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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]