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Lisa Carbone

Publications and source records attributed to Lisa Carbone.

36 records · Page 2Linked to original sources
arXiv2020

A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group

Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra g = g(A) and (adjoint) Kac-Moody group G = G(A)=$\langle\exp(ad(t e_i)), \exp(ad(t f_i)) \,|\, t\in C\rangle$ where $e_i$ and $f_i$ are the simple root vectors. Let $(B^+, B^-, N)$ be the twin BN-pair naturally associated to G and let $(\mathcal B^+,\mathcal B^-)$ be the corresponding twin building with Weyl group W and natural G-action, which respects the usual W-valued distance and codistance functions. This work connects the twin building of G and the Kac-Moody algebra g in a new geometrical way. The Cartan-Chevalley involution, $ω$, of g has fixed point real subalgebra, k, the 'compact' (unitary) real form of g, and k contains the compact Cartan t = k $\cap$ h. We show that a real bilinear form $(\cdot,\cdot)$ is Lorentzian with signatures $(1, \infty)$ on k, and $(1, n -1)$ on t. We define $\{x\in {\rm k} \,|\, (x, x) \leq 0\}$ to be the lightcone of k, and similarly for t. Let K be the compact (unitary) real form of G, that is, the fixed point subgroup of the lifting of $ω$ to G. We construct a K-equivariant embedding of the twin building of G into the lightcone of the compact real form k of g. Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W-tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an $SU(2)$-orbit of chambers stabilized by $U(1)$ which is thus parametrized by a Riemann sphere $SU(2)/U(1)\cong S^2$. For n = 2 the twin building is a twin tree. In this case, we construct our embedding explicitly and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone.

math.GR↗
arXiv2020

Equivariant simplicial reconstruction

We introduce and analyze parallelizable algorithms to compress and accurately reconstruct finite simplicial complexes that have non-trivial automorphisms. The compressed data -- called a complex of groups -- amounts to a functor from (the poset of simplices in) the orbit space to the 2-category of groups, whose higher structure is prescribed by isomorphisms arising from conjugation. Using this functor, we show how to algorithmically recover the original complex up to equivariant simplicial isomorphism. Our algorithms are derived from generalizations (by Bridson-Haefliger, Carbone-Rips and Corson, among others) of the classical Bass-Serre theory for reconstructing group actions on trees.

math.GR↗
arXiv2020

${\boldsymbolπ}$-systems of symmetrizable Kac-Moody algebras

As part of his classification of regular semisimple subalgebras of semisimple Lie algebras, Dynkin introduced the notion of a $π$-system. This is a subset of the roots such that pairwise differences of its elements are not roots. These arise as simple systems of regular semisimple subalgebras. Morita and Naito generalized this notion to all symmetrizable Kac-Moody algebras. In this work, we systematically develop the theory of $π$-systems of symmetrizable Kac-Moody algebras and establish their fundamental properties. We study the orbits of the Weyl group on $π$-systems, and completely determine the number of orbits in many cases of interest in physics. In particular, we show that there is a unique $π$-system of type $HA_1^{(1)}$ (the Feingold-Frenkel algebra) in $E_{10}$ (the rank 10 hyperbolic algebra) up to Weyl group action and negation.

math.RA↗
arXiv2019

Generators and relations for (generalised) Cartan type superalgebras

In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, $A(n-1,0) = \mathfrak{sl}(1|n)$ can be constructed by adding a "gray" node to the Dynkin diagram of $A_{n-1} = \mathfrak{sl}(n)$, corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is $W(n)$, the derivation algebra of the Grassmann algebra on $n$ generators. Here we present a novel construction of $W(n)$, from the same Dynkin diagram as $A(n-1,0)$, but with additional generators and relations.

math.RT↗
arXiv2018

Generators and relations for Lie superalgebras of Cartan type

We give an analog of a Chevalley-Serre presentation for the Lie superalgebras W(n) and S(n) of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W(g) and S(g), where g denotes the Kac-Moody algebra A_r, D_r or E_r. Then W(A_{n-1}) and S(A_{n-1}) are the Lie superalgebras W(n) and S(n). The algebras W(g) and S(g) are constructed from the Dynkin diagram of the Borcherds-Kac-Moody superalgebras B(g) obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W(A_r) and S(A_r) in terms of Chevalley generators and defining relations. We prove that all relations follow from the defining ones at level -2 and higher. The analogous definitions of the algebras in the D- and E-series are given. In the latter case the full set of defining relations is conjectured.

