Irreducible SL Research Articles

Tensor products and intertwining operators for uniserial representations of the Lie algebra [formula omitted

Let gm=sl(2)⋉V(m), m≥1, where V(m) is the irreducible sl(2)-module of dimension m+1 viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial gm-modules consist of a family, say of type Z, containing modules of arbitrary composition length, and some exceptional modules with composition length ≤4.Let V and W be two uniserial gm-modules of type Z. In this paper we obtain the sl(2)-module decomposition of soc(V⊗W) by giving explicitly the highest weight vectors. It turns out that soc(V⊗W) is multiplicity free. Roughly speaking, soc(V⊗W)=soc(V)⊗soc(W) in half of the cases, and in these cases we obtain the full socle series of V⊗W by proving that soct+1(V⊗W)=∑i=0tsoci+1(V)⊗soct+1−i(W) for all t≥0.As applications of these results, we obtain for which V and W, the space of gm-module homomorphisms Homgm(V,W) is not zero, in which case is 1-dimensional. Finally we prove, for m≠2, that if U is the tensor product of two uniserial gm-modules of type Z, then the factors are determined by U. We provide a procedure to identify the factors from U.

Path integrals on sl(2, R) orbits

We quantise orbits of the adjoint group action on elements of the sl Lie algebra. The path integration along elliptic slices is akin to the coadjoint orbit quantization of compact Lie groups, and the calculation of the characters of elliptic group elements proceeds along the same lines as in compact groups. The computation of the trace of hyperbolic group elements in a diagonal basis as well as the calculation of the full group action on a hyperbolic basis requires considerably more technique. We determine the action of hyperbolic one-parameter subgroups of PSL on the adjoint orbits and discuss global subtleties in choices of adapted coordinate systems. Using the hyperbolic slicing of orbits, we describe the quantum mechanics of an irreducible sl representation in a hyperbolic basis and relate the basis to the mathematics of the Mellin integral transform. We moreover discuss the representation theory of the double cover SL of PSL as well as that of its universal cover. Traces in the representations of these groups for both elliptic and hyperbolic elements are computed. Finally, we motivate our treatment of this elementary quantization problem by indicating applications.

The Sl(2, )-Character Variety of a Montesinos Knot

For each Montesinos knot K, we propose an efficient method to explicitly determine the irreducible SL(2, )-character variety, and show that it can be decomposed as χ0(K)⊔χ1(K)⊔χ2(K)⊔χ'(K), where χ0(K) consists of trace-free characters χ1(K) consists of characters of “unions” of representations of rational knots (or rational link, which appears at most once), χ2(K) is an algebraic curve, and χ'(K) consists of finitely many points when K satisfies a generic condition.

Branching from the General Linear Group to the Symmetric Group and the Principal Embedding

Let S be a principally embedded 𝔰𝔩 2 -subalgebra in 𝔰𝔩 n for n≥3. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible 𝔰𝔩 n -representation, V, there exists an irreducible S-representation embedding in V with dimension at most b(n). In a 2017 paper (joint with Hassan Lhou), they prove that b(n)=n is the sharpest possible bound, and also address embeddings other than the principal one.

Character varieties of even classical pretzel knots

Abstract For each even classical pretzel knot P(2k1 + 1, 2k2 + 1, 2k3), we determine the character variety of irreducible SL (2, ℂ)-representations, and clarify the steps of computing its A-polynomial.

Irreducible SL(2, ℂ)-metabelian representations of branched twist spins

An [Formula: see text]-branched twist spin is a fibered [Formula: see text]-knot in [Formula: see text] which is determined by a [Formula: see text]-knot [Formula: see text] and coprime integers [Formula: see text] and [Formula: see text]. For a [Formula: see text]-knot, Nagasato proved that the number of conjugacy classes of irreducible [Formula: see text]-metabelian representations of the knot group of a [Formula: see text]-knot is determined by the knot determinant of the [Formula: see text]-knot. In this paper, we prove that the number of irreducible [Formula: see text]-metabelian representations of the knot group of an [Formula: see text]-branched twist spin is determined up to conjugation by the determinant of the associated [Formula: see text]-knot in the orbit space by comparing a presentation of the knot group of the branched twist spin with the Lin’s presentation of the knot group of the [Formula: see text]-knot.

