Cartan Subalgebra Research Articles

Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

Let U be either the classical or quantized universal enveloping algebra of the Lie algebra [Formula: see text] extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in U over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse form is explicitly constructed.

Heterotic effective action and duality symmetries revisited

The dimensional reduction of heterotic supergravity with gauge fields truncated to the Cartan subalgebra exhibits a continuous O(d,d+16;R) global symmetry, related to the O(d,d+16;Z) T-duality of heterotic strings on a d-torus. The O(d,d+16;R) symmetry is not present, however, if the supergravity reduction is done including the full set of E_8 x E_8 or SO(32) gauge fields. We analyze which duality symmetries are realized to all orders in $\alpha'$ in the proper effective field theories for the massless string states. We find a universal O(d,d;R) symmetry, also predicted by double field theory. We confirm this by giving a novel formulation of the dimensionally reduced supergravity in terms of O(d,d) multiplets, and we discuss cases of symmetry enhancement.

Quantum Magnets and Matrix Lorenz Systems

The Landau-Lifshitz-Gilbert equations for the evolution of the magnetization, in presence of an external torque, can be cast in the form of the Lorenz equations and, thus, can describe chaotic fluctuations. To study quantum effects, we describe the magnetization by matrices, that take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of u(2) = u(1) × su(2). We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.

Cartan subalgebras of amalgamated free product $II_1$ factors

We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $\Gamma$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ arising from a free ergodic probability measure preserving action of $\Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite countable ergodic probability measure preserving equivalence relations, then the II$_1$ factor $L(\mathcal R)$ has a unique Cartan subalgebra, up to unitary conjugacy. Finally, we show that the free product $M=M_1*M_2$ of any two II$_1$ factors does not have a Cartan subalgebra. More generally, we prove that if $A\subset M$ is a diffuse amenable von Neumann subalgebra and $P\subset M$ denotes the algebra generated by its normalizer, then either $P$ is amenable, or a corner of $P$ embeds into $M_1$ or $M_2$.

Simple Modules for Modular Lie SuperalgebrasW(0∣n),S(0∣n), andK(n)

This paper constructs a series of modules from modular Lie superalgebrasW(0∣n),S(0∣n), andK(n)over a field of prime characteristicp≠2. Cartan subalgebras, maximal vectors of these modular Lie superalgebras, can be solved. With certain properties of the positive root vectors, we obtain that the sufficient conditions of these modules are irreducibleL-modules, whereL=W(0∣n),S(0∣n), andK(n).

Simple modules for some Cartan-type Lie superalgebras

The modules are induced from the finite-dimensional Cartan-type modular Lie superalgebrasW , S, H and K over a field of prime characteristic, respectively. We give the Cartan subalgebras of these modular Lie superalgebras. Using certain properties of the positive root vectors, we discuss the simplicity of these modules. 2000 AMS Classification: 17B50, 17B40

More on BPS States in N = 4 $$ \mathcal{N}=4 $$ Supersymmetric Yang-Mills Theory on R × S3

We perform a systematic analysis on supersymmetric states in N=4 supersymmetric Yang-Mills theory (SYM) on R x S^3. We find a new set of 1/16 BPS equations and determine the precise configuration of the supersymmetric states by solving all 1/16 BPS equations when they are valued in Cartan subalgebra of a gauge group and the fermionic fields vanish. We also determine the number of supersymmetries preserved by the supersymmetric states varying the parameters of the BPS solutions. As a byproduct we present the complete set of such supersymmetric states in N=8 SYM on R x S^2 by carrying out dimensional reduction.

