A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical

LetG be a classical algebraic group defined over an algebraically closed field. We classify all instances when a parabolic subgroupP ofG acts on its unipotent radicalP _u , or onp _u , the Lie algebra ofP _u , with only a finite number of orbits.The proof proceeds in two parts. First we obtain a reduction to the case of general linear groups. In a second step, a solution for these is achieved by studying the representation theory of a particular quiver with certain relations.Furthermore, for general linear groups we obtain a combinatorial formula for the number of orbits in the finite cases.     

The Classification of Naturally Graded p-Filiform Leibniz Algebras

In the present article the classification of n-dimensional naturally graded p-filiform (1 ≤ p ≤ n ? 4) Leibniz algebras is obtained. A splitting of the set of naturally graded Leibniz algebras into the families of Lie and non Lie Leibniz algebras by means of characteristic sequences (isomorphism invariants) is proved.     

Group gradings on the Lie and Jordan superalgebras Q(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras Q(n), n ≥ 2, over an algebraically closed field of characteristic different from 2 (and not dividing n + 1 in the Lie case): Fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Algebraic Versus Homological Equivalence for Singular Varieties

Let Y ? ?^N be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimY_sing ≤ min {d + h ? 1, n ? 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H_2(d+h)(Y; ?) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.     

Lie Regular Generators of General Linear Groups

In this article, we introduce the idea of Lie regular elements and study 2 × 2 Lie regular matrices. It is shown that the linear groups GL(2, ?_{2 ⁿ} ), GL(2, ? _{p ⁿ} ), and GL(2, ?_2p ) (where p is an odd prime) can be genrated by Lie regular matrices. Presentations of linear groups GL(2, ?₄), GL(2, ?₆), GL(2, ?₈), and GL(2, ?₁₀) are also given.     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.     

Galois Cohomology of Unipotent Algebraic Groups and Field Extensions

In this note we discuss, in the case of unipotent groups over nonperfect fields k of characteristic p, an analog of a theorem of Steinberg (formely a Serre's conjecture) for unipotent algebraic group schemes, which relates properties of Galois (or flat) cohomology of unipotent group schemes to finite extensions of k of degree divisible by p.     

Stability of Numerical Invariants in Free Groups

Let d be an odd integer, and let k be a field which contains a primitive dth root of unity. Let l ₁ and l ₂ be cyclic field extensions of k of degree d with norms n _{l 1/k} and n _{l 2/k} . Minà?'s approach which showed that quadratic Pfister forms are strongly multiplicative is applied to the form n _{l 1/k}  ? n _{l 2/k} of degree d. Let K = k(X ₁,…, X _{d ²} ). We compute polynomials which are similarity factors of a form of the kind N ? (n _{l 2/k}  ? _k K) over K, where N is the norm of a certain field extension of K of degree d. These polynomials arise by specializing certain indeterminates of the homogeneous polynomial representing the form n _{l 1/k}  ? n _{l 2/k} to be zero. Similar results are obtained for the tensor product of the norm of a cubic division algebra and a cubic norm n _{l 1/k} .     

Branching Rules for General Linear Groups

In this article we give a branching rule for Harish–Chandra restriction from the general linear group Gl _n (q) to the Levi subgroup Gl _n?1(q) × Gl₁(q) in the case of the unipotent block.     

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract

We construct an associative algebra A _k and show that there is a representation of A _k on V ^?k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A _k is the full centralizer of 𝔭(n) on V ^?k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A _k we decompose the tensor space V ^?k , for k = 2 or 3, and show that V ^?k is not completely reducible for any k ≥ 2.     

Specht Modules and Simple Modules for Centralizer Algebras of the Symmetric Group

Let Σ_n be the symmetric group on n letters. For l ≤ n identify Σ_l with a subgroup of Σ_n in the natural way. Let k be an algebraically closed field of characteristic p. This article begins to develop a theory for modules over the centralizer algebras kΣ_n^Σ_l that is analogous to James's theory of permutation modules, Specht modules, and simple modules over kΣ_n. We make a conjecture about how to construct all simple kΣ_n^Σ_l-modules, we develop tools to test the conjecture, and we prove that it is correct for all n when l < p.     

