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A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p , q). In this paper, we study pseudo-Riemannian Lie groups (G , 〈 ⋅ , ⋅ 〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g , then dim ⁡ h ≤ max ⁡ { 0 , min ⁡ { p , q } − 2 }. Secondly, we classify trans-Lorentzian Lie groups (i.e., min ⁡ { p , q } = 2) with non-Killing left-invariant conformal vector fields, and prove that [ g , g ] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat. [ABSTRACT FROM AUTHOR]

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The first author constructed their analogue in case of restricted Lie algebras of characteristic 2 [50] , Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [68]. Thus, we have examples of finitely generated restricted Lie algebras with a nil p -mapping. In characteristic zero, similar examples of Lie and Jordan algebras do not exist by results of Martinez and Zelmanov [43] and [78]. The first author constructed analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of arbitrary characteristic, the virtue of that construction is that Lie superalgebras have clear monomial bases [51] , they have slow polynomial growth. As an analogue of periodicity, Z 2 -homogeneous elements are ad-nilpotent. A recent example of a Lie superalgebra is of linear growth, of finite width 4, just infinite but not hereditary just infinite [13]. By that examples, an extension of the result of Martinez and Zelmanov [43] for Lie superalgebras of characteristic zero is not valid. Now, we construct a just infinite fractal 3-generated Lie superalgebra Q over arbitrary field, which gives rise to an associative hull A , a Poisson superalgebra P , and two Jordan superalgebras J and K , the latter being a factor algebra of J. In case char K ≠ 2 , A has a natural filtration, which associated graded algebra has a structure of a Poisson superalgebra such that gr A ≅ P , also P admits an algebraic quantization using a deformed superalgebra A (t). The Lie superalgebra Q is finely Z 3 -graded by multidegree in the generators, A , P are also Z 3 -graded, while J and K are Z 4 -graded by multidegree in four generators. By virtue of our construction, these five superalgebras have clear monomial bases and slow polynomial growth. We describe multihomogeneous coordinates of bases of Q , A , P in space as bounded by "almost cubic paraboloids". We determine a similar hypersurface in R 4 that bounds monomials of J and K. Constructions of the paper can be applied to Lie (super)algebras studied before to obtain Poisson and Jordan superalgebras as well. The algebras Q , A , and the algebras without unit P o , J o , K o are direct sums of two locally nilpotent subalgebras and there are continuum such decompositions. Also, Q = Q 0 ¯ ⊕ Q 1 ¯ is a nil graded Lie superalgebra, so, Q again shows that an extension of the result of Martinez and Zelmanov for Lie superalgebras of characteristic zero is not valid. In case char K = 2 , Q has a structure of a restricted Lie algebra with a nil p -mapping. The Jordan superalgebra K is nil finely Z 4 -graded, in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk group) of Jordan algebras in characteristic distinct from 2 [78]. Also, K is of slow polynomial growth, just infinite, but not hereditary just infinite. We call the superalgebras Q , A , P , J , K fractal because they contain infinitely many copies of themselves. [ABSTRACT FROM AUTHOR]

In commutative invariant theory, a classical result due to Auslander says that if R = k x 1 , ... , x n and G is a finite subgroup of Au t gr (R) ≅ GL (n , k) which contains no reflections, then there is a natural graded isomorphism R # G ≅ End R G (R). In this paper, we show that a version of Auslander's Theorem holds if we replace R by an Artin-Schelter regular algebra A of global dimension 2, and G by a finite subgroup of Au t gr (A) which contains no quasi-reflections. This extends work of Chan–Kirkman–Walton–Zhang. As part of the proof, we classify all such pairs (A , G) , up to conjugation of G by an element of Au t gr (A). In all but one case, we also write down explicit presentations for the invariant rings A G , and show that they are isomorphic to factors of AS regular algebras. [ABSTRACT FROM AUTHOR]

A complex vector space V is a prehomogeneous G -module if G acts rationally on V with a Zariski-open orbit. The module is called étale if dim ⁡ V = dim ⁡ G . We study étale modules for reductive algebraic groups G with one-dimensional center. For such G , even though every étale module is a regular prehomogeneous module, its irreducible submodules have to be non-regular. For these non-regular prehomogeneous modules, we determine some strong constraints on the ranks of their simple factors. This allows us to show that there do not exist étale modules for G = GL 1 × S × ⋯ × S , with S simple. [ABSTRACT FROM AUTHOR]

