Symplectic Lie Algebras Research Articles

On total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case

Let W be the Weyl group of type BCn. We first provide restriction formulas of the total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case to the maximal parabolic subgroup of W which is of type BCn− 1. Then we show that these two restriction formulas are equivalent, and discuss how the results can be used to examine the existence of affine pavings of Springer fibers corresponding to the symplectic Lie algebra in characteristic 2.

Closed G2-Structures on Unimodular Lie Algebras with Non-trivial Center

We characterize the structure of a seven-dimensional Lie algebra with non-trivial center endowed with a closed G2-structure. Using this result, we classify all unimodular Lie algebras with non-trivial center admitting closed G2-structures, up to isomorphism, and we show that six of them arise as the contactization of a symplectic Lie algebra. Finally, we prove that every semi-algebraic soliton on the contactization of a symplectic Lie algebra must be expanding, and we determine all unimodular Lie algebras with center of dimension at least two that admit semi-algebraic solitons, up to isomorphism.

On a canonical basis of a pair of compatible Poisson brackets on a symplectic Lie algebra

On a canonical basis of a pair of compatible Poisson brackets on a symplectic Lie algebra

Symplectic Novikov Lie algebras

It is well known that a symplectic Lie algebra admits a left symmetric product. In this work, we study the case where this product is Novikov, we show that the left-symmetric product associated to the symplectic Lie algebra is Novikov if and only if it is associative. In this case, the symplectic Lie algebra is called symplectic Novikov Lie algebra (SNLA). We show that any SNLA is completely reducible two-step solvable. The classification of four dimensional and nilpotent six dimensional Lie algebras having an SNLA structure, also some methods for building large classes of examples, are presented. Finally, we give a geometric study of the affine connection associated with an SNLA.

Orthogonal decompositions of Lie algebras over finite commutative rings

Let [Formula: see text] be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over [Formula: see text]. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over [Formula: see text].

Extremal elements in Lie algebras, buildings and structurable algebras

An extremal element in a Lie algebra g over a field of characteristic not 2 is an element x∈g such that [x,[x,g]] is contained in the linear span of x. The linear span of an extremal element, called an extremal point, is an inner ideal of g, i.e. a subspace I satisfying [I,[I,g]]≤I. We show that in characteristic different from 2,3 the geometry with point set the set of extremal points and as lines the minimal inner ideals containing at least two extremal points is a Moufang spherical building, or in case there are no lines a Moufang set.This last result on the Moufang sets is obtained by connecting Lie algebras to structurable algebras, a class of non-associative algebras with involution generalizing Jordan algebras. It is shown that in characteristic different from 2,3 each finite-dimensional simple Lie algebra generated by extremal elements is either a symplectic Lie algebra or can be obtained by applying the Tits-Kantor-Koecher construction to a skew-dimension one structurable algebra. Various relations between the Lie algebra g and its extremal geometry on the one hand and the associated structurable algebra on the other hand are investigated.

On Segal–Sugawara vectors and Casimir elements for classical Lie algebras

We consider the centers of the affine vertex algebras at the critical level associated with simple Lie algebras. We derive new formulas for generators of the centers in the classical types. We also give a new formula for the Capelli-type determinant for the symplectic Lie algebras and calculate the Harish-Chandra images of the Casimir elements arising from the characteristic polynomial of the matrix of generators of each classical Lie algebra.

THE CATEGORY OF WEIGHT MODULES FOR SYMPLECTIC OSCILLATOR LIE ALGEBRAS

The rank n symplectic oscillator Lie algebra 𝔤n is the semidirect product of the symplectic Lie algebra 𝔰𝔭2n and the Heisenberg algebra Hn. In this paper, we first study weight modules with finite-dimensional weight spaces over 𝔤n. When the central charge \( \dot{z} \) ≠ 0, it is shown that there is an equivalence between the full subcategory 𝒪𝔤n \( \left[\dot{z}\right] \) of the BGG category 𝒪𝔤n for 𝔤n and the BGG category 𝒪𝔰𝔭2n for 𝔰𝔭2n. Then using the technique of localization and the structure of generalized highest weight modules, we give the classification of simple weight modules over 𝔤n with finite-dimensional weight spaces. As a byproduct we also determine all simple 𝔤n-modules (not necessarily weight modules) that have a simple Hn-submodule.

Complex symplectic structures on Lie algebras

We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called complex symplectic oxidation , to construct certain complex symplectic Lie algebras of dimension 4 n + 4 from those of dimension 4 n . We specialize this construction to the nilpotent case and apply complex symplectic oxidation to obtain all eight-dimensional nilpotent complex symplectic Lie algebras.

MONOMIAL BASES AND BRANCHING RULES

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra. As an application, we produce monomial bases for representations of the general linear and symplectic Lie algebras associated with natural chains of subalgebras. We also show that our basis in type A is related to both the Gelfand-Tsetlin basis and the Littelmann basis via triangular transition matrices which implies that the triangularity property extends to the matrix connecting the Gelfand-Tsetlin and canonical bases. A similar relationship holds between our basis in type C and a suitably modified version of the basis constructed earlier by the first author.

