Two-step Nilpotent Groups Research Articles

Some properties of relative Rota–Baxter operators on groups

We find connection between relative Rota–Baxter operators and usual Rota–Baxter operators. We prove that any relative Rota–Baxter operator on a group H with respect to ( G , Ψ ) defines a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G . On the other side, we give condition under which a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G defines a relative Rota–Baxter operator on H with respect to ( G , Ψ ) . We introduce homomorphic post-groups and find their connection with λ-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family of post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group.

Уравнения от одной переменной над двуступенно нильпотентными группами и цепочки централизаторов

In the article we prove the equivalence of the concepts of equationally Noetherianity of a torsion-free two-step nilpotent group and the existence in it of an infinite chain of strictly decreasing centralizers and present the idea of a two-way connection between equationally Noetherian partially commutative two-step nilpotent groups and equationally Noetherian graphs with loops.

Densely locally minimal groups

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups Zp of p-adic integers. In [30], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups.We prove that in case that a topological abelian group G is either compact or connected locally compact, then G is densely locally minimal if and only if G either is a Lie group or has an open subgroup isomorphic to Zp for some prime p. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to Zp. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to Zp.

On monomial representations of finitely generated nilpotent groups

ABSTRACTA result of Segal states that every complex irreducible representation of a finitely generated nilpotent group G is monomial if and only if G is abelian-by-finite. A conjecture of Parshin, recently proved affirmatively by Beloshapka and Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture using ideas of Kutzko and Brown. We also give a characterization of the finite-dimensional irreducible representations of two-step nilpotent groups and describe these completely for two-step groups whose center has rank one.

Equations in nilpotent groups

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.

Left invariant Randers metrics on the 3-dimensional Heisenberg group

In the previous paper [MR2430243] we computed some geometric quantities such as curvature and flag curvature for a general left invariant Finsler metric on a two-step nilpotent group. In the present paper we give a more complete description of the Chern--Rund connection defined by a left invariant Randers metric on the 3 dimensional Heisenberg group.

Weil Representations of Symplectic Groups over Non-Principal Rings

Let W : Sp (2n, R) → GL(X) be a Weil representation of the symplectic group of rank 2n over a finite commutative ring R of odd characteristic. This is a complex representation of degree |R|n defined in terms of the action of Sp (2n, R) on a two-step nilpotent group called Heisenberg group. We address the problem of decomposing the Sp (2n, R)-module X into irreducible constituents. The problem can easily be reduced to the case when R is local and quasi-Frobenius. Further, the case when R is a principal ring has already been solved. This was achieved by means of the following recursive property of the Weil representation: precisely two irreducible constituents of X do not admit trivial action by any congruence subgroup of Sp (2n, R); the remaining irreducible constituents lie inside an Sp (2n, R)submodule Y of X that affords a Weil representation for a quotient symplectic group Sp (2n, T ). We show here that this recursive property of Y holds only when R is principal, failing in all other cases. This failure opens the following Pandora box: given any finite commutative quasi-Frobenius local ring R0 of odd characteristic, we can choose R so that R0 is quotient of R and every complex irreducible character of Sp (2n,R0) enters Y when inflated to Sp (2n, R). Thus, the problem of decomposing the Weil module X into irreducible constituents is, in general, as difficult as the problem of finding all complex irreducible characters of all symplectic groups Sp (2n, R0). In spite of this, we manage to identify submodules of X that do admit either a Weil representation or the tensor product of various Weil representations for a quotient symplectic group.

On the stable rank of algebras of operator fields over metric spaces

Let Γ be a finitely generated, torsion-free, two-step nilpotent group. Let C * ( Γ ) denote the universal C * -algebra of Γ . We show that sr ( C * ( Γ ) ) = sr ( C ( ( Γ ^ ) 1 ) , where for a unital C * -algebra A , sr ( A ) is the stable rank of A , and where ( Γ ^ ) 1 is the space of one-dimensional representations of Γ . In process, we give a stable rank estimate for maximal full algebras of operator fields over metric spaces.

