Non-abelian Tensor Research Articles

On the size of the third homotopy group of the suspension of an Eilenberg--MacLane space

The nonabelian tensor square G \otimes G of a group G of G = p^n and G' = p^m (p prime and n,m \ge 1) satisfies a classic bound of the form G \otimes G \le p^{n(n-m)}. This allows us to give an upper bound for the order of the third homotopy group \pi_3(SK(G,1)) of the suspension of an Eilenberg--MacLane space K(G,1), because \pi_3(K(G,1)) is isomorphic to the kernel of \kappa : x \otimes y \in G \otimes G \mapsto [x,y] \in G'. We prove that G \otimes G \le p^{(n-1)(n-m)+2}, sharpening not only G \otimes G \le p^{n(n-m)} but also supporting a recent result of Jafari on the topic. Consequently, we discuss restrictions on the size of \pi_3(SK(G,1)) based on this new estimation.

Remarks on the relative tensor degree of finite groups

The present paper is a note on the relative tensor degree of finite groups. This notion generalizes the tensor degree, introduced recently in literature, and allows us to adapt the concept of relative commutativity degree through the notion of nonabelian tensor square. We show two inequalities, which correlate the relative tensor degree with the relative commutativity degree of finite groups.

Some Properties of Nilpotent Lie Algebras

In this paper we prove an analogue of Robinson Theorem for Lie algebras, which plays an important role to find the results connecting to the idea of nilpotency in Lie algebras. Also we find a criterion such that an extension of Lie algebras can be nilpotent. Keywords: Nilpotent Lie algebras; Non-abelian tensor product 2010 Mathematics Subject Classification: 17B30; 17B60; 17B

Duality-symmetric actions for non-Abelian tensor fields

We construct the duality-symmetric actions for a large class of six-dimensional models describing hierarchies of non-Abelian scalar, vector and tensor fields related to one another by first-order (self-)duality equations that follow from these actions. In particular, this construction provides a Lorentz invariant action for non-Abelian self-dual tensor fields. The class of models includes the bosonic sectors of the $6d$ (1,0) superconformal models of interacting non-Abelian self-dual tensor, vector, and hypermultiplets.

Characterization of Finite p-Groups by Their Non-Abelian Tensor Square

Let G be a finite p-group of order p n with |G′| =p c and G ⊗ G be the non-abelian tensor square of G. In 1991, Rocco proved that |G ⊗ G| =p n(n−c)−l for some integer l ≥ 0. In this article we characterize all finite p-groups, for which |G′| =p and 0 ≤ l ≤ 10.

The Exterior Squares of Some Crystallographic Groups

A crystallographic group is a discrete subgroup G of the set of isometries of Euclidean space En, where the quotient space En/G is compact. A specific type of crystallographic groups is called Bieberbach groups. A Bieberbach group is defined to be a torsion free crystallographic group. In this paper, the exterior squares of some Bieberbach groups with abelian point groups are computed. The exterior square of a group is the factor group of the nonabelian tensor square with the central subgroup of the group.

Non-Abelian tensor towers and (2,0) superconformal theories

With the aim to study six-dimensional (2, 0) superconformal theories with non-Abelian tensor multiplets we propose a five-dimensional superconformal action with eight supersymmetries for an infinite tower of non-Abelian vector, tensor and hypermultiplets. It describes the dynamics of the complete spectrum of the (2, 0) theories compactified on a circle coupled to an additional vector multiplet containing the circle radius and the Kaluza-Klein vector arising from the six-dimensional metric. All couplings are only given in terms of group theoretical constants and the Kaluza-Klein levels. After superconformal symmetry is reduced to Poincaré supersymmetry we find a Kaluza-Klein inspired action coupling super-Yang-Mills theory to an infinite tower of massive non-Abelian tensors. We explore the possibility to restore sixteen supersymmetries by using techniques known from harmonic superspace. Namely, additional bosonic coordinates on a four-sphere are introduced to enhance the R-symmetry group. Maximally supersymmetric Yang-Mills theories and the Abelian (2, 0) tensor theories are recovered as special cases of our construction. Finally, we comment on the generation of an anomaly balancing Wess-Zumino term for the R-symmetry vector at one loop.