Structure of locally conformally symplectic Lie algebras and solvmanifolds

We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups.

On nilpotent generators of the symplectic Lie algebra

Let be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element X in there exists a nilpotent element Y in such that X and Y generate .

Coulomb Branch of a Multiloop Quiver Gauge Theory

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, r loops, one-dimensional framing, and dim V = 2. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank r. Hence it possesses a symplectic resolution with 2r fixed points with respect to a Hamiltonian torus action. We also identify its flavor deformation with a base change of the full Slodowy slice.

Generalized Onsager Algebras

Let mathfrak {g}(A) be the Kac-Moody algebra with respect to a symmetrizable generalized Cartan matrix A. We give an explicit presentation of the fix-point Lie subalgebra mathfrak {k}(A) of mathfrak {g}(A) with respect to the Chevalley involution. It is a presentation of mathfrak {k}(A) involving inhomogeneous versions of the Serre relations, or, from a different perspective, a presentation generalizing the Dolan-Grady presentation of the Onsager algebra. In the finite and untwisted affine case we explicitly compute the structure constants of mathfrak {k}(A) in terms of a Chevalley type basis of mathfrak {k}(A). For the symplectic Lie algebra and its untwisted affine extension we explicitly describe the one-dimensional representations of mathfrak {k}(A).

Orthogonal and Symplectic Yangians and Lie Algebra Representations

Orthogonal or symplectic Yangians are defined by the Yang–Baxter RLL relation involving the fundamental R-matrix with so(n) or sp(2m) symmetry. We investigate the conditions on the first- and second-order evaluations as restrictions imposed on the representation weights.

Symplectic Lie algebras with degenerate center

Every symplectic Lie algebra with degenerate (including non-abelian nilpotent symplectic Lie algebras) has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets. Finally, we give a scheme to classify the isomorphism classes of symplectic Lie algebras with degenerate center. We also give a classification scheme for nilpotent symplectic Lie algebras and compute a complete list of all 6-dimensional nilpotent symplectic Lie algebras, which removes inaccuracies in an older list.

Affine Brauer category and parabolic category $${\mathcal {O}}$$ O in types B, C, D

A strict monoidal category referred to as affine Brauer category $$\mathcal {AB}$$ is introduced over a commutative ring $$\kappa $$ containing multiplicative identity 1 and invertible element 2. We prove that morphism spaces in $$\mathcal {AB}$$ are free over $$\kappa $$ . The cyclotomic (or level k) Brauer category $$\mathcal {CB}^f(\omega )$$ is a quotient category of $$\mathcal {AB}$$ . We prove that any morphism space in $$\mathcal {CB}^f(\omega )$$ is free over $$\kappa $$ with maximal rank if and only if the $${\mathbf {u}}$$ -admissible condition holds in the sense of (1.32). Affine Nazarov–Wenzl algebras (Nazarov in J Algebra 182(3):664–693, 1996) and cyclotomic Nazarov–Wenzl algebras (Ariki et al. in Nagoya Math J 182:47–134, 2006) will be realized as certain endomorphism algebras in $$\mathcal {AB}$$ and $$\mathcal {CB}^f(\omega )$$ , respectively. We will establish higher Schur–Weyl duality between cyclotomic Nazarov–Wenzl algebras and parabolic BGG categories $${\mathcal {O}}$$ associated to symplectic and orthogonal Lie algebras over the complex field $$\mathbb C$$ . This enables us to use standard arguments in (Anderson et al. in Pac J Math 292(1):21–59, 2018; Rui and Song in Math Zeit 280(3–4):669–689, 2015; Rui and Song in J Algebra 444:246–271, 2015), to compute decomposition matrices of cyclotomic Nazarov–Wenzl algebras. The level two case was considered by Ehrig and Stroppel in (Adv. Math. 331:58–142, 2018).

Webs and $q$-Howe dualities in types $\mathbf {BCD}$

We define web categories describing intertwiners for the orthogonal and symplectic Lie algebras, and, in the quantized setup, for certain orthogonal and symplectic coideal subalgebras. They generalize the Brauer category, and allow us to prove quantum versions of some classical type $\mathbf{B}\mathbf{C}\mathbf{D}$ Howe dualities.

Combinatorics of Generalized Exponents

Abstract We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type $A_{n-1}$, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type $C_{n}$, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type $A_{2n-1}$, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig $t$-analogs associated to zero-weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula and discuss some implications to relating two type $C$ branching rules. Our methods are expected to extend to the orthogonal types.

Equivalences between three presentations of orthogonal and symplectic Yangians

We prove the equivalence of two presentations of the Yangian $$Y(\mathfrak {g})$$ of a simple Lie algebra $$\mathfrak {g}$$ , and we also show the equivalence with a third presentation when $$\mathfrak {g}$$ is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians.