , Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the root-involutions and root-subgroups of the Chevalley group E 6 (K) for fields K of characteristic two. The approach we follow is elementary and self-contained depends on the notion of M -sets which we have introduced in [Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type E 6 (K) for fields of characteristic two, Beit. Algebra Geom.58 (2017) 529–534]. The approach is elementary on the account that it consists of little more than naive linear algebra. It is remarkable to mention that Chevalley groups over fields of characteristic two have not much been researched. This work may contribute in this regard. This paper is divided into three main sections: the first section is a combinatorial section, the second section is on relations among M -sets, the last one is on Lie algebra. [ABSTRACT FROM AUTHOR]

Abstract The class of rank one solvable Lie algebras possessing a maximal torus t with eigenvalue spectrum spec (t) = (1 , 4 , 5 , … , n + 2) is studied in the context of rigidity. It is shown that from the value n ≥ 18, three isomorphism classes of rigid Lie algebras exist, two of them being algebraically rigid, and the third being geometrically rigid with a two-dimensional cohomology space H 2 (g , g). [ABSTRACT FROM AUTHOR]

Abstract Given an endomorphism A over a finite dimensional vector space having Jordan–Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's theorem about the characterization of the existence of characteristic non hyperinvariant subspaces. [ABSTRACT FROM AUTHOR]

Abstract 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 Z p , extending the proof given by Clozel for the group Γ 1 (S L 2 (Z p)) , the first congruence kernel of S L 2 (Z p) for primes p > 2. [ABSTRACT FROM AUTHOR]

We describe a simple way to write down explicit derivations of octonions that form a Chevalley basis of g 2 . This uses the description of octonions as a twisted group algebra of the finite field F 8 . Generators of Gal ( F 8 / F 2 ) act on the roots as 120-degree rotations and complex conjugation acts as negation. [ABSTRACT FROM AUTHOR]

We formulate and prove relative versions of several decompositions known in the theory of Chevalley groups over commutative rings. These decompositions are used to obtain factorizations in terms of subsystem subgroups of type A ℓ and upper estimates of the width of principal congruence subgroups with respect to Tits–Vaserstein generators. [ABSTRACT FROM AUTHOR]

In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of n-tuple semigroups. A free n-tuple semigroup of arbitrary rank is constructed and, as a consequence, singly generated free n-tuple semigroups are characterized. Moreover, examples of n-tuple semigroups are presented, the independence of the n-tuple semigroup axioms is proved, and it is shown that the natural semigroups of the constructed free n-tuple semigroup are isomorphic and the automorphism group of this n-tuple semigroup is isomorphic to a symmetric group. [ABSTRACT FROM AUTHOR]

Let G be a group and ω(G) = {o(g)|g ∈ G} be the set of element orders of G. Let k ∈n ω(G) and sk = |{g ∈ G |o(g) = k}|. Let nse(G) = {sk|k ∈ ω(G)}. In this paper, we prove that if G is a group and G2(4) is the Chevalley group such that nse(G) = nse(G2(4)), then G ≅ G2(4). [ABSTRACT FROM AUTHOR]

Abstract: The main result is a general approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups. Further, we contribute to the general theory of L 2‐Betti numbers of totally disconnected groups and provide exact computations of the L 2‐Betti numbers of the Neretin group and Chevalley groups over the field of Laurent series over a finite field and their lattices. [ABSTRACT FROM AUTHOR]

We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with a reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into an acylindrically hyperbolic group has an absolutely elliptic image. This result provides a non-arithmetic generalization of homomorphism superrigidity of Farb-Kaimanovich-Masur and Bridson-Wade. [ABSTRACT FROM AUTHOR]

The enveloping algebra of the niltriangular subalgebra NΦ(K) of the Chevalley algebra of type An−1 is the algebra of niltriangular n × n matrices over K. The enveloping algebras R of other types constructed so far are nonassociative. For classical types, an explicit description of automorphisms of the rings R over any commutative associative ring with an identity is given; in the case where K is a field, all ideals in R are also listed. The enumeration of ideals in R for K = GF(q) leads to a solution of a combinatorial problem concerning ideals of the algebras NΦ(K). [ABSTRACT FROM AUTHOR]

It is proved that every elementary carpet of nonzero additive subgroups which is associated with a Chevalley group of a Lie rank exceeding one over a locally finite field coincides, up to conjugation by a diagonal element, with a carpetwhose additive subgroups are equal to some chosen subfield of the ground field. A similar result is obtained for a full matrix carpet (a full net). [ABSTRACT FROM AUTHOR]