Compatible pairs of Borel subalgebras and shared orbit pairs

Consider a class of pairs (g,g1), where g is a semisimple Lie algebra and g1 is a subalgebra reductive in g, satisfying the following: For any Cartan subalgebra h1⊆g1 there is a unique Cartan subalgebra h⊆g containing h1. Given such a pair (g,g1) and a Borel subalgebra b1⊆g1 we study a (finite) set SBorg(b1) of all Borel subalgebras b⊆g containing b1. In particular we point at the subclass of pairs (g,g1) when SBorg(b1) is a singleton, for every Borel subalgebra b1⊆g1. As a consequence, for such pairs we relate the corresponding flag varieties B(g) and B(g1). As an interesting class of pairs (g,g1), for which we can apply our results on pairs of Borel subalgebras, we study in detail the pairs considered by R. Brylinski and B. Kostant related to shared orbit pairs.

Normalizers inside Amalgamated Free Product von Neumann Algebras

Recently, Adrian Ioana proved that all crossed products L^\infty(X) \rtimes (\Gamma_1*\Gamma_2) by free ergodic probability measure preserving actions of a nontrivial free product group \Gamma_1 * \Gamma_2 have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer \mathcal N_M(A)^{\prime\prime} of an amenable subalgebra A of an amalgamated free product von Neumann algebra M = M_1 *_B M_2 . We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in L^\infty(X) \rtimes (\Gamma_1*\Gamma_2) .

Weyl group orbits on Kac–Moody root systems*

Let be a Dynkin diagram and let be the simple roots of the corresponding Kac–Moody root system. Let denote the Cartan subalgebra, let W denote the Weyl group and let Δ denote the set of all roots. The action of W on , and hence on Δ, is the discretization of the action of the Kac–Moody algebra. Understanding the orbit structure of W on Δ is crucial for many physical applications. We show that for , the simple roots αi and αj are in the same W–orbit if and only if vertices i and j in the Dynkin diagram corresponding to αi and αj are connected by a path consisting only of single edges. We introduce the notion of ‘the Cayley graph of the Weyl group action on real roots’ whose connected components are in one-to-one correspondence with the disjoint orbits of W. For a symmetric hyperbolic generalized Cartan matrix A of rank we prove that any two real roots of the same length lie in the same W–orbit. We show that if the generalized Cartan matrix A contains zeros, then there are simple roots that are stabilized by simple root reflections in W, that is, W does not act simply transitively on real roots. We give sufficient conditions in terms of the generalized Cartan matrix A (equivalently ) for W to stabilize a real root. Using symmetry properties of the imaginary light cone in the hyperbolic case, we deduce that the number of W–orbits on imaginary roots on a hyperboloid of fixed radius is bounded above by the number of root lattice points on the hyperboloid that intersect the closure of the fundamental region for W.

Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras

We show that the continuous core of any type III free product factor has no Cartan subalgebra. This is a complement to previous works due to Houdayer--Ricard and Boutonnet--Houdayer--Raum.

Mordell-Weil torsion and the global structure of gauge groups in F-theory

We study the global structure of the gauge group $G$ of F-theory compactified on an elliptic fibration $Y$. The global properties of $G$ are encoded in the torsion subgroup of the Mordell-Weil group of rational sections of $Y$. Generalising the Shioda map to torsional sections we construct a specific integer divisor class on $Y$ as a fractional linear combination of the resolution divisors associated with the Cartan subalgebra of $G$. This divisor class can be interpreted as an element of the refined coweight lattice of the gauge group. As a result, the spectrum of admissible matter representations is strongly constrained and the gauge group is non-simply connected. We exemplify our results by a detailed analysis of the general elliptic fibration with Mordell-Weil group $\mathbb Z_2$ and $\mathbb Z_3$ as well as a further specialization to $\mathbb Z \oplus \mathbb Z_2$. Our analysis exploits the representation of these fibrations as hypersurfaces in toric geometry.

