Non-abelian Representation Research Articles

Mapping class group dynamics and the holonomy of branched affine structures

We classify, up to few exceptions, the orbit closures of the {mathrm {Mod}}(Sigma )-action on the affine character variety chi ( mathrm {Aff}({mathbb {C}})). We obtain from this classification that the only obstruction for a non-abelian representation rho : pi _1 Sigma longrightarrow mathrm {Aff}({mathbb {C}}) to be the holonomy of a branched affine structure on Sigma is to be Euclidean and not to have positive volume, where Sigma is a closed oriented surface of genus g ge 2.

Existence of non-abelian representations of the near hexagon Q(5,2)⊗Q(5,2)

In [5], a new combinatorial model with four types of points and nine types of lines of the slim dense near hexagon Q(5,2)⊗Q(5,2) was provided and it was then used to construct a non-abelain representation of Q(5,2)⊗Q(5,2) in the extraspecial 2-group \(2_{-}^{1+18}\). In this paper, we give a direct proof for the existence of a non-abelian representation of Q(5,2)⊗Q(5,2) in \(2_{-}^{1+18}\).

Polarized non-abelian representations of slim near-polar spaces

In (Bull Belg Math Soc Simon Stevin 4:299---316, 1997), Shult introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in Sahoo and Sastry (J Algebraic Comb 29:195---213, 2009) for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding.

ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL2(ℂ)

We study the twisted Alexander polynomial ΔK,ρof a knot K associated to a non-abelian representation ρ of the knot group into SL2(ℂ). It is known for every knot K that if K is fibered, then for every non-abelian representation, ΔK,ρis monic and has degree 4g(K) – 2 where g(K) is the genus of K. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot K is non-fibered, then all but finitely many non-abelian representations on some component have ΔK,ρnon-monic and degree 4g(K) – 2. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where ΔK,ρis monic and calculate the number of non-abelian representations where the degree of ΔK,ρis less than 4g(K) – 2.

Non-abelian representations of the slim dense near hexagons on 81 and 243 points

We prove that the near hexagon \(Q(5,2)\times\mathbb{L}_{3}\) has a non-abelian representation in the extra-special 2-group \(2^{1+12}_{+}\) and that the near hexagon Q(5,2)⊗ Q(5,2) has a non-abelian representation in the extra-special 2-group \(2^{1+18}_{-}\). The description of the non-abelian representation of Q(5,2)⊗ Q(5,2) makes use of a new combinatorial construction of this near hexagon.

A non-abelian representation of the dual polar space DQ(2n,2)

We prove that the dual polar space DQ(2n,2), n≥3, of rank n associated with a non-singular parabolic quadric in PG(2n,2) admits a faithful non-abelian representation in the extraspecial 2-group 2+1+2n. The near 2n-gon In (section 2.4) is a geometric hyperplane of DQ(2n,2). For n≥3, we first construct a faithful non-abelian representation of In in 2+1+2n and subsequently extend it to a faithful non-abelian representation of DQ(2n,2) in 2+1+2n.

On the order of a non-abelian representation group of a slim dense near hexagon

We show that, if the representation group $R$ of a slim dense near hexagon $S$ is non-abelian, then $R$ is of exponent 4 and $|R|=2^{\beta}$, $1+NPdim(S)\leq \beta\leq 1+dimV(S)$, where $NPdim(S)$ is the near polygon embedding dimension of $S$ and $dimV(S)$ is the dimension of the universal representation module $V(S)$ of $S$. Further, if $\beta =1+NPdim(S)$, then $R$ is an extraspecial 2-group (Theorem 1.6).

A characterization of finite symplectic polar spaces of odd prime order

A sufficient condition for the representation group for a nonabelian representation (Definition 1.1) of a finite partial linear space to be a finite p-group is given (Theorem 2.9). We characterize finite symplectic polar spaces of rank r at least two and of odd prime order p as the only finite polar spaces of rank at least two and of prime order admitting nonabelian representations. The representation group of such a polar space is an extraspecial p-group of order p 1 + 2 r and of exponent p (Theorems 1.5 and 1.6).