We characterise regular Goursat categories through a specific stability property of regular epimorphisms with respect to pullbacks. Under the assumption of the existence of some pushouts this property can be also expressed as a restricted Beck–Chevalley condition, with respect to the fibration of points, for a special class of commutative squares. In the case of varieties of universal algebras these results give, in particular, a structural explanation of the existence of the ternary operations characterising 3-permutable varieties of universal algebras. [ABSTRACT FROM AUTHOR]

In this article we improve the known uniform bound for subgroup growth of Chevalley groups G(&#120125p[[t]]). We introduce a new parameter, the ridgeline number v(G), and give new bounds for the subgroup growth of G(&#120125p[[t]]) expressed through v(G). We achieve this by deriving a new estimate for the codimension of [U, V ] where U and V are vector subspaces in the Lie algebra of G. [ABSTRACT FROM AUTHOR]

The paper presents the complete classification of Automorphic Lie Algebras based on $${{\mathfrak {sl}}}_{n}(\mathbb {C})$$ , where the symmetry group G is finite and acts on $${{\mathfrak {sl}}}_n(\mathbb {C})$$ by inner automorphisms, $${{\mathfrak {sl}}}_n(\mathbb {C})$$ has no trivial summands, and where the poles are in any of the exceptional G-orbits in $$\overline{\mathbb {C}}$$ . A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on the one hand a powerful tool from the computational point of view; on the other, it opens new questions from an algebraic perspective (e.g. structure theory), which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that this class of Automorphic Lie Algebras associated with the $$\mathbb {T}\mathbb {O}\mathbb {Y}$$ groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only; thus, they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring. [ABSTRACT FROM AUTHOR]

Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q( x , x ,..., x ) = $${\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}$$ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra. [ABSTRACT FROM AUTHOR]

Let g be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the "canonical basis" of the adjoint representation of g. Here, we present a variation of this idea which leads to a new, and quite elementary, construction of g itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of g. [ABSTRACT FROM AUTHOR]

Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p ≥ 0 and let V be an irreducible rational G -module with highest weight λ . When V is self-dual, a basic question to ask is whether V has a non-degenerate G -invariant alternating bilinear form or a non-degenerate G -invariant quadratic form. If p ≠ 2 , the answer is well known and easily described in terms of λ . In the case where p = 2 , we know that if V is self-dual, it always has a non-degenerate G -invariant alternating bilinear form. However, determining when V has a non-degenerate G -invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where G is of classical type and λ is a fundamental highest weight ω i , and in the case where G is of type A l and λ = ω r + ω s for 1 ≤ r < s ≤ l . We also give a solution in some specific cases when G is of exceptional type. As an application of our results, we refine Seitz's 1987 description of maximal subgroups of simple algebraic groups of classical type. One consequence of this is the following result. If X < Y < SL ( V ) are simple algebraic groups and V ↓ X is irreducible, then one of the following holds: (1) V ↓ Y is not self-dual; (2) both or neither of the modules V ↓ Y and V ↓ X have a non-degenerate invariant quadratic form; (3) p = 2 , X = SO ( V ) , and Y = Sp ( V ) . [ABSTRACT FROM AUTHOR]

We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is 2-torsion free.The result is applied to reconstruct the Chevalley Z-supergroups constructed by Fioresi and Gavarini (2012) and by Gavarini (2014). For a wide class of algebraic supergroups we describe their representations by using their superhyperalgebras. [ABSTRACT FROM AUTHOR]

On the adjoint representation of the Chevalley group of type E, cubic forms are constructed so that their partial derivatives are linear combinations of equations on the orbit of the highest weight vector. These forms are described in terms of new combinatorial notions related to maximal squares in root systems of exceptional types. Bibliography 17 titles. [ABSTRACT FROM AUTHOR]

Reduction theorems for triples of short root unipotent subgroups in Chevalley groups of type B and C are proved. The main result asserts that except for one case, any subgroup generated by such a triple is conjugate to a subgroup of G(B, K) U(B, K) or G(C, K)U(C, K), respectively. [ABSTRACT FROM AUTHOR]

