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Bertram W. — The Geometry of Jordan and Lie Structures (Lecture Notes in Mathematics)
Bertram W. — The Geometry of Jordan and Lie Structures (Lecture Notes in Mathematics)



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Название: The Geometry of Jordan and Lie Structures (Lecture Notes in Mathematics)

Автор: Bertram W.

Аннотация:

The geometry of Jordan and Lie structures tries to answer the following question: what is the integrated, or geometric, version of real Jordan algebras, - triple systems and - pairs? Lie theory shows the way one has to go: Lie groups and symmetric spaces are the geometric version of Lie algebras and Lie triple systems. It turns out that both geometries are closely related via a functor between them, called the Jordan-Lie functor, which is constructed in this book.
The reader is not assumed to have any knowledge of Jordan theory; the text can serve as a self-contained introduction to (real finite-dimensional) Jordan theory.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1 edition

Год издания: 2000

Количество страниц: 269

Добавлена в каталог: 09.04.2010

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Предметный указатель
$A : V \to W, x \mapsto A_{x}$ - JTS T in the form $A_{x} (y, z) = T (y, x, z)$      VII.2.4
$Aherm (A, \varepsilon, \mathbb{F})$ - space of A-skew-Hermitian matrices      I.6
$Asym (A, \mathbb{F})$ - space of A-skew-symmetric matrices      I.6
$Co (T)_{00}$ - set of $g \in Co (T)$ with g (0) = 0, $Dg (0) = id_{V}$      VIII.2
$Co (T)_{0}$ - set of $g \in Co (T)$ with g (0) = 0      VIII.2
$d_{g}$ - denominator of a conformal transformation g      VIII.1
$E = (E^{p})_{p \in M}$ - Euler operator on M      VII.1
$Gr_{p, p + q} (\mathbb{F})$ - Grassmannian variety      VIII.4
$Herm (A, \varepsilon, \mathbb{F})$ - space of Hermitian matrices      I.6
$I_{p, q}$ - diagonal matrix with signature (p, q)      I.6
$J^{p}$ - vector field extension of $\mathcal{J}_{p}$      VI.1 VII.1
$j_{p}$ - affine extension of $\mathcal{J}_{p}$      VI. 1
$Lag (A, \varepsilon, \mathbb{F})$ - variety of Lagrangian subspaces      VIII.4
$l_{p} (v)$ - vector field extension of a tangent vector v      I.2 I.A
$M^{(\alpha)}$ - $\alpha$-modification of the space belonging to a JTS T      X.1
$M_{h \mathbb{C}}$ - twisted complexification (or "Hermitification") of M      III.4.3
$M_{ph \mathbb{C}}$ - twisted para-complexification of M      III.4.3
$M_{\mathbb{C}}$ - straight complexification of M      III.4.3
$n_{g}$ - numerator of a conformal transformation      VIII.1
$O (A, \mathbb{F})$ - orthogonal group w.r.t. A      I.6
$P (v) = \frac{1}{2} T (v, \cdot, v)$ - "quadratic representation" of a JTS T      VIII.2
$r_{x}^{\pm}, r_{x, y}$ - "multiplication by $r \in \mathbb{R}^{*}$ w.r.t. a point (x, y)"      VI.3
$Sym (A, \mathbb{F})$ - space of symmetric matrices      I.6
$s_{x}$ - Symmetry w.r.t. a point x in a symmetric space      I.3
$T^{(\alpha)}$ - $\alpha$-modification of a JTS T      III.4
$T_{p}^{2} M$ - second order tangent space      I.B
$t_{x}$ - translation by a vector x      I.A
$U (A, \varepsilon, \mathbb{F})$ - A-unitary group      I.6
$V^{c}$ - conformal completion of V      VIII.3
$W^{c}$ - conformal completion of W      VIII.3
$\gamma$ - Cayley transform      XI.2
$\mathbb{H}$ - skew field of quaternions      I.6
$\mathbf{p}_{v}$ - quadratic vector field $\mathbf{p}_{v} (x) = P (x)v$      VIII.2
$\mathbf{v}$ - constant vector field      I.A.1
$\mathbf{W} = (\mathbf{W}_{p})_{p \in V^{c}$- structure bundle      VIII.3
