Symmetric Lie Algebra Research Articles

Braid group actions on coideal subalgebras of quantized enveloping algebras

We construct braid group actions on coideal subalgebras of quantized enveloping algebras which appear in the theory of quantum symmetric pairs. In particular, we construct an action of the semidirect product of Z n and the classical braid group in n strands on the coideal subalgebra corresponding to the symmetric pair ( sl 2 n ( C ) , sp 2 n ( C ) ) . This proves a conjecture by Molev and Ragoucy. We expect similar actions to exist for all symmetric simple complex Lie algebras. The given actions are inspired by Lusztigʼs braid group action on quantized enveloping algebras and are defined explicitly on generators. Braid group and algebra relations are verified with the help of the package Quagroup within the computer algebra program GAP.

Quiver varieties and cluster algebras

Motivated by a recent conjecture by Hernandez and Leclerc, we embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig’s dual canonical base. The conjectures that cluster monomials are positive and linearly independent (and probably many other conjectures) of Fomin and Zelevinsky follow as consequences when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of “prime” simple ones according to the cluster expansion.

An Algebraic Approach to Weakly Symmetric Finsler Spaces

AbstractIn this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann-Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions 2 and 3. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. Thismeans that reversible non-Berwaldian Finsler spaces with vanishing S-curvaturemay exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.

Generalized Lie Algebras and Cocycle Twists

We use the results of Etingof and Gelaki on the classification of (co)triangular Hopf algebras to extend Scheunert's “discoloration” technique to Lie algebras in the category of (co)modules. As an application, we prove a PBW-type theorem for such Lie algebras. We also discuss the relationship between Lie algebras in the category of (co)modules and symmetric braided Lie algebras introduced by Gurevich. Finally, we construct examples of symmetric braided Lie algebras that are essentially different from Lie coloralgebras.

On symmetric Finsler spaces

In this paper, we study symmetric Finsler spaces. We first study some geometric properties of globally symmetric Finsler spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Then we prove that a globally symmetric Finsler space is a Berwald space. As an application, we use the notion of Minkowski symmetric Lie algebras to give an algebraic description of symmetric Finsler spaces and obtain the formulas for flag curvature and Ricci scalar. Finally, some rigidity results of locally symmetric Finsler spaces related to the flag curvature are also given.

Isotropy subalgebras of elliptic orbits in semisimple Lie algebras, and the canonical representatives of pseudo-Hermitian symmetric elliptic orbits

We give a method of determining the centralizer of an elliptic element in a real semisimple Lie algebra g , in relation with the maximal compact subalgebra of g and the compact dual of g . Moreover, we determine a special central element (called the H -element) of the isotropy subalgebra of each simple irreducible pseudo-Hermitian symmetric Lie algebra.

Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Let \(\mathfrak g\) be a semisimple Lie algebra over a field \(\mathbb K\), \(\text{char}\left( \mathbb{K} \right)=0\), and \(\mathfrak g_1\) a subalgebra reductive in \(\mathfrak g\). Suppose that the restriction of the Killing form B of \(\mathfrak g\) to \(\mathfrak g_1 \times \mathfrak g_1\) is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra \(\mathfrak h_1\) of \(\mathfrak g_1\) there is a unique Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) containing \(\mathfrak h_1\); ( 2) \(\mathfrak g_1\) is self-normalizing in \(\mathfrak g\); ( 3) The B-orthogonal \(\mathfrak p\) of \(\mathfrak g_1\) in \(\mathfrak g\) is simple as a \(\mathfrak g_1\)-module for the adjoint representation. We give some answers to this natural question: For which pairs \((\mathfrak g,\mathfrak g_1)\) do ( 1), ( 2) or ( 3) hold? We also study how \(\mathfrak p\) in general decomposes as a \(\mathfrak g_1\)-module, and when \(\mathfrak g_1\) is a maximal subalgebra of \(\mathfrak g\). In particular suppose \((\mathfrak g,\sigma )\) is a pair with \(\mathfrak g\) as above and σ its automorphism of order m. Assume that \(\mathbb K\) contains a primitive m-th root of unity. Define \(\mathfrak g_1:=\mathfrak g^{\sigma}\), the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) \((\mathfrak g,\mathfrak g_1)\) satisfies ( 1); (b) For m prime, \((\mathfrak g,\mathfrak g_1)\) satisfies ( 2).