Non-Abelian braid statistics versus projective permutation statistics

Recent papers by Finkelstein, Galiautdinov, and co-workers [J. Math. Phys. 42, 1489 (2001); 42, 3299 (2001)] discuss a suggestion by Wilczek that non-Abelian projective representations of the permutation group can be used as a new type of particle statistics, valid in any dimension. Wilczek’s suggestion was based in part on an analysis by Nayak and Wilczek (NW) of the non-Abelian representation of the braid group in a quantum Hall system. We point out that projective permutation statistics is not possible in a local quantum field theory as it violates locality, and show that the NW braid group representation is not equivalent to a projective representation of the permutation group. The structure of the finite image of the braid group in a 2n/2−1-dimensional representation is obtained.

Non-abelian electrodynamics and the vacuumB (3)

By using an 0(3) gauge group, a non-Abelian theory of vacuum electrodynamics is developed in which the newly discovered longitudinal vacuum fieldsB(3) andiE(3) appear self-consistently with the usual plane wavesB(1),B(2),E(1), andE(2) in the circular basis (1), (2), (3), a complex representation of space. Using the charge quantization condition Open image in new window the vacuum Maxwell equations are given in the non-Abelian representation.

Algebra for Heckoid Groups

We introduce an infinite collection of (Laurent) polynomials associated with a $2$-bridge knot or link normal form $K = (\alpha ,\beta )$. Experimental evidence suggests that these "Heckoid polynomials" define the affine representation variety of certain groups, the Heckoid groups, for $K$ . We discuss relations which hold in the image of the generic representation for each polynomial. We show that, with a certain change of variable, each Heckoid polynomial divides the nonabelian representation polynomial of $L$ , where $L$ belongs to an infinite collection of $2$-bridge knots/links determined by $K$ and the Heckoid polynomial. Finally, we introduce a "precusp polynomial" for each $2$-bridge knot normal form, and show it is the product of two (possibly reducible) non-constant polynomials. We are preparing a sequel on the Heckoid groups and the evidence for some of the geometrical assertions stated in the introduction.

ANYON EQUATION ON A TORUS

Starting from the quantum field theory of nonrelativistic matter on a torus interacting with Chern-Simons gauge fields, we derive the Schrödinger equation for an anyon system. The nonintegrable phases of the Wilson line integrals on a torus play an essential role. In addition to generating degenerate vacua, they enter in the definition of a many-body Schrödinger wave function in quantum mechanics, which can be defined as a regular function of the coordinates of anyons. It obeys a non-Abelian representation of the braid group algebra, being related to Einarsson’s wave function by a singular gauge transformation.

Algebra for Heckoid groups

We introduce an infinite collection of (Laurent) polynomials associated with a 2 2 -bridge knot or link normal form K = ( α , β ) K = (\alpha ,\beta ) . Experimental evidence suggests that these "Heckoid polynomials" define the affine representation variety of certain groups, the Heckoid groups, for K K . We discuss relations which hold in the image of the generic representation for each polynomial. We show that, with a certain change of variable, each Heckoid polynomial divides the nonabelian representation polynomial of L L , where L L belongs to an infinite collection of 2 2 -bridge knots/links determined by K K and the Heckoid polynomial. Finally, we introduce a "precusp polynomial" for each 2 2 -bridge knot normal form, and show it is the product of two (possibly reducible) non-constant polynomials. We are preparing a sequel on the Heckoid groups and the evidence for some of the geometrical assertions stated in the introduction.

Constrained systems, OSP(1,1|2) and string theory

The underlying OSP invariance of the Fradkin-Vilkovisky formalism is discussed. Ghost degrees of freedom are interpreted as negative dimensional phase space variables that eliminate unphysical degrees of freedom by the Parisi-Sourlas mechanism, ensuring manifest covariance. The formalism makes use of subsidiary constraints, extending the usual algebra of constraints. A relations between abelian and nonabelian constraint algebras is established, and exploited to construct a nonabelian representation of the OSP generators. For theories based entirely on constraints such as string theories, the natural Fradkin-Vilkovisky hamiltonian is a manifestly OSP invariant squared length of a graded phase space vector. As an application, the OSP covariant formulation of bosonic strings is discussed.