$\mathcal{J} = (\mathcal{J}_{p})_{p \in M}$ - almost complex structure or polarization on M      III.1
$\mathfrak{co} (T)$ - conformal Lie algebra of a JTS T      VII.1
$\mathfrak{co} (V)$ - conformal Lie algebra of a Jordan algebra V      VII.1
$\mathfrak{g} (\mathcal{J})$ - invariance algebra of $\mathcal{J}$      VI.A
$\mathfrak{g}^{b}$ - inner conformal Lie algebra      VII.1
$\mathfrak{p}$ - Lie algebra of P      VIII.3
$\mathfrak{q}$ - -1-eigenspace in an involutive Lie algebra; LTS      I.1
$\mathfrak{X} (M)$ - Lie algebra of vector fields on M      I.A.1
$\mu$ - multiplication map of a symmetric space      I.4
$\nabla$ - affine connection      I.B
$\Omega$ - open symmetric orbit in a vector space      II.1
$\Theta$ - involution of Co (T) and of $\mathfrak{co} (T)$      VII.2 VIII.2
$\tilde{r}_{x}, \tilde{r}_{x, y}$ - exponential of a twisted polarization      VI.2
$\tilde{t}_{x}$ - exponential of the quadratic vector field $\mathbf{p}_{x}$      VIII.2
Affine connection      I.B
Affine locally symmetric space      I.2
Affine map      I.B
Affinization      VIII.3
Almost complex structure      III.1
Associative algebra      II.3
Associative form      V.4
B (x, y) - Bergman operator      VIII.2
Bergman operator      VIII.2
Borel imbedding      VII.3 X.1
c-dual      I.1
Canonical connection      I.2
Cartan involution      X.6
Casual diffeomorphism      IX.2.3
Casual group      IX.2
Casual intervals      XI.3
Casual Makarevic space      XI.3
Casual structure      IX.2
Casual symmetric space      XI.3
Causal, compactly-, non-compactly-      XI.3
Cayley transform      I.6 XI.2
Cayley type spaces      XI.3
Circled space      VI.1
Classical group      I.6
Co (T) - conformal group of a JTS T      VIII.1
Co (V) - conformal group of a Jordan algebra V      VIII.2
Co' (T) - set of $g \in Co (T)$ with $d_{g} (0) \ne 0$      VIII.2
Complexification diagram      III.4
Complexification, straight      III.1
Complexification, twisted      III.2
Cone, regular-      IX.2.3
Cone, symmetric      V.5
Conformal compactification      VIII.3
Conformal completion      VIII.3
Conformal group      VIII.1
Conformal Lie algebra      VII.1
Conformal map (T-, G-)      VIII.1
Conformally equivalent      XI.5
Conjugation, complex      III.1
Connected pair      III.3
Curvature tensor      I.2 I.B
Denominator      VIII.1
density      X.6
Der (V, A) - derivation algebra of a bilinear map $A : V \otimes V \to V$      II.1
Displacement group      I.3
Elliptic realization of a cone      XI.3
Elliptic tripotent      X.5
Equivariant map      I.1
Euclidean Jordan algebra      V.5
Euler operator      I.A VII.1
Exp - exponential map of a connection or of a symmetric space      I.B
exp - exponential map of a Lie group      I.5
Exponential map      I.B X.4
Extension of a tangent map      I.A
Extension of a vector field      I.A
F - symmetric matrix      I.6
Faithful JTS      VII.2
First kind      XI.1
Fundamental formula      II.2 VIII.C
Fundamental theorem      IX.1 IX.2
G (M) - group of displacements      I.3
Geodesic symmetry      I.3
Global polarized space      VIII.3
Global space      X.2
Graded Lie algebra      III.3
Graph      VIII.4
Grassmannian      VIII.4
Harish-Chandra imbedding      VII.2 X.2
Helwig-space      II.4 XI.4
Hermitian (para-)complexification      III.4
Hermitian JTS      III.2
Hermitian symmetric space      V.5
Homomorphism of symmetric spaces      I.2 I.4
Hyperbolic functions on a JTS      X.4
Hyperbolic realization of a cone      XI.3
Hyperbolic space      I.6
Hyperbolic tripotent      X.5
Integrability      VI.A
Interval (causal)      XI.3
Invertible element      XI.1
J - complex structure or polarization on a LTS      III.1 III.3