math.RT↗
arXiv2016

Root subsystems of rank 2 hyperbolic root systems

Let $Δ$ be a rank 2 hyperbolic root system. Then $Δ$ has generalized Cartan matrix $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\right)$ indexed by $a,b\in\mathbb{Z}$ with $ab\geq 5$. If $a\neq b$, then $Δ$ is non-symmetric and is generated by one long simple root and one short simple root; whereas if $a= b$, $Δ$ is symmetric and is generated by two long simple roots. We prove that if $a\neq b$, then $Δ$ contains an infinite family of symmetric rank 2 hyperbolic root subsystems $H(k,k)$ for certain $k\geq 3$, generated by either two short or two long simple roots. We also prove that $Δ$ contains non-symmetric rank 2 hyperbolic root subsystems $H(a',b')$, for certain $a',b'\in\mathbb{Z}$ with $a'b'\geq 5$. One of our tools is a characterization of the types of root subsystems that are generated by a subset of roots. We classify these types of subsystems in rank 2 hyperbolic root systems.

math-ph↗
arXiv2016

Uniqueness of representation--theoretic hyperbolic Kac--Moody groups over $\Z$

For a simply laced and hyperbolic Kac--Moody group $G=G(R)$ over a commutative ring $R$ with 1, we consider a map from a finite presentation of $G(R)$ obtained by Allcock and Carbone to a representation--theoretic construction $G^λ(R)$ corresponding to an integrable representation $V^λ$ with dominant integral weight $λ$. When $R=\Z$, we prove that this map extends to a group homomorphism $ρ_{λ,\Z}: G(\Z) \to G^λ(\Z).$ We prove that the kernel $K^λ$ of the map $ρ_{\lam,\Z}: G(\Z)\to G^{\lam}(\Z)$ lies in $H(\C)$ and if the group homomorphism $φ:G(\Z)\to G(\C)$ is injective, then $K^λ\leq H(\Z)\cong(\Z/2\Z)^{rank(G)}$.

math.GR↗
arXiv2016

Kac--Moody groups and automorphic forms in low dimensional supergravity theories

Kac--Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravity theories dimensionally reduced to dimensions less than 3, and their integral forms $G(\mathbb{Z})$ conjecturally encode quantized symmetries. In this review paper, we briefly introduce the conjectural symmetries of Kac--Moody groups in supergravity as well as the known evidence for these conjectures. We describe constructions of Kac--Moody groups over $\R$ and $\Z$ using certain choices of fundamental modules that are considered to have physical relevance. Eisenstein series on certain finite dimensional algebraic groups are known to encode quantum corrections in the low energy limit of superstring theories. We describe briefly how the construction of Eisenstein series extends to Kac--Moody groups. The constant terms of Eisenstein series on $E_9$, $E_{10}$ and $E_{11}$ are predicted to encode perturbative string theory corrections.

hep-th↗
arXiv2015

Presentation of hyperbolic Kac-Moody groups over rings

Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis-Tits presentation of the finite groups of Lie type. When the ground ring is finitely generated, we derive the finite presentability of the Steinberg group, and similarly for the Kac-Moody group when the ground ring is a Dedekind domain of arithmetic type. These finite-presentation results need slightly stronger hypotheses when the rank is smallest possible, namely 4. The presentations simplify considerably when the ground ring is Z, a case of special interest because of the conjectured role of the Kac-Moody group E10(Z) in superstring theory.

math.GR↗
arXiv2015

Eisenstein series on rank 2 hyperbolic Kac--Moody groups

We define Eisenstein series on rank 2 hyperbolic Kac--Moody groups over R, induced from quasi--characters. We prove convergence of the constant term and hence the almost everywhere convergence of the Eisenstein series. We define and calculate the degenerate Fourier coefficients. We also consider Eisenstein series induced from cusp forms and show that these are entire functions.

