Symmetric Lie Algebra Research Articles

Geometry of Poisson brackets and methods of Liouville integration of systems on symmetric spaces

A survey is given of various methods of integrating Hamiltonian systems on symmetric spaces and Lie algebras, and an analysis is also presented of applications of the technique developed to some mechanical problems. Some questions of nonintegrability of Hamiltonian systems on symplectic manifolds are considered, and, in particular, the Morse theory of completely integrable Hamiltonian systems is expounded.

Kac-Moody symmetry of generalized non-linear Schr�dinger equations

The classical non-linear Schrodinger equation associated with a symmetric Lie algebra ℊ=k⊕m is known to possess a class of conserved quantities which from a realization of the algebrak⊗ℂ [λ]. The construction is now extended to provide a realization of the Kac-Moody algebrak⊗ℂ[λ, λ−1] (with central extension). One can then define auxiliary quantities to obtain the full algebra ℊ⊗ℂ[λ, λ−1]. This leads to the formal linearization of the system.

Analytic continuation of local representations of symmetric spaces

We prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space ( G, K, σ). The problem of passing from the Euclidean group to the Poincaré group appears first to have been addressed and solved this way by Klein and Landau. Let G be a Lie group, K a closed subgroup, and σ an involutive automorphism with K as fixed-point subgroup. If g = k + m is the corresponding symmetric Lie algebra, we form g ∗ = k + im , and let G ∗ denote the simply connected Lie group with g ∗ as Lie algebra. We consider virtual representations π of G on a fixed complex Hilbert space H , adopting the definitions due to J. Fröhlich, K. Osterwalder, and E. Seiler; in particular, π(g −1) ⊂ π(σ(g)) ∗ (possibly unbounded operators) for g in a neighborhood of e in G. We prove that every such π continues analytically to a strongly continuous unitary representation of G ∗ on H . Our theorem extends results due to Klein-Landau, Fröhlich et al., and others, earlier, for special cases. Previous results were known only for special ( G, K, σ), and then only for certain π.

Symmetric Lie algebras

Lie Rootsystems R are introduced, with axioms which reflect properties of the rootset of a Lie algebra L as structured by representations of compatible simple restricted rank 1 subquotients of L. The rank 1 Lie rootsystems and the rank 2 Lie rootsystems defined over Z p are classified up to isomorphism. Base, closure and core are discussed. The rootsystems of collapse on passage from R to Core R are shown to be of type S m . Given any Lie rootsystem R, its independent root pairs are shown to fall into eleven classes. Where the eleventh (anomoly) pair T 2 never occurs, it is shown that R is contained in R 0 + S (not always equal), where R 0 is a Witt rootsystem and S is a classical rootsystem. This result is of major importance to two papers (D. J. Winter, Generalized classical-Albert-Zassenhaus Lie algebras, to appear; Rootsystems of simple Lie algebras, to appear), since it implies that the rootsystems of the simple nonclassical Lie algebras considered there are Witt rootsystems. Toral Lie algebras and symmetric Lie algebras are introduced and studied as generalizations of classical-Albert-Zassenhaus Lie algebras. It is shown that their rootsystems are Lie rootsystems. The cores of toral Lie algebras are shown to be classical-Albert-Zassenhaus Lie algebras. These results form the basis for the abovementioned papers on rootsystems of simple Lie algebras and the classification of the rootsystems of two larger classes of Lie algebras, the generalized classical-Albert-Zassenhaus Lie algebras and the classical-Albert-Zassenhaus-Kaplansky Lie algebras. Symmetric Lie algebras are introduced as generalizations of classical-Albert-Zassenhaus Lie algebras. It is shown that their rootsets R are Lie rootsystems. Consequently, symmetric Lie algebras can be studied locally using the classification of rank 2 Lie rootsystems. This is done in detail for toral Lie algebras.

Parallel submanifolds of complex space forms II

This is a continuation of Part I, which appeared in this journal.In the previous paper I we have defined the following notions: orthogonal Jordan triple system (OJTS), orthogonal symmetric graded Lie algebra (OSGLA), orthogonal Jordan algebra (OJA), Hermitian symmetric graded Lie algebra (HSGLA). And we have shown that equivalent classes of OJTS naturally correspond to equivalent classes of OSGLA and through this correspondence we have naturally constructed HSGLA’s from the OJTS’s associated with OJA’s with unity.