j - Jordan inverse      II.2
J - standard symplectic matrix      I.6
jet      VIII.2
Jordan algebra      II.2
Jordan coordinates      VII.2
Jordan extension      III.4
Jordan inverse      II.2
Jordan pair      III.3
Jordan triple system (JTS)      III.2
Jordan — Lie functor      III.4
L (v) - operator of left multiplication by v      II.1
Lagrangian      I.6
Lie functor      I.1
Lie group      I.1
Lie triple algebra      II.1
Lie triple system (LTS)      I.1
Linear relation      IX.2
Liouville theorem      IX.1
Makarevic space      XI.1
Modification (of a JTS)      III.4
Multiplication map      I.4
Non-degenerate JTS      V.4
Numerator      VIII.1
Orbit, open symmetric-      II.1 X.1
Orbit, symmetric-      II.1
P - stabilizer of 0 in Co (T)      VIII.3
P' - intersection of Co (T) with the affine group of V      VIII.3
Para-conjugation      III.3
Para-Hermitian symmetric space      III.4
Para-real form      III.3
Parabolic realization of a cone      XI.3
Paracomplex structure      III.3
Peirce decomposition      X.5
Polarisation      III.3
Polarized LTS, JTS      III.3
Positive JTS      V.5
Power associative      II.2 X.A
Powers in a Jordan algebra      II.2
Powers in a symmetric space      I.5
Prehomogeneous symmetric space      II.1
Prehomogeneous vector space      II.1
Projective (T-)      VII.2 VIII.
Pseudo-Hermitian symmetric space      V.4
Pseudo-metric      V.1 X.6
Q - quadratic map of a symmetric space      I.5
Quadratic map      I.5
Quadratic prehomogenous symmetric space      II. 2
Quadratic representation      II.2
Quasi-inverse      VIII.2
R - curvature tensor, Lie triple product      I.1 I.2
Real Cayley transform      I.6 XI.2
Real form      III.1
Relative invariant      V.4
Representation of a symmetric space      I.5
Representation, polynomial      II.2
Ric - Ricci-tensor      V.1
Ricci form      V.1
Ruled space      VI.2
Semisimple JTS      V.4
Semisimple LTS      V.1
Semisimple symmetric space      V.1
Siegel space      I.6
Simple JTS      V.4
Simple LTS      V.1
Sphere      I.6 IV.1
Standard imbedding      I.1
Str (T) - structure group of a JTS T      VIII.1
Str (V) - structure group of a Jordan algebra V      II.3
Straight complexification      III.1
Structure algebra      II.2 VII.2
Structure bundle      VIII.3
Structure group      II.2 VIII.
Structure monoid      VIII.1
Structure tensor      III.2
Structure variety      III.4
Svar (T) - structure variety of a JTS T      III.4
Symmetric cone      V.5
Symmetric Lie algebra      I.1
Symmetric pair      I.1
Symmetric R-space      V.5 X.6
Symmetric space with twist      III.4
Symmetric space, (pseudo-)Hermitian      V.4
Symmetric space, (pseudo-)Riemannian      V.1
Symmetric space, (straight) complex      I.1 III.1
Symmetric space, algebraic      I.4
Symmetric space, para-Hermitian      V.4
Symmetric space, polarized      III.3
Symmetric space, topological      I.4
Symmetric space, twisted complex      III. 2
Symmetric submanifold      II.4
T - structure tensor, Jordan triple product      III.2 III.3 III.4
Trace form      V.2 V.4
Tripotent      X.5
Tube domain      XI.2
Twisted complex structure      III.1
Twisted complexification      III.4
Twisted para-complexification      III.4
Twisted polarization      III.3
Underlying Jordan pair      IV.2
V' - set of invertible elements in a Jordan algebra V      II.2
Vector field extension      I.A
W - subspace ${T ( \cdot, v, \cdot) | v \in V}$ of $Hom (S^{2} V, V)$      VII.2
W - vector space V with Str (T)-action by $g.w = \Theta (g) (w)$      II.2
[X, Y, Z] - Lie triple product      I.1
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