Three-step Harmonic Solvmanifolds

The Lichnerowicz conjecture asserted that every harmonic Riemannian manifold is locally isometric to a two-point homogeneous space. In 1992, E. Damek and F. Ricci produced a family of counter-examples to this conjecture, which arise as abelian extensions of two-step nilpotent groups of type-H. In this paper we consider a broader class of Riemannian manifolds: solvmanifolds of Iwasawa type with algebraic rank one and two-step nilradical. Our main result shows that the Damek–Ricci spaces are the only harmonic manifolds of this type.

On Primitive Representations of Minimax Nilpotent Groups

We show, in particular, that in the class of minimax two-step nilpotent groups only finitely generated groups can admit exact irreducible primitive representations over a finitely generated field of characteristic zero. We also suggest some approaches to studying induced representations of nilpotent groups of finite rank.

Free two-step nilpotent groups whose automorphism group is complete

According to the result by Dyer and Formanek [4], the automorphism group of a finitely generated free two-step nilpotent group is complete except in the case when this group is a one- or three-generator (the three-generator groups have automorphism tower of height 2). The purpose of this paper is to prove that the automorphism group of an infinitely generated free two-step nilpotent group is also complete. (Recall that a group G is said to be complete if G is centreless and every automorphism is inner.)The paper may be considered as a contribution to the study of automorphism towers of relatively free groups.

The equivariant brauer group and twisted transformation group $C^{\ast}$-algebras

Twisted transformation group $C^{\ast}$-algebras associated to locally compact dynamical systems $(X = Y/N,G)$ are studied, where $G$ is abelian, $N$ is a closed subgroup of $G$, and $Y$ is a locally trivial principal $G$-bundle over $Z = Y/G$. An explicit homomorphism between $H^{2}(G,C(X,\mathbb{T}))$ and the equivariant Brauer group of Crocker, Kumjian, Raeburn and Williams, $Br_{N}(Z)$, is constructed, and this homomorphism is used to give conditions under which a twisted transformation group $C^{\ast}$-algebra $C_{0}(X) \times_{\tau,\omega}G$ will be strongly Morita equivalent to another twisted transformation group $C^{\ast}$-algebra $C_{0}(Z) \times_{Id,\omega}N$. These results are applied to the study of twisted group $C^{\ast}$-algebras $C^{\ast}(\Gamma,\mu)$ where $\Gamma$ is a finitely generated torsion free two-step nilpotent group.

Solvability for a class of non-homogeneous differential operators on two-step nilpotent groups

The purpose of this article is to extend our investigations on the problem of solvability for second order lefl-invariant differential operators on twostep nilpotent Lie groups from the case of homogeneous operators, which was treated in a complete way in [MR3], to the case of non-homogeneous operators. More precisely, on a connected, simply connected two-step nilpotent Lie group G we shall consider lefi-invariant differential operators of the form

The Primitive Ideal Space of Two-Step Nilpotent Group C*-Algebras

Let N be a two-step nilpotent, locally compact, second countable group having center Z and quotient A = N/ Z. We study the Jacobson topology on the primitive ideal space Prim C*( N) of the group C*-algebra of N. We are able to describe this topology in terms of convergence of subgroup-representation pairs, as used by the first author in an earlier work. Under appropriate conditions on N, we are able to describe Prim C*( N) globally as the quotient of a principal  bundle over Ẑ modulo an equivalence relation determined entirely by the group structure. We use this second result to compute the primitive ideal spaces of several examples, including all finitely generated, non-torsion two-step nilpotent discrete groups of rank less than or equal to five. Applications of our methods to more general central twisted crossed products are discussed.

Zentrale Distributionen auf nilpotenten Liegruppen

We contribute some results to the following question discussed in several papers ([3], [9], [12]): To what extent the characters of the irreducible representations of a nilpotent Lie groupN determine all central tempered distributions onN. A hereditary property is proved, which enables us to give a negative answer for a large class of nilpotent Lie groups. However, we give a positive answer for a certain series of two-step nilpotent groups. Furthermore, proving a generalisation of the Schwartz Kernel Theorem we decide the question specially for all groups of dimension<5.