math.RT↗
arXiv2015

Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories

Kac-Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravities in dimensions less than 3, and their integer forms $G(\mathbb{Z})$ are conjecturally U-duality groups. Mathematical descriptions of $G(\mathbb{Z})$, due to Tits, are functorial and not amenable to computation or applications. We construct Kac-Moody groups over $\mathbb{R}$ and $\mathbb{Z}$ using an analog of Chevalley's constructions in finite dimensions and Garland's constructions in the affine case. We extend a construction of Eisenstein series on finite dimensional semisimple algebraic groups using representation theory, which appeared in the context of superstring theory, to general Kac-Moody groups. This coincides with a generalization of Garland's Eisenstein series on affine Kac-Moody groups to general Kac-Moody groups and includes Eisenstein series on $E_{10}$ and $E_{11}$. For finite dimensional groups, Eisenstein series encode the quantum corrections in string theory and supergravity theories. Their Kac-Moody analogs will likely also play an important part in string theory, though their roles are not yet understood.

hep-th↗
arXiv2015

Integral group actions on symmetric spaces and discrete duality symmetries of supergravity theories

For $G(\mathbb{R})$ a split, simply connected, semisimple Lie group of rank $n$ and $K$ the maximal compact subgroup of $G$, we give a method for computing Iwasawa coordinates of $G/K$ using the Chevalley generators and the Steinberg presentation. When $G/K$ is a scalar coset for a supergravity theory in dimensions $\geq 3$, we determine the action of the integral form $G(\mathbb{Z})$ on $G/K$. We give explicit results for the action of the discrete $U$--duality groups $SL_2(\mathbb{Z})$ and $E_7(\mathbb{Z})$ on the scalar cosets $SL_2(\mathbb{R})/SO_2(\mathbb{R})$ and $E_{7(+7)}(\mathbb{R})/[SU(8,\mathbb{R})/\{\pm Id\}]$ for type IIB supergravity in ten dimensions and 11--dimensional supergravity in $D=4$ dimensions, respectively. For the former, we use this to determine the discrete U--duality transformations on the scalar sector in the Borel gauge and we describe the discrete symmetries of the dyonic charge lattice. We determine the spectrum--generating symmetry group for fundamental BPS solitons of type IIB supergravity in $D=10$ dimensions at the classical level and we propose an analog of this symmetry at the quantum level. We indicate how our methods can be used to study the orbits of discrete U--duality groups in general.

hep-th↗
arXiv2015

Weyl group orbits on Kac--Moody root systems

Let $\mathcal{D}$ be a Dynkin diagram and let $Π=\{α_1,\dots ,α_{\ell}\}$ be the simple roots of the corresponding Kac--Moody root system. Let $\mathfrak{h}$ denote the Cartan subalgebra, let $W$ denote the Weyl group and let $Δ$ denote the set of all roots. The action of $W$ on $\mathfrak{h}$, 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 $i\neq j$, the simple roots $α_i$ and $α_j$ are in the same $W$--orbit if and only if vertices $i$ and $j$ in the Dynkin diagram corresponding to $α_i$ and $α_j$ are connected by a path consisting only of single edges. We introduce the notion of `the Cayley graph $\mathcal{P}$ 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 $\geq 4$ we prove that any 2 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 ${\mathcal D}$) 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 $W$.

math.GR↗
arXiv2012

Fundamental domains for congruence subgroups of SL2 in positive characteristic

In this work, we construct fundamental domains for congruence subgroups of $SL_2(F_q[t])$ and $PGL_2(F_q[t])$. Our method uses Gekeler's description of the fundamental domains on the Bruhat- Tits tree $X = X_{q+1}$ in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma.

math.GR↗
arXiv2011

Groups acting simply transitively on hyperbolic buildings

We construct and classify all groups, given by triangular presentations associated to the smallest thick generalized quadrangle, that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification shows 23 non-isomorphic torsion free groups (obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the Euclidean case, we find both torsion and torsion free groups acting on the same building.

math.GR↗
arXiv2010

Abstract simplicity of complete Kac-Moody groups over finite fields

Let $G$ be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix $A$, as constructed by Tits. It is known that $G$ admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group $\hat{G}$ which is defined to be the closure of $G$ in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of $\hat{G}$ was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices $A$ of rank at least four. Our proof uses Tits' simplicity theorem for groups with a BN-pair and methods from the theory of pro-$p$ groups.

math.GR↗
arXiv2010

Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits

We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Saçlioglu, we present the 238 hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.

math.RT↗