Real Simple Lie Algebras Research Articles

Cayley-Klein algebras as graded contractions of so(N+1)

We study $\Bbb Z_2^{\otimes N}$ graded contractions of the real compact simple Lie algebra $so(N+1)$, and we identify within them the Cayley-Klein algebras as a naturally distinguished subset.

A UNIFIED GAUGE FIELD LAGRANGIAN FOR ELECTROWEAK STRONG AND GRAVITATIONAL INTERACTION

Gravity can be descibed as a gauge field theory where connection and curvature are so(2,3)-valued. In the standard gauge field theory for strong and electroweak interaction corresponding quantities take their value in the su(3)⊕su(2)⊕u(1) algebra. Therefore, unification of gravity with the other fundamental interactions is obtained by using the non-compact simple real Lie algebra so*(14)⊃so(2,3)⊕su(3)⊕su(2)⊕u(1) as a unifying algebra. The so*(14) gauge field defined by a connection one-form on the SO*(14) principal fiber bundle unifies the fundamental interactions in particle physics, gravity included. The unified gauge field Lagrangian is defined by the Yang-Mills Weil form on the SO*(14) principal bundle.

THE UNIVERSAL LINK POLYNOMIAL

The solution of the non-Abelian SU (N) quantum Chern–Simons field theory defined in R3 is presented. It is shown how to compute the expectation values of the Wilson line operators, associated with oriented framed links, in closed form. The main properties of the universal link polynomial, defined by these expectation values, are derived in the case of a generic real simple Lie algebra. The resulting polynomials for some simple examples of links are reported.

On induced scalar products and unitarization

Infinitesimal unitarity of representations of the simple real Lie algebras su(2), su(1,1), and so(4) is discussed with respect to scalar products induced by sesquilinear forms on their universal enveloping algebras. Sesquilinear forms are explicitly calculated. The Verma modules, their irreducible quotients, their irreducible submodules, and their infinitesimal unitarizability are discussed.

Matched pairs of lie groups and Hopf algebra bicrossproducts

We construct a new class of examples of non-commutative and non-cocommutative Hopf algebras. There is essentially one for every Lie bialgebra, and several other examples including one for every simple real Lie algebra. The latter are obtained from a canonical solution of the Classical Yang-Baxter Equations. Physically, these Hopf algebras arise as the algebra of observables of quantum mechanics on homogeneous spacetimes.

Classification of nilpotent elements in simple real Lie algebras E6(6) and E6(−26) and description of their centralizers

Classification of nilpotent elements in simple real Lie algebras E6(6) and E6(−26) and description of their centralizers

Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers

Let g0 be a connected Lie group whose Lie algebra g0 is a simple exceptional non-compact real Lie algebra of inner type (up to isomorphism there are 10 such algebras). We classify the nilpotent orbits of G0 in g0 (for the adjoint action) by using Kostant's conjecture which was recently proved by J. Sekiguchi and the author (independently). For each nilpotent element Eϵg0, E ≠ 0, we determine the dimension of the orbit and the Levi factor of the centralizer of E in g0.

A METHOD OF CLASSIFICATION OF REAL IRREDUCIBLE REPRESENTATIONS OF REAL SEMI-SIMPLE LIE ALGEBRAS

In this article an n. a. s. c. of relativity of complex finite dimensional representations of areal semi-simple Lie algebra and an n. a. s. c. in which an autocorrelative representation admits anantiinvolution of the first kind have simply and precisely been given by using the Satake diagramsand the knowledge of the highest weights of the representations. By the way, in [9] an n. a. s. c.in which finite dimensional complex irreducible representations of real simple Lie algebras are realones has been computed by means of the mothod of the article. The results heve emended theoversight of [1] about the(2m-1)(?) basic representation of D2m2m and of [3] about E6(1) and E6(3).

Semi-simple real subalgebras of non-compact semi-simple real Lie algebras IV

The investigation of the problem of embedding a semi-simple real Lie algebra L ′ in a non-compact semi-simple real Lie algebra L is extended to the case in which at least one of the real Lie algebras has a semi-simple complex extension, which consists of the direct sum of two simple complex Lie algebras. Detailed procedures are given, which together with those given previously, allow the construction of all embeddings of L ′ in L when their complex extensions are A 1, B 1, C 1, D 1 or a direct sum of any two of these. The procedures are illustrated by considering examples corresponding to complex Lie algebra embeddings A 1⊂( A 2⊕ A 2), ( A 1⊕ A 1)⊂( A 2⊕ A 2), ( A 1⊕ A 1)⊂ A 3, ( A 1⊕ A 1)⊂⊂( A 3⊕ A 3) and ( A 1⊕ A 1)⊂( A 3⊕ A 2). Because of its physical significanc embeddings of SL(2, C) in simple and semi-simple real Lie algebras are studied in detail.

On the geometry and classification of absolute parallelisms. I

8. The irreducible case Let (M, ds2) be a simply connected globally symmetric pseudo-riemannian manifold, and φ an absolute parallelism on M consistent with ds2. We assume (M, ds2) to be irreducible. Our standing notation is the LTS of ^-parallel vector fields on M, the Lie algebra of all Killing vector fields on M, conjugation of g by the symmetry sx at x e M, g = I + m: eigenspace decomposition under σx. The irreducibility says that m is a simple noncommutative LTS, and thus (Lemma 6.2) says the same for p. 8.1. Lemma. Either [p, p] = p or [p, p] Π p = 0. Proof. Let t = [p, p] Π p. Then [!>,£],£)] C p implies [x,p] C t and so [tfψ] C i. Thus i is a LTS ideal in p. By simplicity, either t = 0 or t = p. If t = 0, then [p, p] Π p = 0. If t == p, then p c [p, jj]. As [i, p] C i, also [p, p] C p. Hence [p, p] = p. q.e.d. We do the group manifolds immediately. 8.2. Proposition. Let (M, ds2) be irreducible simply connected and globally symmetric, with consistent absolute parallelism φ such that the LTS of φparallel fields satisfies [p, p] ΓΊ p Φ 0. Then [p, p] — p, p is a simple real Lie algebra, and (M, φ, ds2) ^ (P, λ, da2) where (i) P is the simply conncted group for p, (ii) λ is the parallelism of left translation on P, and (iii) dσ2 is the bi-invariant metric induced by a nonzero multiple of the Killing form of p. The symmetry of (P, dσ2) at 1 eP is given by s(x) = x~ι. The group G of all isometries of (P, dσ2) has isotropy subgroup K at 1 given by

Isomorphisms between simple real Lie algebras

Two new necessary conditions for isomorphism between simple real Lie algebras are stated and proved, and are used, together with previously known necessary conditions and sufficient conditions, to investigate a new type of isomorphism between simple real Lie algebras. It is shown that three and only three examples of this type exist, of which one, the isomorphism between the Lie algebras of the groups SO(6,2) and ND 8, has not been noted previously. The existence of these examples is traced to certain exceptional properties of the groups of automorphisms of the corresponding simple complex Lie algebras A 1 and D 4.

CLASSIFICATION OF THE REAL INFINITE SIMPLE AND REAL INFINITE PRIMITIVE LIE ALGEBRAS

In this paper the real infinite simple and real infinite primitive Lie algebras are classified.

On simple groups associated with the real simple Lie algebras

On simple groups associated with the real simple Lie algebras