Baer Invariants and Cohomology of Precrossed Modules

In this paper we study Baer invariants of precrossed modules relative to the subcategory of crossed modules, following Frohlich and Furtado-Coelho’s general theory on Baer invariants in varieties of Ω-groups and Modi’s theory on higher dimensional Baer invariants. Several homological invariants of precrossed and crossed modules were defined in the last two decades. We show how to use Baer invariants in order to connect these various homology theories. First, we express the low-dimensional Baer invariants of precrossed modules in terms of a new non-abelian tensor product of a precrossed module. This expression is used to analyze the connection between the Baer invariants and the homological invariants of precrossed modules defined by Conduche and Ellis. Specifically we prove that the second homological invariant of Conduche and Ellis is in general a quotient of the first component of the Baer invariant we consider. The definition of classical Baer invariants is generalized using homological methods. These generalized Baer invariants of precrossed modules are applied to the construction of five term exact sequences connecting the generalized Baer invariants with the cohomology theory of crossed modules considered by Carrasco, Cegarra and R.-Grandjean and the cohomology theory of precrossed modules.

Six-dimensional superconformal couplings of non-abelian tensor and hypermultiplets

Abstract We construct six-dimensional superconformal models with non-abelian tensor and hypermultiplets. They describe the field content of (2, 0) theories, coupled to (1, 0) vector multiplets. The latter are part of the non-abelian gauge structure that also includes non-dynamical three- and four-forms. The hypermultiplets are described by gauged nonlinear sigma models with a hyper-Kähler cone target space. We also address the question of constraints in these models and show that their resolution requires the inclusion of abelian factors. These provide couplings that were previously considered for anomaly cancellations with abelian tensor multiplets and resulted in the selection of ADE gauge groups.

SOME ALGEBRAIC AND TOPOLOGICAL PROPERTIES OF THE NONABELIAN TENSOR PRODUCT

In the nonabelian tensor product $G\otimes H$ of two groups $G$ and $H$ many properties pass from $G$ and $H$ to $G\otimes H$. There is a wide literature for different properties involved in this passage. We look at weak conditions for which such a passage may happen.

On some Metacycli cp -groups and their Nonabelian Tensor Square

In this paper, we develop appropriate programme using Groups, Algorithms and Programming (GAP) software enables performing different computations on various characteristics of all finite nonabelianmetacyclic p -groups, p is an odd prime, of nilpotency class at least three. Such programme enables to compute structure of the group, order of the group, structure of the center, the number of conjugacy classes, structure of commutator subgroup, abelianization, Whitehead`s universal quadratic functor and other characteristics. In addition, structures of some other groups such as the nonabelian tensor square and various homological functors including Schur multiplier and exterior square can be computed using this programme. Furthermore, by computing the epicenter order or the exterior center order the capability can be determined. In our current article, we only compute the nonabelian tensor square of certain order groups, as an example, and give GAP codes for computing other characteristics and some subgroups.

On some metacycli cp-groups and their nonabelian tensor square

On some metacycli cp-groups and their nonabelian tensor square

Computing the Nonabelian Tensor Square of Metacyclic p–Groups of Nilpotency Class Two

In this paper, we develop appropriate programme using Groups, Algorithms and Programming (GAP) software enables performing different computations on various characteristics of all finite nonabelian metacyclic p–groups, p is prime, of nilpotency class 2. Such programme enables to compute structure of the group, order of the group, structure of the center, the number of conjugacy classes, structure of commutator subgroup, abelianization, Whitehead’s universal quadratic functor and other characteristics. In addition, structures of some other groups such as the nonabelian tensor square and various homological functors including Schur multiplier and exterior square can be computed using this programme. Furthermore, by computing the epicenter order or the exterior center order the capability can be determined. In our current article, we only compute the nonabelian tensor square of certain order groups, as an example, and give GAP codes for computing other characteristics and some subgroups.

