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Robert Yuncken

Publications and source records attributed to Robert Yuncken.

26 records · Page 2Linked to original sources
arXiv2014

Equivariant Fredholm modules for the full quantum flag manifold of $SU_q(3)$

We introduce $C^*$-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct $SL_q(3,\mathbb{C})$-equivariant Fredholm modules for the full quantum flag manifold $X_q = SU_q(3)/T$ of $SU_q(3)$, based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold $ X_q $ satisfies Poincaré duality in equivariant $ KK $-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to $SU_q(3)$.

math.KT↗
arXiv2013

Quantum subgroups of the compact quantum group SU_{-1}(3)

We study the (compact) quantum subgroups of the compact quantum group $SU_{-1}(3)$: we show that any non-classical such quantum subgroup is a twist of a compact subgroup of SU(3) or is isomorphic to a quantum subgroup of $U_{-1}(2)$.

math.QA↗
arXiv2011

Regular tessellations of the hyperbolic plane by fundamental domains of a Fuchsian group

For positive integers p and q with 1/p + 1/q < 1/2, a tessellation of type {p,q} is a tessellation of the hyperbolic plane by regular p-gons with q p-gons meeting at each vertex. In this paper, a necessary and sufficient condition on the integers p and q is established to determine when a tessellation of type {p,q} can be realized as a tessellation of the hyperbolic plane by fundamental domains of some Fuchsian group. Specifically, a tessellation of type {p,q} is a tessellation by fundamental domains if and only if q has a prime divisor less than or equal to p.

math.GR↗
arXiv2011

Foliation C*-algebras on multiply fibred manifolds

Motivated by index theory for semisimple groups, we study the relationship between the foliation C^*-algebras on manifolds admitting multiple fibrations. Let F_1,...,F_r be a collection of smooth foliations of a manifold X. We impose a condition of local homegeneity on these foliations which ensures that they generate a foliation F under Lie bracket of tangential vector fields. We then show that the product of longitudinal smoothing operators along each F_j belongs to the C*-closure of the smoothing operators along F. An application to noncommutative harmonic analysis on compact Lie groups is presented.

math.FA↗
arXiv2010

On Coarse Spectral Geometry in Even Dimension

Let $σ$ be the involution of the Roe algebra $\Roe{\RR}$ which is induced from the reflection $\RR\to\RR; x\mapsto -x$. A graded Fredholm module over a separable $C^*$-algebra $A$ gives rise to a homomorphism $\tildeρ:A\to\Roe{\RR}^σ$ to the fixed-point subalgebra. We use this observation to give an even-dimensional analogue of a result of Roe. Namely, we show that the $K$-theory of this symmetric Roe algebra is $K_0(\Roe{\RR}^σ)\cong\ZZ$, $K_1(\Roe{\RR})=0$, and that the induced map $\tildeρ_*:K_0(A) \to \ZZ$ on $K$-theory gives the index pairing of $K$-homology with $K$-theory.

math.KT↗
arXiv2009

The Bernstein-Gelfand-Gelfand complex and Kasparov theory for SL(3,C)

Let $G=\mathrm{SL}(3,\mathbb{C})$. We construct an element of $G$-equivariant $K$-homology from the Bernstein-Gelfand-Gelfand complex for $G$. This furnishes an explicit splitting of the restriction map from the Kasparov representation ring $R(G)$ to the representation ring $R(K)$ of its maximal compact subgroup, and the splitting factors through the equivariant $K$-homology of the flag variety of $G$. In particular, we obtain a new model for the gamma element of $G$. The proof makes extensive use of earlier results of the author concerning harmonic analysis of longitudinal psuedodifferential operators on the flag variety.

math.OA↗
arXiv2008

Products of longitudinal pseudodifferential operators on flag varieties

Associated to each set $S$ of simple roots for $SL(n,\mathbb{C})$ is an equivariant fibration $X\to X_S$ of the space $X$ of complete flags of $\mathbb{C}^n$. To each such fibration we associate an algebra $J_S$ of operators on $L^2(X)$ which contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. These form a lattice of operator ideals whose common intersection is the compact operators. As a consequence, the product of fibrewise smoothing operators (for instance) along the fibres of two such fibrations, $X\to X_S$ and $X\to X_T$, is a compact operator if $S\cup T$ is the full set of simple roots. The construction uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as `essential orthogonality of subrepresentations'.

math.FA↗