Eigenvalue Distributions of Reduced Density Matrices

Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

Partition Functions of $${\mathcal{N}=(2,2)}$$ N = ( 2 , 2 ) Gauge Theories on S 2 and Vortices

We apply localization techniques to compute the partition function of a two-dimensional \({\mathcal{N}=(2,2)}\) R-symmetric theory of vector and chiral multiplets on S 2. The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet–Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. For applications, we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

A Method for Constructing Invariants of a Lie Group

For the adjoint action of a Lie group G (or a subgroup of G) on the Lie algebra Lie (G), we suggest a method for constructing its invariants. The method is easy to implement and may shed light on the algebraical independence of invariants. The main idea is to extend automorphisms of the Cartan subalgebra to automorphisms of the whole Lie algebra Lie (G). The corresponding matrices in the linear space V ∼ = Lie (G) define the Reynolds operator, which “gathers” invariants of the torus T ⊂ G into special polynomials. The condition for a linear combination of polynomials to be G-invariant is equivalent to the existence of a solution for a certain system of linear equations in the coefficients of the combination. As an example, we consider the adjoint action of the Lie group SL(3) (and its subgroup SL(2)) on the Lie algebra sl(3).

Conjugacy theorems for loop reductive group schemes and Lie algebras

The conjugacy of split Cartan subalgebras in the finite dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras --extended affine Lie algebras-- that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings

A study of saturated tensor cone for symmetrizable Kac–Moody algebras

Let $$\mathfrak {g}$$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $$\mathfrak {h}$$ and the Weyl group $$W$$ . Let $$P_+$$ be the set of dominant integral weights. For $$\lambda \in P_+$$ , let $$L(\lambda )$$ be the integrable, highest weight (irreducible) representation of $$\mathfrak {g}$$ with highest weight $$\lambda $$ . For a positive integer $$s$$ , define the saturated tensor semigroup as $$\begin{aligned} \Gamma _s:= \{(\lambda _1, \dots , \lambda _s,\mu )\in P_+^{s+1}: \exists \, N\ge 1 \,\text {with}\,L(N\mu )\subset L(N\lambda _1)\otimes \dots \otimes L(N\lambda _s)\}. \end{aligned}$$ The aim of this paper is to begin a systematic study of $$\Gamma _s$$ in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by $$\Gamma _s$$ . These inequalities are indexed by products in $$H^*(G^{\mathrm{min }}/B; \mathbb {Z})$$ for $$B$$ the standard Borel subgroup, where $$G^{\mathrm{min }}$$ is the ‘minimal’ Kac-Moody group with Lie algebra $$\mathfrak {g}$$ . The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that $$\mathfrak {g}$$ is affine of rank 2, we show that these inequalities are necessary and sufficient. We further prove that any integer $$d>0$$ is a saturation factor for $$A^{(1)}_1$$ and 4 is a saturation factor for $$A^{(2)}_2$$ .

A II1 Factor Approach to the Kadison–Singer Problem

We show that the Kadison-Singer problem, asking whether the pure states of the diagonal subalgebra $\ell^\infty\Bbb N\subset \Cal B(\ell^2\Bbb N)$ have unique state extensions to $\Cal B(\ell^2\Bbb N)$, is equivalent to a similar statement in II$_1$ factor framework, concerning the ultrapower inclusion $D^\omega \subset R^\omega$, where $D$ is the Cartan subalgebra of the hyperfinite II$_1$ factor $R$, and $\omega$ is a free ultraflter. While we do not settle the problem in this latter form, we prove that if $A$ is any singular maximal abelian subalgebra of $R$, then the inclusion $A^\omega \subset R^\omega$ does satisfy the Kadison-Singer property.

Restrictions of Symmetric Invariants of a Complex Semisimple Lie Algebra and Some Applications