On the non-abelian tensor square of groups of order p4 where p is an odd prime

In this paper, we determine the non-abelian tensor square, G G, for non-abelian groups of order p 4 , where

The Non-abelian Tensor Square and Schur Multiplier of Groups of Orders p2q and p2qr

The aim of this paper is to determine the non-abelian tensor square and Schur multiplier of groups of square-free orders and of groups of orders p2q, pq2and p2qr, where p, q and r are primes and p < q < r.

New gauge anomalies and topological invariants in various dimensions

In the model of extended non-Abelian tensor gauge fields we have found new metric-independent densities: the exact (2n+3)-forms and their secondary characteristics, the (2n+2)-forms as well as the exact 6n-forms and the corresponding secondary (6n−1)-forms. These forms are the analogs of the Pontryagin densities: the exact 2n-forms and Chern–Simons secondary characteristics, the (2n−1)-forms. The (2n+3)- and 6n-forms are gauge invariant densities, while the (2n+2)- and (6n−1)-forms transform non-trivially under gauge transformations, which we compare with the corresponding transformations of the Chern–Simons secondary characteristics. This construction allows to identify new potential gauge anomalies in various dimensions.

A theory of non-abelian tensor gauge field with non-abelian gauge symmetry [formula omitted

The Chern–Simons action of the ABJM theory is not gauge invariant in the presence of a boundary. In Chu and Smith (2010) [1], this was shown to imply the existence of a Kac–Moody current algebra on the theory of multiple self-dual strings. In this paper we conjecture that the Kac–Moody symmetry induces a U(N)×U(N) gauge symmetry in the theory of N coincident M5-branes. As a start, we construct a G×G gauge symmetry algebra structure which naturally includes the tensor gauge transformation for a non-abelian 2-form tensor gauge field. The gauge covariant field strength is constructed. This new G×G gauge symmetry algebra allows us to write down a theory of a non-abelian tensor gauge field in any dimensions. The G×G gauge bosons can be either propagating, in which case the 2-form gauge fields would interact with each other through the 1-form gauge field; or they can be auxiliary and carry no local degrees of freedom, in which case the 2-form gauge fields would be self-interacting non-trivially. We finally comment on the possible application to the system of multiple M5-branes. We note that the field content of the G×G non-abelian tensor gauge theory can be fitted nicely into (1,0) supermultiplets; and we suggest a construction of the theory of multiple M5-branes with manifest (1,0) supersymmetry.

Two generalizations of the nonabelian tensor product

The purpose of this paper is two-fold. First we introduce the box-tensor product of two groups as a generalization of the nonabelian tensor product of groups. We extend various results for nonabelian tensor products to the box-tensor product such as the finiteness of the product when each factor is finite. This would give yet another proof of Ellisʼs theorem on the finiteness of the nonabelian tensor product of groups when each factor is finite. Secondly, using the methods developed in proving the finiteness of the box-tensor product, we prove the finiteness of Inassaridzeʼs tensor product under some additional hypothesis which generalizes his results on the finiteness of his product. In addition, we prove an Ellis like finiteness theorem under weaker assumptions, which is a generalization of his theorem on the finiteness of nonabelian tensor product. As a consequence, we prove the finiteness of low dimensional nonabelian homology groups.

A note on the nonabelian tensor square

In this paper, we determine the nonabelian tensor square G ⊗G for special orthogonal groups SOn (Fq) and spin groups Spinn (Fq), where Fq is a field with q elements.

A Note on the Nonabelian Tensor Square

In this paper, we determine the nonabelian tensor square G ⊗G for special orthogonal groups SOn (Fq) and spin groups Spinn (Fq), where Fq is a field with q elements.