arXiv subjects

Kenro Furutani

Publications and source records attributed to Kenro Furutani.

At least 19 recordsLinked to original sources

arXiv2025

Geodesic orbit pseudo-Riemannian H-type nilmanifolds: case of minimal admissible Clifford modules

We investigate the geodesic orbit property of pseudo-Riemannian nilmanifolds, specifically those known in the literature as pseudo $H$-type Lie groups -- i.e., 2-step nilpotent Lie groups of Heisenberg type equipped with a left invariant pseudo-Riemannian metric. The study of homogeneous geodesics on Riemannian $H$-type Lie groups was completed by C.~Riehm in 1984. In this work, we extend these results to the pseudo-Riemannian $H$-type Lie groups and provide a complete characterization of the geodesic orbit property for the case where the underlying Lie algebras are constructed from the admissible Clifford modules of minimal dimension.

math.DG↗

arXiv2023

Invariant integral structures in pseudo $H$-type Lie algebras: construction and classification

Pseudo $H$-type Lie algebras are a special class of 2-step nilpotent metric Lie algebras, intimately related to Clifford algebras $\Cl_{r,s}$. In this work we propose the classification method for integral orthonormal structures of pseudo $H$-type Lie algebras. We apply this method for the full classification of these structures for $r\in\{1,\ldots,16\}$, $s\in \{0,1\}$ and irreducible Clifford modules. The latter cases form the basis for the further extensions by making use of the Atiyah-Bott periodicity. The existence of integral structures gives rise to the integral discrete uniform subgroups of the pseudo $H$-type Lie groups.

math.RA↗

arXiv2021

Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane

Our purpose is to show the existence of a Calabi-Yau structure on the punctured cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ of the Cayley projective plane $P^{2}\mathbb{O}$ and to construct a Bargmann type transformation from a space of holomorphic functions on $T^{*}_{0}(P^2\mathbb{O})$ to $L_{2}$-space on $P^{2}\mathbb{O}$. The space of holomorphic functions corresponds to the Fock space in the case of the original Bargmann transformation. A Kähler structure on $T^{*}_{0}(P^{2}\mathbb{O})$ was shown by identifying it with a quadrics in the complex space $\mathbb{C}^{27}\backslash\{0\}$ and the natural symplectic form of the cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ is expressed as a Kähler form. Our method to construct the transformation is the pairing of polarizations, one is the natural Lagrangian foliation given by the projection map ${\bf q}:T^{*}_{0}(P^2\mathbb{O})\longrightarrow P^{2}\mathbb{O}$ and the polarization given by the Kähler structure. The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators whose canonical relations are defined by the graph of the geodesic flow action at each time. It turn out that for the Cayley projective plane the results are not same with other cases of the original Bargmann transformation for Euclidean space, spheres and other projective spaces.

math.DG↗

arXiv2019

Spectral theory of a class of nilmanifolds attached to clifford modules

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-diffeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-diffeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.

math.SP↗

arXiv2019

Automorphism groups of pseudo $H$-type algebras

In the present paper, we determine the group of automorphisms of pseudo $H$-type Lie algebras, which are two-step nilpotent Lie algebras closely related to the Clifford algebras $\Cl(\mathbb R^{r,s})$.

math.RA↗

arXiv2017

Lie algebras attached to Clifford modules and simple graded Lie algebras

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo $H$-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type $B_n$ with $|2|$-grading do not contain non-Heisenberg pseudo $H$-type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types $A_n$, $C_n$ and $D_n$ provide such a possibility. Among exceptional algebras only $F_4$ and $E_6$ contain non-Heisenberg pseudo $H$-type Lie algebras as their negative part of $|2|$-grading. An analogous question addressed to real simple graded Lie algebras is more difficult, and we give results revealing the main differences with the complex situation.

math.DG↗

arXiv2017

Complete classification of pseudo $H$-type algebras: II

We classify a class of 2-step nilpotent Lie algebras related to the representations of the Clifford algebras in the following way. Let $J\colon \Cl(\mathbb R^{r,s})\toU$ be a representation of the Clifford algebra $\Cl(\mathbb R^{r,s})$ generated by the pseudo Euclidean vector space $\mathbb R^{r,s}$. Assume that the Clifford module $U$ is endowed with a bilinear symmetric non-degenerate real form $\la\cdot\,,\cdot\ra_U$ making the linear map $J_z$ skew symmetric for any $z\in\mathbb R^{r,s}$. The Lie algebras and the Clifford algebras are related by $\la J_zv,w\ra_U=\la z,[v,w]\ra_{\mathbb R^{r,s}}$, $z\in \mathbb R^{r,s}$, $v,w\in U$. We detect the isomorphic and non-isomorphic Lie algebras according to the dimension of $U$ and the range of the non-negative integers~$r,s$.

math.RT↗

arXiv2016

A co-dimensional 3 sub-Riemannian structure on Gromoll-Meyer exotic sphere

We construct a co-dimension $3$ completely non-holonomic sub-bundle on the Gromoll-Meyer exotic $7$ sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a $4n+3$ dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated simple proof for the standard sphere $\mathbb{S}^7$.