In his seminal paper, half a century ago, Hyman Bass established commutator formulas for a (stable) general linear group, which were the key step in defining the group K . Namely, he proved that for an associative ring A with identity, where GL( A) is the stable general linear group and E( A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings for classical groups, algebraic groups, and their analogs, and mostly in relation to subnormal subgroups of these groups. The basic classical theorems and methods developed for their proofs are associated with the names of the heroes of classical algebraic K-theory: Bak, Quillen, Milnor, Suslin, Swan, Vaserstein, and others. One of the dominant techniques in establishing commutator type results is localization. In the present paper, some recent applications of localization methods to the study (higher/relative) commutators in the groups of points of algebraic and algebraic-like groups, such as general linear groups GL( n, A), unitary groups GU(2 n, A, Λ), and Chevalley groups G(Φ, A), are described. Some auxiliary results and corollaries of the main results are also stated. The paper provides a general overview of the subject and covers the current activities. It contains complete proofs borrowed from our previous papers and expositions of several main results to give the reader a self-contained source. [ABSTRACT FROM AUTHOR]

The non-Abelian symmetries of the half-infinite XXZ spin chain for all possible types of integrable boundary conditions are classified. For each type of boundary conditions, an analog of the Chevalley-type presentation is given for the corresponding symmetry algebra. In particular, two new algebras arise that are, respectively, generated by the symmetry operators of the model with triangular and special U q ( g l 2 ) -invariant integrable boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we show that general homogeneous manifolds G / P satisfy Conjecture O of Galkin, Golyshev and Iritani which ‘underlies’ Gamma conjectures I and II of them. Our main tools are the quantum Chevalley formula for G / P and a theory on nonnegative matrices including Perron–Frobenius theorem. [ABSTRACT FROM AUTHOR]

Let q be a power of a prime p , let G be a finite Chevalley group over F q and let U be a Sylow p -subgroup of G ; we assume that p is not a very bad prime for G . We explain a procedure of reduction of irreducible complex characters of U , which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F 4 , where we observe that the parametrization is “uniform” over good primes p > 3 , but differs for the bad prime p = 3 . We also explain how it has been applied for all groups of rank 4 or less. [ABSTRACT FROM AUTHOR]

Let G = G(Φ, K) be a Chevalley group of type Φ over a field K, where Φ is a simply laced root system. By studying the extraspecial unipotent radical of G, it is proved that any its element is a product of at most three root elements. Moreover, it is shown that up to conjugation by an element of the Levi subgroup, any element of the radical is the product of six elementary root elements. [ABSTRACT FROM AUTHOR]

We extend some general properties of automorphisms and derivations known for the Lie algebras to finite-dimensional complex Leibniz algebras. The analogs of the Jordan-Chevalley decomposition for derivations and the multiplicative decomposition for automorphisms of finite-dimensional complex Leibniz algebras are obtained. [ABSTRACT FROM AUTHOR]

Let H be a Hecke algebra arising as an endomorphism algebra of the representation of a Chevalley group G over $$F_{q}$$ induced by a unipotent cuspidal representation of a Levi quotient L of a parabolic subgroup. We assume that L is not a torus. In this paper we outline a geometric interpretation of the coefficients of the canonical basis of H in terms of perverse sheaves. We illustrate this in detail in the case where the Weyl group of G is ot type $$B_{4}$$ and that of L is of type $$B_{2}$$ . [ABSTRACT FROM AUTHOR]

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups. [ABSTRACT FROM AUTHOR]

Let be a finite simple group of Lie type. In this paper, we study characters of that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of . Such characters can be viewed as a natural generalization of the Steinberg character. For groups of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements. [ABSTRACT FROM AUTHOR]

We consider the question: what is the contragredient in terms of L-homomorphisms? We conjecture that it corresponds to the Chevalley automorphism of the L-group, and prove this in the case of real groups. The proof uses a characterization of the local Langlands correspondence over R. We also consider the related notion of Hermitian dual, in the case of GL(n,R). [ABSTRACT FROM AUTHOR]

Let $R$ be a commutative ring, let $F$ be a locally compact non-archimedean field of finite residual field $k$ of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. We show that the pro-$p$-Iwahori Hecke $R$-algebra of $G=\mathbf{G}(F)$ admits a presentation similar to the Iwahori–Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was previously known only for a $F$-split group $\mathbf{G}$. [ABSTRACT FROM AUTHOR]

In the current article we study the structure of a Chevalley group G ( R ) over a commutative ring R . We generalize and improve the following results on: • the standard, relative, and multi-relative commutator formulas; • the nilpotent structure of [relative] K 1 ; • the bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for the principal congruence subgroup, corresponding to a principal ideal. [ABSTRACT FROM AUTHOR]