A classification of certain symmetric Lie algebras

A classification of certain symmetric Lie algebras

Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism

This paper is a natural sequel to [6(a)]. H ere we complete the solution of [l, Problem 35, p. 3361, in the intended algebraic spirit. (See [6(c)] for another kind of solution of the same problem.) Specifically, let g = k @ p be a Cartan decomposition of a real semisimple Lie algebra, a a Cartan subspace of p, g = k @ a @ n a corresponding Iwasawa decomposition, G the universal enveloping algebra of g, Gk the centralizer of k in G, A the universal enveloping algebra of a, and p: Gk + A the HarishChandra homomorphism, defined to be the projection to A with respect to the decomposition G = A @ (kG + Gn). The problem is to prove “purely algebraically” Harish-Chandra’s theorem [2, Sect. 41 that the image of p equals the algebra Aw of translated (by half the sum of positive restricted roots) restricted Weyl group invariants in A. The proof should be valid for semisimple symmetric Lie algebras with splitting Cartan subspaces, over arbitrary fields of characteristic zero (see [l, Sect. I.131 and [6(a)]). In [6(a)], such a proof was given for the fact that p(Gk) C A, . (The subalgebra n will be denoted u in the body of this paper, since the symbol n has another natural use; see Section 2.) Now the assertion that ,p(Gk) is all of A, is equivalent to the assertion of Chevalley’s polynomial restriction theorem [3(a), p. 430, Theorem 6.10]-that the restriction map from p to a takes the algebra of k-invariant polynomial functions on p onto the algebra of polynomial functions on a invariant under the restricted Weyl group W. In the paper, we give two suitably general proofs of Chevalley’s theorem; see Theorem 5.1. Actually, one of these proofs is nothing more than a modification of Harish-Chandra’s classical proof presented in [3(a), pp. 4334341; see the Appendix. In the body of this paper, we present a proof similar in spirit to the Kostant-Steinberg-Varadarajan proof of Chevalley’s

Linear Factorization of Conical Polynomials Over Certain Nonassociative Algebras

Conical polynomials are defined as certain polynomials in quadratic elements of the universal enveloping algebra of a semisimple symmetric Lie algebra over a field of characteristic zero. These polynomials were used in an earlier paper to describe the conical vectors in certain induced modules. Here it is shown that when the base field is extended to a certain type of nonassociative algebra, the conical polynomials can be factored “linearly". One such nonassociative algebra is discussed in detail—an (alternative) composition algebra intimately related to the structure of the Lie algebra and studied earlier by B. Kostant in the context of real semisimple Lie algebras. The linear factorization leads in a later paper to an extension of the earlier work on conical vectors in induced modules.

Cartan subspaces of symmetric Lie algebras

A symmetric Lie algebra is defined, following J. Dixmier, to be a Lie algebra g \mathfrak {g} with a decomposition g = k ⊕ p \mathfrak {g} = \mathfrak {k} \oplus \mathfrak {p} such that k \mathfrak {k} is a subalgebra of g , [ k , p ] ⊂ p \mathfrak {g},[\mathfrak {k},\mathfrak {p}] \subset \mathfrak {p} and [ p , p ] ⊂ k [\mathfrak {p},\mathfrak {p}] \subset \mathfrak {k} . A definition of Cartan subspace of a symmetric Lie algebra is given, and a theory is presented which parallels the standard theory of Cartan subalgebras of Lie algebras, and which generalizes the classical results for real and complex semisimple symmetric Lie algebras.

Linear factorization of conical polynomials over certain nonassociative algebras

Conical polynomials are defined as certain polynomials in quadratic elements of the universal enveloping algebra of a semisimple symmetric Lie algebra over a field of characteristic zero. These polynomials were used in an earlier paper to describe the conical vectors in certain induced modules. Here it is shown that when the base field is extended to a certain type of nonassociative algebra, the conical polynomials can be factored “linearly". One such nonassociative algebra is discussed in detail—an (alternative) composition algebra intimately related to the structure of the Lie algebra and studied earlier by B. Kostant in the context of real semisimple Lie algebras. The linear factorization leads in a later paper to an extension of the earlier work on conical vectors in induced modules.

A generalization of H. Weyl’s “unitary trick”

H. Weyl’s “unitary trick” is generalized to the context of semisimple symmetric Lie algebras with Cartan subspaces, over fields of characteristic zero. As an illustration of its usefulness, the result is used to transfer to characteristic zero an important theorem in the representation theory of real semisimple Lie algebras.

On the uniqueness of conical vectors

Certain conical vectors in induced modules for semisimple symmetric Lie algebras have been determined in a previous paper. Here the uniqueness aspects of those results are proved in a simpler way, and a refinement of one of the earlier results is given.