math.DG↗

arXiv2015

Complete classification of $H$-type algebras: I

Let $\mathscr N$ be a 2-step nilpotent Lie algebra endowed with non-degenerate scalar product $\langle.\,,.\rangle$ and let $\mathscr N=V\oplus_{\perp}Z$, where $Z$ is the centre of the Lie algebra and $V$ its orthogonal complement with respect to the scalar product. We study the classification of the Lie algebras for which the space $V$ arises as a representation space of a Clifford algebra $\Cl(\mathbb R^{r,s})$ and the representation map $J\colon \Cl(\mathbb R^{r,s})\to(V)$ is related to the Lie algebra structure by $\langle J_zv,w\rangle=\langle z,[v,w]\rangle$ for all $z\in \mathbb R^{r,s}$ and $v,w\in V$. The classification is based on the range of parameters $r$ and $s$ and is completed for the Clifford modules $V$, having minimal possible dimension, that are not necessary irreducible. We find the necessary condition for the existence of a Lie algebra isomorphism according to the range of integer parameters $0\leq r,s<\infty$. We present the constructive proof for the isomorphism map for isomorphic Lie algebras and defined the class of non-isomorphic Lie algebras.

math.RT↗

arXiv2015

Pseudo-metric 2-step nilpotent Lie algebras

The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that any 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with Lie brackets. This choice of the standard pseudo-metric form allows to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo $H$-type algebras have bases with rational structural constants, which implies that the corresponding pseudo $H$-type groups admit lattices.

math.RT↗

arXiv2015

Free nilpotent and $H$-type Lie algebras. Combinatorial and orthogonal designs

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz-Radon orthogonal matrices.

math.DG↗

arXiv2014

Classification of pseudo $H$-type algebras

We present a partial classification of the pseudo $H$-type algebras with minimal admissible Clifford modules. Furthermore, we prove that the subspace $\mathfrak{v}_{r,s}$ of $\mathfrak{n}_{r,s}$ is strongly bracket generating if and only if $r=0$ or $s=0$. Additionally, we discuss the classification of pseudo $H$-type algebras related to non-equivalent irreducible Clifford modules.

math.RT↗

arXiv2013

Existence of the lattice on general $H$-type groups

Let $\mathscr N$ be a two step nilpotent Lie algebra endowed with non-degenerate scalar product $\langle\cdot\,,\cdot\rangle$ and let $\mathscr N=V\oplus_{\perp}Z$, where $Z$ is the center of the Lie algebra and $V$ its orthogonal complement with respect to the scalar product. We prove that if $(V,\langle\cdot\,,\cdot\rangle_V)$ is the Clifford module for the Clifford algebra $\Cl(Z,\langle\cdot\,,\cdot\rangle_Z)$ such that the homomorphism $J\colon \Cl(Z,\langle\cdot\,,\cdot\rangle_Z)\to\End(V)$ is skew symmetric with respect to the scalar product $\langle\cdot\,,\cdot\rangle_V$, or in other words the Lie algebra $\mathscr N$ satisfies conditions of general $H$-type Lie algebras ~\cite{Ciatti, GKM}, then there is a basis with respect to which the structural constants of the Lie algebra $\mathscr N$ are all $\pm 1$ or 0.

math.RT↗

arXiv2004

Atiyah-Patodi-Singer boundary condition and a splitting formula of a spectral flow

We describe a relation between Atiyah-Patodi-Singer boundary condition and a global elliptic boundary condition which naturally appears in formulating a splitting formula for a spectral flow, when we decompose the manifold into two components. Then we give a variant of the splitting formula with the Hoermander index as a correction term.

math.SG↗

arXiv2003

Fredholm-Lagrangian-Grassmannian and the Maslov index

We explain the topology of the space, so called, Fredholm-Lagrangian-Grassmannain and the quantity ``Maslov index'' for paths in this space based on the standard theory of Functional Analysis. Our standing point is to define the Maslov index for arbitrary paths in terms of the fundamental spectral property of the Fredholm operators, which was first recognized by J. Phillips and used to define the ``Spectral flow''. We tried to make the arguments to be all elementary and we summarize basic facts for this article from Functional Analysis in the Appendix.

math.DG↗

arXiv2003

On the Quillen determinant

We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line bundles and trivial on some subspaces. Also we remark a relation of the {\it Quillen determinant line bundle} and the {\it Maslov line bundle}.

math.DG↗

arXiv2003

Quantization of the Geodesic flow on Quaternion Projective Spaces

We study a problem of the geometric quantization for the quaternion projective space. First we explain a Kaehler structure on the punctured cotangent bundle of the quaternion projective space, whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and show that the canonical line bundle of this complex structure is holomorphically trivial by explicitly constructing a nowhere vanishing holomorphic global section. Then we construct a Hilbert space consisting of a certain class of holomorphic functions on the punctured cotangent bundle by the method of pairing polarization and incidentally we construct an operator from this Hilbert space to the $L_2$ space of the quaternion projective space. Also we construct a similar operator between these two Hilbert spaces through the Hopf fiberation. We prove that these operators quantize the geodesic flow of the quaternion projective space to the one parameter group of the unitary Fourier integral operators generated by the square root of the Laplacian plus suitable constant. Finally we remark that the Hilbert space above has the reproducing kernel.

math.DG↗

arXiv2003

A Kaehler structure on the punctured cotangent bundle of the Cayley projective plane

We construct a Kaehler structure on the punctured cotangent bundle of the Cayley projective plane whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and we show that the geodesic flow action is holomorphic and is expressed in a quite explicit form. We also give an embedding of the punctured cotangent bundle of the Cayley projective plane into the space of 8 times 8 complex matrices.

math.DG↗