local isomorphism of compact connected lie groups - Project Euclid
Recommend Documents
Abstract. The 2-rank of a compact Lie group G is the maximal possible rank of ... any prime p) of compact Lie groups was initiated in 1953 by A. Borel and J.-P.
d v in § 3. Then § 4 contains our main results, including the fact. (Corollary 5) ...... E. H. Connell, D. Montgomery and C. T. Yang, Compact groups in En, Ann. of.
IIflILp(X, Rn) (IX Ifl p dx) 1/p" If S is a linear operator which maps R n valued L p ...... (ii) There exists C>0 such that fs n 1 J f dms~ ~=C fwk f dmwk for all inte-.
types are absolutely closed. In \S 2, the problem (b) will be treated mainly. Though the problem (1) is still open in general case, the following reduction is ...
May 27, 1986 - the subalgebras of a given finite dimensional real Lie algebra L and local subgroups ... neighborhood on which the Campbell-Hausdorff-Baker ...
We will now show that the line Lgm(am),gm(bm) has an increasingly large ... using first the triangle inequality and then Lemmas 2.2 and 2.1 we obtain: Ï(xm ...
(G, K, Ï) is as follows: A complex Hubert space Jf, with dense linear ..... [16] K. Osterwalder, and R. Schrader, Axioms for Euclidean Green's functions, I, II,. Comm ...
Two subgroups H and K of a finite group G are said to be. N-connected if the ... permutes with every subgroup of K. Products of totally permutable groups have since ...... 4, Walter de Gruyter & Co., Berlin, 1992. MR 93k:20033. [11] P. Hauck, A. Mart
Apr 15, 1981 - Thus, using th$e$ uniform continuity of $D^{r}\phi(\lambda(t)+\theta\mu(t))$ ..... $\lim_{n\rightarrow\infty}\sigma_{\Delta_{\hslash}}$ converges ...
Oct 29, 2010 - Therefore ff induces a biholomorphic mapping of base spaces ..... Then note that the triplet Ùl l glÙ1Ù, f n a, VÙ is a PV of type IFPS if and only.
Sep 20, 1982 - DAVID HANDELMAN2 AND WULF ROSSMANN. Introduction ...... in G, so t(yg)= 0 for all g in G, and thus Q(y)= 0; hence ker c kerQ. Since M is ...
the compact coregular linear groups of transformations of 1RÎ In particular we show that a contra variant metric matrix P(p) can be defined in the interior of
Oct 29, 2010 - sums of special linear Lie algebras slً2ق l slًm1ق l ءءء l slًmkق admit left invariant flat complex projective structures if the equality 4 m2. 1 ءءء m2.
B is trivial and v8 (by) = dim PM. .... for aG, PQMO- To prove local triviality of 13' we may by the .... 2) If G does not act transitively on M, i.e . if dim O/G'> 0, then.
Mar 18, 1971 - be the totality of $u\in\Gamma(E)$ such that $u(x)\in W_{x}$ for every .... Define a mapping $\Phi:\Gamma(W, F)/N\times V\rightarrow\Gamma(F)$ ...... {\prime}}v)(y)=(dExp_{\nu})_{w^{\prime}(y)}v(y),\\u(x)=(dExp_{y})_{w(y},u(y).
made to describe SUq(2) as a RieffeÎs strict deformation quantization of Poisson Lie .... Algebra of quantum functions ^ol(G) (G = SO(N), Sp(N)) is defined as the.
The Johns Hopkins University Press is collaborating with JSTOR to digitize, ...... [4] Ronnie Lee, "Differentiable classification of some topologically linear actions,".
Let 01 and 02 denote the equivalence classes of objects of C, and C2 defined ... reflexion in a line but are not C' equivalent to a linear action [4]. Also ..... then for any compact Ck G-manifold M' CGk(M', M) is dense in CGr(M1' M), 0 ? r ? k. 00 .
Jan 30, 2007 - that connected semisimple real Lie groups with finite center have property RD ... stability of property RD under some central extensions, a result ...
Jul 8, 1993 - We show that cl{I)£ cl*(I) and that the two closures coincide ... For every Lie algebra L, and every ideal I of L, cl(I) ^ ..... University of Warwick.
Jan 7, 2016 - differential Geometry, in memory of. John Nash. ISSN: 1736-4337 GLTA, an open access journal. Chief Factors of Lie Algebras. Towers DA*.
