Symplectic Lie Algebras Research Articles

The representation theory of Brauer categories II: Curried algebra

A representation of \mathfrak{gl}(V)=V \otimes V^{\ast} is a linear map \mu \colon \mathfrak{gl}(V) \otimes M \rightarrow M satisfying a certain identity. By currying, giving a linear map \mu is equivalent to giving a linear map a \colon V \otimes M \rightarrow V \otimes M , and one can translate the condition for \mu to be a representation into a condition on a . This alternate formulation does not use the dual of V and makes sense for any object V in a tensor category \mathcal{C} . We call such objects representations of the curried general linear algebra on V . The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category is the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.

Double extensions of left-symmetric structures

The notion of double extension for symplectic Lie algebras was introduced by Medina and Revoy in 1985. In 2021, Valencia gave a condition for symplectic Lie algebras which are obtained through double extension to admit compatible left-symmetric structures. In this paper, we formulate the notion of double extension for left-symmetric structures and give a condition for left-symmetric structures to be obtained through double extension. Moreover, we prove that right nilpotent left-symmetric structures on some solvable Lie algebras are obtained through double extension. We also give a condition for left-symmetric structures which are obtained through double extension to be complete.

On the N-waves hierarchy with constant boundary conditions. Spectral properties

This paper is devoted to [Formula: see text]-wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators [Formula: see text], whose potentials [Formula: see text] tend to constants [Formula: see text] for [Formula: see text]. For special choices of [Formula: see text], we outline the spectral properties of [Formula: see text], the direct scattering transform and construct its fundamental analytic solutions. We generalize Wronskian relations for the case of CBC — this allows us to analyze the mapping between the scattering data and the [Formula: see text]-derivative of the potential [Formula: see text]. Next, using the Wronskian relations, we derive the dispersion laws for the [Formula: see text]-wave hierarchy and describe the NLEE related to the given Lax operator.

AN EXACT SYMPLECTIC STRUCTURE OF LOW DIMENSIONAL 2-STEP SOLVABLE LIE ALGEBRAS

In this paper, we study a Lie algebra equipped by an exact symplectic structure. This condition implies that the Lie algebra has even dimension. The research aims to identify and to contruct 2-step solvable exact symplectic Lie algebras of low dimension with explicit formulas for their one-forms and symplectic forms. For case of four-dimensional, we found that only one class among three classes is 2-step solvable exact symplectic Lie algebra. Furthermore, we also give more examples for case six and eight dimensional of Lie algebras with exact symplectic forms which is included 2-step solvable exact sympletic Lie algebras. Moreover, it is well known that a 2-step solvable Lie algebra equipped by an exact symplectic form is nothing but it is called a 2-step solvable Frobenius Lie algebra.

The Virtual Cactus Group and Littelmann Paths

We define a virtual cactus group and show that the cactus group action on Littelmann paths is compatible with the virtualization map defined by Pan-Scrimshaw. To show this we use the combinatorics of Dynkin diagram foldings. Our definition generalizes the group with the same name recently defined for the symplectic Lie algebra in recent joint work with Azenhas and Tarighat-Feller.

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Let M be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n). We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for G on M can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for G are diagonalizable on the space spanned by singular vectors of M and hence on M. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.

Classification of Orbit Closures in the Variety of 4-Dimensional Symplectic Lie Algebras

Classification of Orbit Closures in the Variety of 4-Dimensional Symplectic Lie Algebras

Howe duality in the toroidal setting

In this paper, we construct and study various dual pairs acting on the oscillator modules of the symplectic toroidal Lie algebras coordinated by irrational quantum tori. This extends the classical Howe dual pairs to the toroidal setup.

Eight-dimensional symplectic non-solvable Lie algebras

In this paper, we classify eight-dimensional non-solvable Lie algebras that support a symplectic structure. As well as a complete classification is given, up to symplectomorphism, of eight-dimensional symplectic non-solvable Lie algebras.

Some Soliton Hierarchies Associated with Lie Algebras mathfrak {sp}(4) and mathfrak {so}(5)

Based on the symplectic Lie algebra mathfrak {sp}(4), we obtain two integrable hierarchies of mathfrak {sp}(4), and by using the trace identity, we give their Hamiltonian structures. Then, we use 2times 2 Kronecker product, and construct integrable coupling systems of one soliton equation. Next, we consider two bases of Lie algebra mathfrak {so}(5), and we get the corresponding two integrable hierarchies. Finally, we discuss the relation between the integrable hierarchies of two different bases associated with Lie algebra mathfrak {so}(5).