Let 𝔤 be a finite-dimensional complex semisimple Lie algebra and σ an arbitrary semisimple automorphism of 𝔤. Let 𝔱 be a Cartan subalgebra of 𝔨 = 𝔤σ and 𝔥 =Z 𝔤(𝔱) be the centralizer of 𝔱 in 𝔤. Then 𝔥 is a σ-invariant Cartan subalgebra of 𝔤 and 𝔱 = 𝔥σ. Let W(𝔤, 𝔥) be the Weyl group. One knows that Δ(𝔤, 𝔱), the set of roots of 𝔤 in 𝔱, is also a root system. It is proved that the corresponding Weyl group W(𝔤, 𝔱) is isomorphic to W(𝔤, 𝔥)σ, which is the subgroup of W(𝔤, 𝔥) consisting of those elements commuting with σ. It is also shown that the image of the restriction map S(𝔥*) W(𝔤, 𝔥) → S(𝔱*) W(𝔨, 𝔱), where S(𝔥*) and S(𝔱*) are the polynomial algebras on 𝔥 and 𝔱, respectively, is exactly S(𝔱*) W(𝔤, 𝔱). Based on the above result, we also get a complete classification of the pairs (𝔤, σ) such that 𝔤σ is noncohomologous to zero in 𝔤.

Vanishing of the First Continuous L2-Cohomology for II1 Factors

We prove that the continuous version of the Connes-Shlyakhtenko first L-cohomology for II1 factors, as proposed by A. Thom in [Th06], always vanishes. In [CS03], A. Connes and D. Shlyakhtenko developed an L2-cohomology theory for finite von Neumann algebras M , and more generally for weakly dense ∗-subalgebras A ⊂ M of such von Neumann algebras. Then in [Th06], A. Thom provided an alternative, Hochschild-type characterization of the first such L2-cohomology of M as the quotient of the space of derivations δ : M → Aff(M ⊗Mop) by the space of inner derivations, where Aff(M ⊗Mop) denotes the ∗-algebra of operators affiliated with M ⊗Mop. Thom also proposed in [Th06] a continuous version of the first L2-cohomology, by considering the (smaller) space of derivations δ that are continuous from M with the operator norm to Aff(M ⊗Mop) with the topology of convergence in measure. He noted that in many cases (e.g., when M has a Cartan subalgebra, or when M is not prime), this cohomology vanishes, i.e. any continuous derivation of M into Aff(M ⊗Mop) is inner. Following up on this work, V. Alekseev and D. Kyed have shown in [AK11] that the first continuous L2-cohomology also vanishes whenM has property (T), whenM is finitely generated with nontrivial fundamental group, or when M has property Gamma. Recently, V. Alekseev proved in [Al13] that this is also the case for the free group factors L(Fn). In this article, we prove that in fact the first continuous L2-cohomology vanishes for all finite von Neumann algebras. The starting point of our proof is a key calculation in the proof of [Al13, Proposition 3.1], which provides a concrete sequence of elements yn in the II1 factor M = L(F3) of the free group F3 with generators a, b, c, that tends to 0 in operator norm, but has the property that if a derivation δ : M → Aff(M ⊗Mop) satisfies δ(ua) = ua ⊗ 1 and δ(ub) = δ(uc) = 0, then δ(yn) does not tend to 0 in measure. More precisely, the yn’s in [Al13] are scalar multiples of words wn in a, b, c with the property that δ(wn) is a larger and larger sum of free independent Haar unitaries. In the case of an arbitrary II1 factor M , we fix a hyperfinite II1 factor R ⊂ M with trivial relative commutant, and then use [Po92] to “simulate” (in distribution) L(F3) inside M , with a any fixed unitary in M and bm, cm Haar unitaries in R such that a, bm, cm are asymptotically free. If now δ is a continuous derivation on M , then by subtracting an inner derivation, we may assume δ vanishes on R, thus on bm, cm. If δ(a) 6= 0, and if we formally define yn’s via the same formula as Alekseev’s, with a, bm, cm in lieu of a, b, c, then a careful estimation of norms of yn and δ(yn), which uses results in [HL99], shows that one still has ‖yn‖ → 0, while δ(yn) 6→ 0 in measure. Mathematics Department, UCLA, CA 90095-1555 (United States), popa@math.ucla.edu Supported in part by NSF Grant DMS-1101718 KU Leuven, Department of Mathematics, Leuven (Belgium), stefaan.vaes@wis.kuleuven.be Supported by Research Programme G.0639.11 of the Research Foundation – Flanders (FWO) and KU Leuven BOF research grant OT/13/079.