Verma Modules and Preprojective Algebras

AbstractWe give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra g in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra Λ.

Minkowski Symmetric Lie Algebras and Symmetric Berwald Spaces

In this paper, we introduce the notion of Minkowski symmetric Lie algebras to give an algebraic description of symmetric Berwald spaces. As an application, we study some geometric properties of these spaces. In particular, we generalize an elegant result of E. Cartan on the sectional curvature of Riemannian symmetric spaces to the flag curvature of symmetric Berwald spaces.

Symmetric identities and restricted Lie algebras

We use symmetric identities to study centres of reduced enveloping algebras of restricted Lie algebras. We formulate a criterion for a reduced centre to map injectively into a reduced enveloping algebra.

Quiver varieties and tensor products

In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety &?tilde; in a quiver variety, and show the following results: (1) The homology group of &?tilde; is a representation of a symmetric Kac-Moody Lie algebra ?, isomorphic to the tensor product V(λ1)⊗...⊗V(λ N ) of integrable highest weight modules. (2) The set of irreducible components of &?tilde; has a structure of a crystal, isomorphic to that of the q-analogue of V(λ1)⊗...⊗V(λ N ). (3) The equivariant K-homology group of &?tilde; is isomorphic to the tensor product of universal standard modules of the quantum loop algebra U q (L?), when ? is of type ADE. We also give a purely combinatorial description of the crystal of (2). This result is new even when N=1.

K-Loops, gyrogroups and symmetric spaces

In this paper we present a new class of K-Loops consisting of homogeneous symmetric spaces of the noncompact type. The most famous physical example of a representative of this class is found in special relativity: The set of all relativistically admissible velocities, R c 3:= {υ ∈ R3: ¦v¦< c} (c the speed of light), together with the relativistic velocity composition law forms a K-Loop, as was already shown by A. Ungar in 1988. The set of the corresponding boosts turns out to be the effective homogeneous symmetric space SO (1,3)0/SO(3), where SO(1,3)0 denotes the proper orthochronous Lorentz group and SO(3) the subgroup of proper rotations in three dimensions, which is also found to be the set of fixed points of an automorphism σ: SO(1,3)0 → SO(1,3)0, whose action on a matrix A in the defining representation is given by σ(A) = (AT)−1. Moreover, the triple (so(l,3),so(3),s) consisting of the Lie algebras of SO(1,3) and SO(3) and the induced automorphism s = (dσ)e is an orthogonal symmetric Lie algebra of the noncompact type. This property of the Lie algebra makes SO(l, 3)/SO(3) an (effective) homogeneous symmetric space of the noncompact type. The method of exact decomposition allows endowing each homogeneous symmetric space of the noncompact type with the structure of a K-Loop in a similar manner. Finally, we treat some alternative approaches which lead to K-Loops that are isomorphic to the one described above.

Curve motion inducing modified Korteweg-de Vries systems

A straightforward vector generalization of the modified Korteweg-deVries equation is shown to be intimately related to the geometry of curves in n -dimensional Euclidean spaces and spheres. This mKdV system, which is coupled in a particularly a simple way, describes the dynamics of the natural curvature vector of a unit speed curve subject to an elementary geometric evolution equation. The underlying structure of these equations is related to generalizations of the nonlinear Schrödinger and localized induction equations in the context of Hermitian symmetric Lie algebras.

Values of indefinite quadratic forms at integral points and flows on spaces of lattices

This mostly expository paper centers on recently proved conjectures in two areas: A) A conjecture of A. Oppenheim on the values of real indefinite quadratic forms at integral points. B) Conjectures of Dani, Raghunathan, and Margulis on closures of orbits in spaces of lattices such as SL„(R)/SL„(Z). At first sight, A) belongs to analytic number theory and B) belongs to ergodic and Lie theory, and they seem to be quite unrelated. They are discussed together here because of a very interesting connection between the two pointed out by M. S. Raghunathan, namely, a special case of B) yields a proof of A). The first main goal of this talk is to describe the Oppenheim conjecture and various refinements and to derive them from one statement about closures of orbits in the space of unimodular lattices in R3 (see Proposition 2 in §2.3). In §§3 and 4 we put this statement in context and describe more general conjectures and results on orbit closures and invariant probability measures on quotients of Lie groups by discrete subgroups. §5 gives some brief comments on the proofs and further developments. §§6, 7, and 8 are devoted to the so-called S-arithmetic setting, where we consider products of real and p-adic groups. §6 is concerned with a generalized Oppenheim conjecture; §7 with a generalization of the orbit closure theorem proved by M. Ratner [R8]; and §8 with applications to quadratic forms. Since the subject matter of that last section has not been so far discussed elsewhere, we take this opportunity to present proofs, obtained jointly with G. Prasad. Finally, §9 gives the proof of a lemma on symmetric simple Lie algebras, a special case of which is used in §8. I am glad to thank M. Ratner and G. Prasad for a number of remarks on, and corrections to, a preliminary version of this paper, thanks to which many typos and inaccuracies have been eliminated.

Ol'shanskiǐ wedges in symmetric lie algebras

Ol'shanskiǐ's semigroup plays a prominent role in the discussion of symmetric spaces. There are two important applications: the construction of the discrete series in representation theory and analysis on symmetric spaces. In this article the notion of an Ol'shanskiǐ wedge in a symmetric Lie algebra is defined. The tangent wedge of Ol'shanskiǐ's semigroup is an example of such a wedge. In Section 1 of this paper, a geometric characterization of Ol'shanskiǐ wedges is given and their relation to special Lie wedges is established. Section 2 deals with invariant Ol'shanskiǐ wedges and Ol'shanskiǐ semialgebras. We give a complete classification of symmetric Lie algebras supporting invariant Ol'shanskiǐ wedges, resp., Ol'shanskiǐ semialgebras.

Matrix nonlinear Schrödinger equations and moment maps into loop algebras

It is shown how Darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear Schrödinger equations are realized as isospectral Hamiltonian flows. The parametrization follows from a moment map embedding of the symplectic vector space, reduced by suitable group actions, into the dual ■g̃+*̃ of the algebra ■g̃+ of positive frequency loops in a Lie algebra 𝔤 . The resulting phase space is identified with a Poisson subspace of ■g̃+* consisting of elements that are rational in the loop parameter. Reduced coordinates associated to the various Hermitian symmetric Lie algebras (𝔤 ,𝔨) corresponding to the classical Lie algebras are obtained.

S-ALGEBRAS AND HARISH-CHANDRA MODULES OVER SYMMETRIC LIE ALGEBRAS

Structural relations in Mickelsson S-algebras over symmetric Lie algebras are investigated. A description of regular (generalized) Harish-Chandra modules over symmetric Lie algebras of inner type is given.

The structure of symmetric Lie algebras

In Winter [7], a certain class of Lie algebras, s~*mmrtric Lie ab and then we use the theory of algebraic groups and the theory of Lie rootsystems to pro\.p the following theorem, which expresses the structure of a symmetric Lie aigebra L in terms of a classica! Lie algebra L,S, a semisimple symmetric Lie algebra Lw whose root system is a Witt rootsystem (defined below) and solvable ideals. In the case of a ground field of characteristic 0, the theorem simply says that a symmetric Lie algebra of characteristic 0 is of the form L = L, @ Solv L with L, semisimple, which follows from Levi’s Theorem. So. we restrict ourselves in this paper to the much more difficult context of a ground field k of prime characteristic p > 3 (and sometimes 7).

R-matrices and symmetric spaces

Non-antisymmetric R-matrices which play a central role in the description of the Poisson structure of integrable models, are investigated. A wide class of such solutions is described. They are associated with the symmetric Lie algebras constructed from simple Lie algebras. This extends the classification of skew-symmetric matrices achieved by Belavin and Drinfel'd.

5. The bi-Hamiltonian property of Euler equations and symmetric Lie algebras

5. The bi-Hamiltonian property of Euler equations and symmetric Lie algebras