Mar 26, 2014 - OA] 26 Mar 2014. Rigidity of action of compact quantum groups on compact, connected manifolds. Biswarup Das1, Debashish Goswami2, ...
Feb 21, 2015 - arXiv:1502.06911v1 [math.RT] 21 Feb 2015 .... g : K â S is a map which is not a homomorphism such that g(1) = 1, the set. {(k, 1,g(k)); k â K} ...
Oct 22, 2009 - DG] 22 Oct 2009. RIEMANNIAN SUBMERSIONS FROM SIMPLE,. COMPACT LIE GROUPS. MARTIN KERINâ AND KRISHNAN SHANKARâ.
local isomorphism of compact connected lie groups - Project Euclid
Given a Lie group G it is natural to ask: "What are all the groups locally isomorphic to G?" The answer to this is well known. If G denotes the simply connected ...
PACIFIC JOURNAL OF MATHEMATICS Vol. 22, No. 2, 1967
LOCAL ISOMORPHISM OF COMPACT CONNECTED LIE GROUPS P. F.
BAUM
It is shown that two nonisomorphic compact connected Lie groups can be covering groups of each other. This is examined in detail and is related to the question of determining all the compact connected Lie groups belonging to a fixed Lie algebra. Given a Lie group G it is natural to ask: "What are all the groups locally isomorphic to G?" The answer to this is well known. If G denotes the simply connected covering group of G, then any group locally isomorphic to G is obtained by forming the quotient group G/Γ where Γ is a discrete subgroup of the center of G. (For this see e.g. [3]). We shall say that a Lie group Gλ covers a Lie group G2 if there exists a continuous homomorphism of Gλ onto G2 with discrete kernel. We can then pose the question: "Let G be a Lie group and let Gu G2, be Lie groups such that any group locally isomorphic to G is isomorphic to one and only one of the G{. Order the set Gl9 G2, by setting G{ ^ Gά if G{ covers G, . How can one describe the structure of this ordered set?" In the case when G is compact connected we give a precise answer to this question. One surprising result of the investigation is that two compact connected Lie groups can cover each other but not be isomorphic. The simplest example of this is provided by the groups U(5) and U(5)/Γ2, where U(5) denotes the 5 x 5 unitary group and Γ2 is the subgroup of Ϊ7(5) consisting of the two matrices ± 7 , I— identity matrix. Closely related to this example is the: PROPOSITION. In U(n) let Γk denote the subgroup of all diagonal matrices λ l where λ is a complex number such that λfc = 1. Then U(n)/Γkl and U(n)/Γk2 are isomorphic if and only if kλ = ±& 2 modw.
1* Notation* If G is a compact connected Lie then G shall denote the covering group of G of the form Hλ x H2 x x Ht x Tn where each Hi is a compact connected simply connected group in one of the Cartan series An, Bn, Cn, Dn or is a compact connected simply connected exceptional group, and Tn is the ^-dimensional torus group. Tn = {(zl9 z29 - , zn) I Zi e C and | z{ \ = 1}. If K is a subgroup of the center of Hx x H2 x x Ht and φ is a homomorphism of K into Tn, then (K, φ) denotes the subgroup of G consisting of all elements (g, G/Γ2 is a covering homomorphism then it is not in general true that h can be lifted to a homomorphism h:G—>G.
6* Corollary* G/(KU φx) covers (is isomorphic to) G/(K2, φ2) if and only if there exists an automorphism a of Hλx x Ht and a n covering homomorphism β of T onto itself (and an automorphism β of Tn) such that a maps Kx into (onto) K2 and the diagram
a\Ki
is commutative. Proof. By applying the previous proposition and the fact that any homomorphism of G onto itself is of the form a x β where a is an automorphism of Hλ x x Ht and β is a covering homomorphism n of T onto itself. 7* Definition* A quasi-ordered set is a set S together with a transitive relation ^ , such that for all xe S, x^x. Two quasi ordered sets Si and S2 are isomorphic if there exists a one-to-one and onto mapping of sets / : S i —>S2 such that for all x9yeSux^y if and only if f(x) ^ /(#). 8* Definition* Let G be a compact connected Lie group and let Gl9 G2, , Gι be compact connected Lie groups such that any compact connected Lie group locally isomorphic to G is isomorphic to one and only one of the G{. Define a transitive relation on the Gi by setting Gi^Gj if Gi covers Gj9 The set Gl9 G2, , Gt together with the relation Ξ> is called the local isomorphism system of G. 9* Definition* Let G be a compact connected Lie group and let (Kl9