Stretching the parameters of a Littlewood–Richardson coefficient of value 2 by a factor of n results in a coefficient of value n + 1 [12,9,19] . We give a geometric proof of a generalization for representations of quivers. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the stable rank of any pullback (fibre product) does not exceed the stable ranks of factors. We obtain also another result bounding the stable rank of a ring by the stable ranks of its factor rings. [Copyright &y& Elsevier]

Abstract: An element g of a group is called reversible if it is conjugate in the group to its inverse. An element is an involution if it is equal to its inverse. This paper is about factoring elements as products of reversibles in the group of formal maps of , i.e. formally-invertible n-tuples of formal power series in n variables, with complex coefficients. The case was already understood [25]. Each product F of reversibles has linear part of determinant ±1. The main results are that for each map F with is the product of reversibles, and may also be factored as the product of involutions (where the ceiling of x is the smallest integer ⩾x). [Copyright &y& Elsevier]

Abstract: We consider a family of parabolic subalgebras of a simple Lie algebra of type and give a full description of the adjoint homology of the nilradicals of as a module over its Levi factor. All the nilradicals considered are 2-step nilpotent. [Copyright &y& Elsevier]

Abstract: Let be the projective variety of binary forms of degree d whose linear factors are distributed according to the partition λ of d. We determine minimal sets of local generators of , where is the normalization of , and we show that the local Jacobian matrices of contain the product of the identity matrix of maximal rank with a unit. We use this to fill a gap in a crucial proof in Chipalkattiʼs “On equations defining Coincident Root Loci”. Also, we give a new description of the singular locus of and a criterion for the smoothness of . [Copyright &y& Elsevier]

Abstract: In this paper we attempt to generalize the notion of “unique factorization domain” in the spirit of “half-factorial domain”. It is shown that this new generalization of UFD implies the now well-known notion of half-factorial domain. As a consequence, we discover that one of the standard axioms for unique factorization domains is slightly redundant. [Copyright &y& Elsevier]

Abstract: We give criteria to characterize abnormal, pronormal and locally pronormal subgroups of a direct product of two finite groups , under hypotheses of solvability for at least one of the factors, either A or B. [Copyright &y& Elsevier]

Abstract: The structure of a finitely presented monomial algebra over a field K is described. Here X is a finitely generated free monoid and I is a prime ideal of X that is finitely generated. As an application, a new structural proof of the recent result of Bell and Pekcagliyan [J. Bell, P. Pekcagliyan, Primitivity of finitely presented monomial algebras, preprint, arXiv: 0712.0815v1] on the primitivity of such algebras is presented, which yields a positive solution to the trichotomy problem, raised by Bell and Smoktunowicz [J. Bell, A. Smoktunowicz, The prime spectrum of algebras of quadratic growth, J. Algebra 319 (2008) 414–431], in the finitely presented case. Our approach is based on a new result on the form of prime Rees factors of semigroups satisfying the ascending chain condition on one-sided annihilators and on its refinement in the case of finitely presented factors of the form . [Copyright &y& Elsevier]

Abstract: Parabolic subalgebras of semi-simple Lie algebras decompose as where is a Levi factor and the corresponding nilradical. By Richardson''s theorem [R.W. Richardson, Bull. London Math. Soc. 6 (1974) 21–24], there exists an open orbit under the action of the adjoint group P on the nilradical. The elements of this dense orbits are known as Richardson elements. In this paper we describe a normal form for Richardson elements in the classical case. This generalizes a construction for of Brüstle et al. [Algebr. Represent. Theory 2 (1999) 295–312] to the other classical Lie algebra and it extends the authors normal forms of Richardson elements for nice parabolic subalgebras of simple Lie algebras to arbitrary parabolic subalgebras of the classical Lie algebras [K. Baur, Represent. Theory 9 (2005) 30–45]. As applications we obtain a description of the support of Richardson elements and we recover the Bala–Carter label of the orbit of Richardson elements. [Copyright &y& Elsevier]

Abstract: We show that ideal-length defines a length function on almost-Noetherian integral domains. This length function is a sum of multiplicities and in favourable cases, a linear combination on of discrete valuations. As a consequence, -Noetherian domains have length functions. Infra-Krull domains are weakly Krull almost-Noetherian domains. We characterize these integral domains and establish their properties, placing emphasis on integral closures and length functions. Descent properties are shown. Applications to the computation of elasticities and factorization properties are given. [Copyright &y& Elsevier]

It is shown that if in the factorization of a finite cyclic group each factor has either prime power order or order equal to the product of two primes then one factor must be periodic. This is shown to be the best possible result of this type. [Copyright &y& Elsevier]