Triangulation independent Ptolemy varieties

The Ptolemy variety for SL(2,C) is an invariant of a topological ideal triangulation of a compact 3-manifold M. It is closely related to Thurston's gluing equation variety. The Ptolemy variety maps naturally to the set of conjugacy classes of boundary-unipotent SL(2,C)-representations, but (like the gluing equation variety) it depends on the triangulation, and may miss several components of representations. In this paper, we define a Ptolemy variety, which is independent of the choice of triangulation, and detects all boundary-unipotent irreducible SL(2,C)-representations. We also define variants of the Ptolemy variety for PSL(2,C)-representations, and representations that are not necessarily boundary-unipotent. In particular, we obtain an algorithm to compute all irreducible SL(2,C)-characters as well as the full A-polynomial. All the varieties are topological invariants of M.

The classification of the perfect cyclic [formula omitted]-modules

A perfect cyclic module of a perfect Lie algebra g with solvable radical r and Levi decomposition g=s⋉r is a finite dimensional module of g generated by an irreducible module of the semisimple Lie algebra s. In this paper we classify all the perfect cyclic finite dimensional indecomposable modules of the perfect Lie algebras sl(n+1)⋉Cn+1, given by the semidirect sum of the simple Lie algebra An with its standard representation. Furthermore, using the embedding of the Lie algebra sl(n+1)⋉Cn+1 in sl(n+2), we show that any finite dimensional irreducible module of sl(n+2) restricted to sl(n+1)⋉Cn+1 is a perfect cyclic module and that any perfect cyclic sl(n+1)⋉Cn+1-module can be constructed as quotient module of the restriction to sl(n+1)⋉Cn+1 of some finite dimensional irreducible sl(n+2)-module. This explicit realization of the perfect cyclic sl(n+1)⋉Cn+1-modules plays a role in their classification.

Jet Bundles on Projective Space II

In previous papers the structure of the jet bundle as P-module has been studied using different techniques. In this paper we use techniques from algebraic groups, sheaf theory, generliazed Verma modules, canonical filtrations of irreducible SL(V)-modules and annihilator ideals of highest weight vectors to study the canonical filtration Ul (g)Ld of the irreducible SL(V)-module H0 (X, iX(d))* where X = i‡(m, m + n). We study Ul (g)Ld using results from previous papers on the subject and recover a well known classification of the structure of the jet bundle il (i(d)) on projective space i(V*) as P-module. As a consequence we prove formulas on the splitting type of the jet bundle on projective space as abstract locally free sheaf. We also classify the P-module of the first order jet bundle iX1 (iX (d)) for any d ≥ 1. We study the incidence complex for the line bundle i(d) on the projective line and show it is a resolution of the ideal sheaf of I l (i(d)) - the incidence scheme of i(d). The aim of the study is to apply it to the study of syzygies of discriminants of linear systems on projective space and grassmannians.

METABELIAN SL(N, C) REPRESENTATIONS OF KNOT GROUPS, III: DEFORMATIONS

Given a knot K and an irreducible metabelian SL(n,C) representation we establish an equality for the dimension of the first twisted cohomology. In the case of equality, we prove that the representation must have finite image and that it is conjugate to an SU(n) representation. In this case we show it determines a smooth point x in the SL(n,C) character variety, and we use a deformation argument to establish the existence of a smooth (n-1)-dimensional family of characters of irreducible SL(n,C) representations near x. Combining this with our previous existence results, we deduce the existence of large families of irreducible SU(n) and SL(n,C) non-metabelian representation for knots K in homology 3-spheres S with nontrivial Alexander polynomial. We then relate the condition on twisted cohomology to a more accessible condition on untwisted cohomology of a certain metabelian branched cover of S branched along K.

The classification of uniserial [formula omitted]-modules and a new interpretation of the Racah–Wigner 6j-symbol

All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let V(m) be the irreducible sl(2)-module with highest weight m⩾1 and consider the perfect Lie algebra g=sl(2)⋉V(m). Recall that a g-module is uniserial when its submodules form a chain. In this paper we classify all uniserial g-modules. The main family of uniserial g-modules is actually constructed in greater generality for the perfect Lie algebra g=s⋉V(μ), where s is a semisimple Lie algebra and V(μ) is the irreducible s-module with highest weight μ≠0. The fact that the members of this family are, but for a few exceptions of lengths 2, 3 and 4, the only uniserial sl(2)⋉V(m)-modules depends in an essential manner on the determination of certain non-trivial zeros of Racah–Wigner 6j-symbol.

Twisted Alexander polynomials for SL(2;C)-irreducible representations of torus knots

We prove that the twisted Alexander polynomial of a torus knot with an irreducible SL(2,C)-representation is locally constant. In the case of a (2, q) torus knot, we can give an explicit formula for the twisted Alexander polynomial and deduce Hirasawa-Murasugi’s formula for the total twisted Alexander polynomial. We also give examples which address a mis-statement in a paper of Silver and Williams.

