arXiv subjects

Patrick Eberlein

Publications and source records attributed to Patrick Eberlein.

5 recordsLinked to original sources

arXiv2017

Isometries of Clifford Algebras I

Let $V$ be a finite dimensional vector space over a field $F$ of characteristic different from 2, and let $Q$ be a nondegenerate, symmetric, bilinear form on $V$. Let $C\ell(V,Q)$ be the Clifford algebra determined by $V$ and $Q$. The bilinear form $Q$ extends in a natural way to a nondegenerate, symmetric, bilinear form $\bar{Q}$ on $C\ell(V,Q)$. Let $G$ be the group of isometries of $C\ell(V,Q)$ relative to $\bar{Q}$, and let $LG$ be the Lie algebra of infinitesimal isometries of $C\ell(V,Q)$ relative to $\bar{Q}$. We derive some basic structural information about $LG$, and we compute $G$ in the case that $F = R, V = R^{n}$ and $Q$ is positive definite on $R^{n}$. In a sequel to this paper we determine $LG$ in the case that $F = R, V = R^{n}$ and $Q$ is nondegenerate on $R^{n}$.

math.DG↗

arXiv2017

Isometries of Clifford algebras II

Let F be a field of characteristic different from 2, and let $F^{n}$ denote the vector space of n-tuples of elements in F. Let ${e_{1}, ... , e_{n}}$ denote the canonical basis of $F^{n}$. Let r and s be nonnegative integers such that r + s = n, and let Q denote the nondegenerate bilinear form on $F^{n}$ such that $Q(e_{i}, e_{j}) = 0$ if i,j are distinct, $Q(e_{i},e_{i}) = 1$ if $1 \leq i \leq r$ and $Q(e_{r+j},e_{r+j}) = -1$ if $1 \leq j \leq s$. Let $C\ell(r,s)$ denote the Clifford algebra determined by Q and $F^{n}$. There is a canonical extension of Q to a nondegenerate, symmetric, bilinear form $\bar{Q}$ on $C\ell(r,s)$. An element g of $C\ell(r,s)$ will be called an isometry of $C\ell(r,s)$ if left and right translations by g preserve $\bar{Q}$. Let $G_{r,s}$ denote the group of all isometries of $C\ell(r,s)$. We construct a Lie algebra $LG_{r,s}$ over F that equals the Lie algebra of $G_{r,s}$ in the case that F = R or C. The Lie algebra $LG_{r,s}$ admits an involutive automorphism whose +1 and -1 eigenspaces determine a Cartan decomposition $LG_{r,s} = K_{r,s} \oplus P_{r,s}$. We compute the bracket relations for a natural system of generators of $LG_{r,s}$. Finally, we determine $LG_{r,s}$ in the case that F = R.

math.DG↗

arXiv2015

Central conjugate locus of 2-step nilpotent Lie groups

The goals of this article are twofold : 1) to compute the conjugate locus of a geodesic that lies in the center of a simply connected, 2-step nilpotent Lie group with a left invariant metric 2) compare the isometry types of two such nilpotent Lie groups whose conjugate loci for central geodesics are "the same" in a suitable sense. The first goal is achieved. The second is elusive, but we obtain a partial result.

math.DG↗

arXiv2010

Growth Estimates for Orbits of Self Adjoint Groups

Let G denote a closed, connected, self adjoint, noncompact subgroup of GL(n,R), and let d_{R} denote the canonical right invariant Riemannian metric on G. For v in R^{n} let G_{v} = {g in G : g(v) = v}. We obtain algebraically defined upper and lower bounds for the asymptotic growth rate of g --> log |g(v)| / d_{R}(g,G_{v}), and these bounds are sharp if G_{v} is compact. If the lower bound is positive, then the orbit G(v) is closed in R^{n}. The results apply to representations of noncompact semisimple Lie groups G on finite dimensional real vector spaces.

math.DG↗

arXiv2008

2-step nilpotent Lie groups arising from semisimple modules

Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie algebra.

math.DG↗