Conical Vectors in Induced Modules

Let $\mathfrak {g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n}$, and let $\mathfrak {m}$ be the centralizer of $\mathfrak {a}$ in $\mathfrak {k}$. A conical vector in a $\mathfrak {g}$-module is defined to be a nonzero $\mathfrak {m} \oplus \mathfrak {n}$-invariant vector. The $\mathfrak {g}$-modules which are algebraically induced from one-dimensional $(\mathfrak {m} \oplus \mathfrak {a} \oplus \mathfrak {n})$-modules on which the action of $\mathfrak {m}$ is trivial have “canonical generators” which are conical vectors. In this paper, all the conical vectors in these $\mathfrak {g}$-modules are found, in the special case $\dim \mathfrak {a} = 1$. The conical vectors have interesting expressions as polynomials in two variables which factor into linear or quadratic factors. Because it is too difficult to determine the conical vectors by direct computation, metamathematical “transfer principles” are proved, to transfer theorems about conical vectors from one Lie algebra to another; this reduces the problem to a special case which can be solved. The whole study is carried out for semisimple symmetric Lie algebras with splitting Cartan subspaces, over arbitrary fields of characteristic zero. An exposition of the Kostant-Mostow double transitivity theorem is included.

On the Harish-Chandra Homomorphism

Using the Iwasawa decomposition $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n}$ of a real semisimple Lie algebra $\mathfrak {g}$, Harish-Chandra has defined a now-classical homomorphism from the centralizer of $\mathfrak {k}$ in the universal enveloping algebra of $\mathfrak {g}$ into the enveloping algebra $\mathcal {A}$ of $\mathfrak {a}$. He proved, using analysis, that its image is the space of Weyl group invariants in $\mathcal {A}$. Here the weaker fact that the image is contained in this space of invariants is proved “purely algebraically". In fact, this proof is carried out in the general setting of semisimple symmetric Lie algebras over arbitrary fields of characteristic zero, so that Harish-Chandra’s result is generalized. Related results are also obtained.

On the Harish-Chandra homomorphism

Using the Iwasawa decomposition g = k ⊕ a ⊕ n \mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n} of a real semisimple Lie algebra g \mathfrak {g} , Harish-Chandra has defined a now-classical homomorphism from the centralizer of k \mathfrak {k} in the universal enveloping algebra of g \mathfrak {g} into the enveloping algebra A \mathcal {A} of a \mathfrak {a} . He proved, using analysis, that its image is the space of Weyl group invariants in A \mathcal {A} . Here the weaker fact that the image is contained in this space of invariants is proved “purely algebraically". In fact, this proof is carried out in the general setting of semisimple symmetric Lie algebras over arbitrary fields of characteristic zero, so that Harish-Chandra’s result is generalized. Related results are also obtained.

Conical vectors in induced modules

Let g \mathfrak {g} be a real semisimple Lie algebra with Iwasawa decomposition g = k ⊕ a ⊕ n \mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n} , and let m \mathfrak {m} be the centralizer of a \mathfrak {a} in k \mathfrak {k} . A conical vector in a g \mathfrak {g} -module is defined to be a nonzero m ⊕ n \mathfrak {m} \oplus \mathfrak {n} -invariant vector. The g \mathfrak {g} -modules which are algebraically induced from one-dimensional ( m ⊕ a ⊕ n ) (\mathfrak {m} \oplus \mathfrak {a} \oplus \mathfrak {n}) -modules on which the action of m \mathfrak {m} is trivial have “canonical generators” which are conical vectors. In this paper, all the conical vectors in these g \mathfrak {g} -modules are found, in the special case dim ⁡ a = 1 \dim \mathfrak {a} = 1 . The conical vectors have interesting expressions as polynomials in two variables which factor into linear or quadratic factors. Because it is too difficult to determine the conical vectors by direct computation, metamathematical “transfer principles” are proved, to transfer theorems about conical vectors from one Lie algebra to another; this reduces the problem to a special case which can be solved. The whole study is carried out for semisimple symmetric Lie algebras with splitting Cartan subspaces, over arbitrary fields of characteristic zero. An exposition of the Kostant-Mostow double transitivity theorem is included.

Symmetric Lie algebras of non-linear transformations of conformal type in quantum mechanics

The Koecher construction of simple symmetric Lie algebras is used to realize colineation and conformai Lie algebras of non-linear transformations of a pseudo-orthogonal vector space in the canonical Weyl algebras, which are used in the Schrodinger representation. The realization maps the linear sub-algebras onto symmetrized polynomials of second degree, whereas the non-linear parts are mapped onto polynomials of first and third degree. For the two examples the Meyberg Jordan algebras are explicitly given.