The simple non-Lie Malcev algebra as a Lie–Yamaguti algebra

The simple 7-dimensional Malcev algebra M is isomorphic to the irreducible sl(2,C)-module V(6) with binary product [x,y]=α(x∧y) defined by the sl(2,C)-module morphism α:Λ2V(6)→V(6). Combining this with the ternary product (x,y,z)=β(x∧y)⋅z defined by the sl(2,C)-module morphism β:Λ2V(6)→V(2)≈sl(2,C) gives M the structure of a generalized Lie triple system, or Lie–Yamaguti algebra. We use computer algebra to determine the polynomial identities of low degree satisfied by this binary–ternary structure.

RETRACTED ARTICLE: Irreducible SL n+1–Representations Remain Indecomposable Restricted to Some Abelian Subalgebras

RETRACTED ARTICLE: Irreducible SL n+1–Representations Remain Indecomposable Restricted to Some Abelian Subalgebras

Metabelian SL(n,ℂ) representations of knot groups, II: Fixed points

Given a knot K in an integral homology sphere with exterior N_K, there is a natural action of the cyclic group Z/n on the space of SL(n,C) representations of the knot group \pi_1(N_K), and this induces an action on the SL(n,C) character variety. We identify the fixed points of this action in terms of characters of metabelian representations, and we apply this to show that the twisted Alexander polynomial associated to an irreducible metabelian SL(n,C) representation is actually a polynomial in t^n.

Generalized projective representations for [formula omitted

It is well known that n-dimensional projective group gives rise to a non-homogeneous representation of the Lie algebra sl ( n + 1 ) on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known as Larsson functor), we generalize the above representation of sl ( n + 1 ) to a non-homogeneous representation on the tensor space of any finite-dimensional irreducible g l ( n ) -module with the polynomial space. Moreover, the structure of such a representation is completely determined by employing projection operator techniques and well-known Kostant's characteristic identities for certain matrices with entries in the universal enveloping algebra. In particular, we obtain a new one parameter family of infinite-dimensional irreducible sl ( n + 1 ) -modules, which are in general not highest-weight type, for any given finite-dimensional irreducible sl ( n ) -module. The results could also be used to study the quantum field theory with the projective group as the symmetry.

A WEIL REPRESENTATION OF sp(4) REALIZED BY DIFFERENTIAL OPERATORS IN THE SPACE OF SMOOTH FUNCTIONS ON S2 × S1

In the space of complex-valued smooth functions on S2 × S1, we explicitly realize a Weil representation of the real Lie algebra sp(4) by means of differential generators. This representation is a rare example of highest weight irreducible representation of sp(4) all whose weight spaces are 1-dimensional. We also show how this space splits into the direct sum of irreducible sl(2)-submodules. Selected applications: complete classification of yrast-band energies in even-even nuclei, the dynamical symmetry in some collective models of nuclear structure, the mapping methods for simplifying initial problem Hamiltonians.

A Clebsch–Gordan formula for [formula omitted] and applications to rationality

If R, S, T are irreducible SL 3 ( C ) -representations, we give an easy and explicit description of a basis of the space of equivariant maps R ⊗ S → T (Theorem 3.1). We apply this method to the rationality problem for invariant function fields. In particular, we prove the rationality of the moduli space of plane curves of degree 34. This uses a criterion which ensures the stable rationality of some quotients of Grassmannians by an SL-action (Proposition 5.4).

Classification of gradient space of dimension 8 by a reducible sℓ(2, C) action

This paper deals with a reducible sl(2,C) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau conjecture: Suppose that sl(2,C) acts on the formal power series ring via (1.1). Then I(f) = (li1) ⊕ (li2) ⊕... ⊕ (lis) modulo some one dimensional sl(2,C) representations where (li) is an irreducible sl(2,C) representation of li dimension and {li1li2,...,lis} ⊆ {l1, l2...,lr}. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat the reducible action and higher dimensional representations successively.

Metabelian SL(n, ℂ) representations of knot groups

We give a classification of irreducible metabelian representations from a knot group into SL(n,C) and GL(n,C). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n,C) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for this number in terms of the Alexander polynomial of the knot. These results are the higher rank generalizations of a result of Nagasato, who recently studied irreducible, metabelian SL(2,C) representations of knot groups. Finally we deduce the existence irreducible metabelian SL(n,C) representations of the knot group for any knot with nontrivial Alexander polynomial.