Para-Kahler and Para-Hermitian Structures on Six-Dimensional Unsolvable Lie Algebras

In this paper, we investigate into the matter of the existence of para-Kahlerian and para-Hermitian structures on six-dimensional unsolvable Lie algebras that are semidirect products. According to the classification results, there are four Lie algebras that are semidirect products of the Lie algebras so(3), sl(2, R) and three soluble Lie algebras A3.1=R3, A3.3 and A3.5. We show that only g=A3.5⋉sl(2, R) has a symplectic structure, and it admits a para-Kahlerian structure of zero Ricci curvature. The paper presents calculated curvature characteristics and the method to find other para-Kahler structures based on deformations of some initial para-Kahler structure. Other Lie algebras admit para-Hermitian structures, i.e. integrable paracomplex structures consistent with the natural non-degenerate 2-form. It follows from the results of the paper that the sixdimensional symplectic Lie algebra must be solvable except for one case when g=A3.5⋉sl(2, R). It complements the well-known result of Chu Bon-Yao that a four-dimensional symplectic Lie algebra must be solvable.

A geometric characterization of the symplectic Lie algebra

A geometric characterization of the symplectic Lie algebra

Complex symplectic Lie algebras with large Abelian subalgebras

We present two constructions of complex symplectic structures on Lie algebras with large Abelian ideals. In particular, we completely classify complex symplectic structures on almost Abelian Lie algebras. By considering compact quotients of their corresponding connected, simply connected Lie groups we obtain many examples of complex symplectic manifolds which do not carry (hyper)kähler metrics. We also produce examples of compact complex symplectic manifolds endowed with a fibration whose fibers are Lagrangian tori.

Quadratic Lie algebras with 2-plectic structures

We study the existence of 2-plectic structures on Lie algebras which admit an ad-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.

On the Notion of Metaplectic Barbasch–Vogan Duality

Abstract In analogy with the Barbasch–Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this duality notion with the theory of primitive ideals and extend the notion of special unipotent representations to the real metaplectic groups. We also interpret the duality map in terms of double cells of Weyl group representations.

Flat symplectic Lie algebras

Let be a symplectic Lie group, i.e., a Lie group endowed with a left invariant symplectic form. If is the Lie algebra of G then we call a symplectic Lie algebra. The product on defined by extends to a left invariant connection on G which is torsion free and symplectic (. When has vanishing curvature, we call a flat symplectic Lie group and a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to ω. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection of a flat symplectic Lie group is always complete. We prove that the flat symplectic double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of flat symplectic double extension of flat symplectic Lie algebras starting from {0}. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension .

A Whittaker category for the symplectic Lie algebra and differential operators

A Whittaker category for the symplectic Lie algebra and differential operators

Symmetric Spaces Associated to Classical Groups with Even Characteristic

Let G=GL(V) for an N-dimensional vector space V over an algebraically closed field k, and Gθ the fixed point subgroup of G under an involution θ on G. In the case where Gθ=O(V), the generalized Springer correspondence for the unipotent variety of the symmetric space G/Gθ was described in [SY], assuming that ch k≠2. The definition of θ given there, and of the symmetric space arising from θ, make sense even if ch k=2. In this paper, we discuss the Springer correspondence for those symmetric spaces with even characteristic. We show, if N is even, that the Springer correspondence is reduced to that of symplectic Lie algebras in ch k=2, which was determined by Xue. While if N is odd, the number of Gθ-orbits in the unipotent variety is infinite, and a very similar phenomenon occurs as in the case of exotic symmetric space of higher level, namely of level r=3.

Manin matrices of type C: Multi-parametric deformation

We constructed a multi-parametric deformation of the Brauer algebra representation related with the symplectic Lie algebras. The notion of Manin matrix of type C was generalised to the case of the multi-parametric deformation by using this representation and corresponding quadratic algebras. We derived pairing operators for these quadratic algebras and minors for the considered Manin matrices. The rank of pairing operators and dimensions of components of quadratic algebras were calculated.

SYMPLECTIC PBW DEGENERATE FLAG VARIETIES; PBW TABLEAUX AND DEFINING EQUATIONS

We define a set of PBW-semistandard tableaux that is in a weight-preserving bijection with the set of monomials corresponding to integral points in the Feigin–Fourier–Littelmann–Vinberg polytope for highest weight modules of the symplectic Lie algebra. We then show that these tableaux parametrize bases of the multihomogeneous coordinate rings of the complete symplectic original and PBW degenerate flag varieties. From this construction, we provide explicit degenerate relations that generate the defining ideal of the PBW degenerate variety with respect to the Plücker embedding. These relations consist of type Α degenerate Plücker relations and a set of degenerate linear relations that we obtain from De Concini’s linear relations.