Okubo algebras in characteristic 3 and their automorphisms

Associated to any eight-dimensional non-unital composition algebra with associative norm, there are outer automorphisms of order 3 of the corresponding spin group, such tiat the fixed subgroup is the automorphism group of the composition algebra. Over fields of characteristic ≠ 3 these are simple algebraic groups of types G ₂ or A ₂, related respectively to the para-octonion and the Okubo algebras

A connection between the Okubo algebras over fields of characteristic 3 with some simple noncommutative Jordan algebras will be used to compute explicitly the automorphism groups and Lie algebras of derivations of these algebras. In contrast to the other characteristics, ths groups will no longer be of type A ₂ and will either be trivial or contain a large unipotent radical.     

A Generalized Global Cartan Decomposition: A Basic Example

Suppose G ₁ ?  G are complex linear simple Lie groups. Let  ₁ ?  be the corresponding pair of Lie algebras. For the Killing-orthogonal  of  ₁ in  we have a vector space direct sum  =  ₁⊕ , which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G ₁) = (SL (4,?),Sp (2,?)); here  =  (4,?),  ₁ =  (2,?), and  = {X ?  | X ^? = X}, where X? X ^? is the symplectic involution. We prove that G  =  G ₁exp  ∪ i G ₁exp . The key point of the proof is to study in detail the set exp ; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ^? = X, which is here a natural counterpart of the standard Pfaffian.     

Power Values of Derivations with Annihilator Conditions on Lie Ideals in Prime Rings

Let R be a prime ring of char R ≠ 2, d a nonzero derivation of R, U a noncentral Lie ideal of R, and a ∈ R. If au ^{n ₁} d(u) ^{n ₂} u ^{n ₃} d(u) ^{n ₄} u ^{n ₅} … d(u) ^{n _k?1} u ^{n _k}  = 0 for all u ∈ U, where n ₁, n ₂,…,n _k are fixed non-negative integers not all zero, then a = 0 and if a(u ^s d(u)u ^t ) ⁿ  ∈ Z(R) for all u ∈ U, where s ≥ 0, t ≥ 0, n ≥ 1 are some fixed integers, then either a = 0 or R satisfies S ₄, the standard identity in four variables.     

Generalized Whittaker vectors for holomorphic and quaternionic representations

The conjugacy class of parabolic subgroups with Heisenberg unipotent radical in a simple Lie groups over ³ not of type C_nC_{n} contains an element defined over  for each quaternionic real form. In this paper we study the Whittaker models for quaternionic discrete series of these real forms and prove results analogous and by analogous methods to the case of simple Lie groups over  that are the automorphism groups of tube type Hermitian symmetric domain and (so-called Bessel models) for holomorphic representations. In particular we calculate the decomposition of the space of Whittaker vectors under the action of the stabilizer of the corresponding character in a Levi factor of the Heisenberg parabolic subgroup.     

Identities of Group Algebras

We consider polynomial identities of group algebras over a field F of characteristic zero. We prove that any PI group algebra satisfies the same identities as a matrix algebra M _n (F ), where n is the maximal degree of finite dimensional representations of the group over algebraic extensions of F.     

Representations of algebraic groups of type <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> with small weight multiplicities

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of type C _n in characteristic p ≤ 7 are found. If n ≥ 8 and p ≠ 2 , then for an irreducible representation either such a multiplicity is at least n− 4 − [n]₄,where [n]₄ is the residue of n modulo 4, or all the weight multiplicities are equal to 1.For p = 2, the situation is more complicated, and for every n and l there exists a class of representations with the maximal weight multiplicity equal to 2 ^l . For symplectic groups in characteristic p > 7 and spinor groups similar results were obtained earlier. Bibliography: 15 titles.     

Commutative Schur Rings of Maximal Dimension

A commutative Schur ring over a finite group G has dimension at most s _G  = d ₁ + … +d _r , where the d _i are the degrees of the irreducible characters of G. We find families of groups that have S-rings that realize this bound, including the groups SL(2, 2 ⁿ ), metacyclic groups, extraspecial groups, and groups all of whose character degrees are 1 or a fixed prime. We also give families of groups that do not realize this bound. We show that the class of groups that have S-rings that realize this bound is invariant under taking quotients. We also show how such S-rings determine a random walk on the group and how the generating function for such a random walk can be calculated using the group determinant.