Symplectic Lie Algebras Research Articles

PBW Filtration and Bases for Symplectic Lie Algebras

We study the PBW filtration on the highest weight representations $V(\la)$ of $\msp_{2n}$. This filtration is induced by the standard degree filtration on $U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr V(\la)$ in terms of generators and relations. We also construct a basis of $gr V(\la)$. As an application we derive a graded combinatorial formula for the character of $V(\la)$ and obtain a new class of bases of the modules $V(\la)$.

On tensor spaces for Birman–Murakami–Wenzl algebras

Let l , n ∈ N . Let sp 2 l be the symplectic Lie algebra over the complex number field C . Let V be the natural representation of the quantized enveloping algebra U q ( sp 2 l ) and B n , q the specialized Birman–Murakami–Wenzl algebra with parameters − q 2 l + 1 , q . In this paper, we construct a certain element in the annihilator of V ⊗ n in B n , q , which comes from some one-dimensional two-sided ideal of Birman–Murakami–Wenzl algebra and is explicitly characterized (modulo the determination of some constants). We prove that the two-sided ideal generated by this element is indeed the whole annihilator of V ⊗ n in B n , q and conjecture that the same is true over arbitrary ground fields and for any specialization of the parameter q. The conjecture is verified in the case when q is specialized to 1 (i.e., the Brauer algebra case) and the case when n = l + 1 .

Representations of twisted q-Yangians

The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with $${\mathfrak{gl}_N}$$ . We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras $${\mathfrak{sp}_{2n}}$$ . The representations are parameterized by their highest weights satisfying certain dominance-type conditions. In the simplest case of $${\mathfrak{sp}_2}$$ , we give an explicit description of all the representations as tensor products of evaluation modules. We give new proofs of the (well-known) Poincaré–Birkhoff–Witt theorem for the quantum affine algebra and for the twisted q-Yangians in their RTT-presentations. We also reproduce Tarasov’s proof of the classification theorem for finite-dimensional irreducible representations of the quantum affine algebra by relying on its R-matrix presentation.

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple $(G,\omega,\nabla)$ such that $G$ is a finite-dimensional real Lie group and $\omega$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $\nabla$. In this paper starting from a special symplectic Lie group we show how to ``deform" the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure $\nabla$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.

Symplectic geometry of Maxwell theory and the photon concept

For an arbitrary multiply connected cavity a solution theory for the Maxwell equations is developed under ideal conductor boundary conditions. The trajectories of the electromagnetic fields are decomposed into their Helmholtz–Hodge components, including so–called cohomological fields which correspond to the harmonic differential forms. A canonical formalism is formulated in the Coulomb gauge, which comprises only the transversal and cohomological components. The canonical vector potentials and their conjugate fields are smeared in terms of smooth test functions. The Poisson bracket for the smeared fields is based on a symplectic form in the test function space and the corresponding infinite dimensional symplectic Lie algebra is introduced. The conservation quantities of Maxwell theory arise in this manner as co–momentum images of the Lie algebra. By means of a symplectic transformation the transversal dynamical generator is diagonalized and defines a special complexification of the test function space. A diagonalization of the cohomological dynamical generator is proved impossible.Since the Weyl quantization provides a symplectic equivariant map of the classical fields onto the quantum fields (affiliated with our C*–Weyl algebra), we have in quantum electrodynamics the same freedom of symplectic transformations and complexifications in the test function space (and not in the representing Hilbert space). Using the diagonalizing complexification we introduce the (smeared) transversal creation and annihilation operators and arrive at a unique particle structure for the transversal fields in a Fock representation. The symplectic Lie algebra elements, multiplied by iℏ, constitute the one–photon observables, where especially the diagonal dynamical Maxwell generator generalizes Einstein's photon energy to our general set up. The quantum co–momentum map acquires the form of a second quantization of the one-photon observables and leads to particle conserving multi–photon Hamiltonians for the complex linear Lie algebra elements, whereas the anti-linear Lie algebra elements generate particle non–conserving squeezing Hamiltonians. Since there is no particle structure for the quantized cohomological fields their collective nature is confirmed, illustrating the difference between quantization and particle discretization.

PREFACE TO THE SPECIAL ISSUE OF JNMP IN MEMORY OF F. A. BEREZIN

Felix Alexandrovich Berezin was born on April 25, 1931 and perished on July 14, 1980 (or several days before) under obscure circumstances while serving — during his summer holidays — as an auxiliary worker for a geological party near the Kolyma river. Wikipedia currently lists several researchers as discoverers of what we now call supersymmetry. Interestingly, Berezin is not even mentioned in this Wikipedia article. However, none of the authors mentioned so far as “pioneers of supersymmetry” qualify as such: Before Wess and Zumino have explained the meaning and importance of supersymmetry none of the earlier discoverers did fully appreciate the meaning and value of their own works, and some of them do not realize its full meaning even now judging by their later works. Contrariwise, Berezin definitely was the first to realize the real value of his discoveries, the fact that he stands on the threshold of the new branch of mathematics, currently referred to as “supermathematics” and guess its possible worthiness for physics, first consciously demonstrated by Wess and Zumino. In early 1960s, Berezin obtained a parallel description of the second quantization for Boze and Fermi particles summarized in his book [1] (where, among other things, Berezin introduces the Berezin integral, and gives a parallel description of the spinor and oscillator representations of, respectively, infinite dimensional orthogonal o(2∞) and symplectic Lie algebras sp(2∞), later rediscovered many times; for an overview of Berezin’s works, see the “scientific part” of [5]). In May 2007, World Scientific published the book “FELIX BEREZIN: Life and Death of the Mastermind of Supermathematics” edited by M. Shifman, [8]. The book consists of contributions of Berezin’s friends, colleagues and students; recollections of Berezin as a person, and some scientific reviews of his results and ideas, their role in mathematical physics and their developments. The core of the book is the skillfully written heart-breaking story of Berezin’s widow, Elena Karpel; and the two Berezin’s letters — to the Rector of Moscow State University and the Moscow Mathematical Society — on the situation in these institutions and in mathematical community in the Soviet Union in general (the problems pointed at in these letters did not lose their timeliness today).

Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence

Let G G be a simply connected algebraic group of type B B , C C or D D over an algebraically closed field of characteristic 2 2 . We construct a Springer correspondence for the dual vector space of the Lie algebra of G G . In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic 2 2 .

A Class of Integrable Flows on the Space of Symmetric Matrices

For a given skew symmetric real n × n matrix N, the bracket [X, Y] N = XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system \({\dot{X} = [X^2, N]}\), or equivalently in Lax form, the equation \({\dot{X} = [X, XN + NX]}\)on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure.If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra \({\mathfrak{sp}(n, N^{-1})}\)associated to the symplectic form on \({\mathbb{R}^n}\)given by N −1. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N −1). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from Sym(n, N) to \({\mathfrak{sp}(n, N^{-1})}\)maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with \({\mathfrak{sp}(n, N^{-1})}\), our system becomes a Mischenko- Fomenko system directly on Sym(n, N).If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve.

TWO PERMANENTS IN THE UNIVERSAL ENVELOPING ALGEBRAS OF THE SYMPLECTIC LIE ALGEBRAS

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra.

Twisted Yangians and Finite W-Algebras

We construct an explicit set of generators for the finite W-algebras associated to nilpotent matrices in the symplectic or orthogonal Lie algebras whose Jordan blocks are all of the same size. We use these generators to show that such finite W-algebras are quotients of twisted Yangians.

Abelian ideals and cohomology of symplectic type

Let b be a Borel subalgebra of the symplectic Lie algebra sp (2n, C) and let n be the corresponding maximal nilpotent subalgebra. We find a connection between the abelian ideals of b and the cohomology of n with trivial coefficients. Using this connection, we are able to enumerate the number of abelian ideals of b with given dimension via the Poincare polynomials of Weyl groups of types A n- 1 and C n .

LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

A Clifford Algebraic Framework for $$\mathfrak{sp}(m)$$ -Invariant Differential Operators

We introduce a framework for studying differential operators which are invariant with respect to the real (complex) symplectic Lie algebra \(\mathfrak{sp}(m)\) (\(\mathfrak{sp}_{2m}({\mathbb{C}})\)), associated to a quaternionic structure on a vector space \({\mathbb{R}}^{4m}\). To do so, these algebras are realized within the orthogonal Lie algebra \(\mathfrak{so}(4m)\). This leads in a natural way to a refinement of the recently introduced notion of complex Hermitean Clifford analysis, in which four variations of the classical Dirac operator play a dominant role.

Ensemble averaged coherent state path integral for disordered bosons with a repulsive interaction (Derivation of mean field equations)

AbstractWe consider bosonic atoms with a repulsive contact interaction in a trap potential for a Bose‐Einstein condensation (BEC) and additionally include a random potential. The ensemble averages for two models of static (I) and dynamic (II) disorder are performed and investigated in parallel. The bosonic many body systems of the two disorder models are represented by coherent state path integrals on the Keldysh time contour which allow exact ensemble averages for zero and finite temperatures. These ensemble averages of coherent state path integrals therefore present alternatives to replica field theories or super‐symmetric averaging techniques. Hubbard‐Stratonovich transformations (HST) lead to two corresponding self‐energies for the hermitian repulsive interaction and for the non‐hermitian disorder‐interaction. The self‐energy of the repulsive interaction is absorbed by a shift into the disorder‐self‐energy which comprises as an element of a larger symplectic Lie algebra sp(4M) the self‐energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product of M × sp(2); ‘M’ is the number of discrete time intervals of the disorder‐self‐energy in the generating function ). After removal of the remaining Gaussian integral for the self‐energy of the repulsive interaction, the first order variations of the coherent state path integrals magnified image result in the exact mean field or saddle point equations, solely depending on the disorder‐self‐energy matrix. These equations can be solved by continued fractions and are reminiscent to the `Nambu‐Gorkov' Green function formalism in superconductivity because anomalous terms or pair condensates of the bosonic atoms are also included into the selfenergies. The derived mean field equations of the models with static (I) and dynamic (II) disorder are particularly applicable for BEC in d = 3 spatial dimensions because of the singularity of the density of states at vanishing wavevector. However, one usually starts out from restricted applicability of the mean field approach for d = 2; therefore, it is also pointed out that one should consider different HST's in d = 2 spatial dimensions with the block diagonal densities as ‘hinge’ functions and that one has to introduce a coset decomposition Sp (4M)\U (2M) into densities and anomalous terms of the total disorder‐self‐energy sp(4M) for deriving a nonlinear sigma model.

Symplectic structures on quadratic Lie algebras

We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T ∗ -extension of a nilpotent Lie algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic Lie algebra by the one-dimensional Lie algebra. Finally, we prove that every symplectic quadratic Lie algebra is a special symplectic Manin algebra and we give an inductive description in terms of symplectic quadratic double extensions.

Two determinants in the universal enveloping algebras of the orthogonal Lie algebras

This paper gives a direct proof for the coincidence of the following two central elements in the universal enveloping algebra of the orthogonal Lie algebra: an element recently given by A. Wachi in terms of the column-determinant in a way similar to the Capelli determinant, and an element given by T. Umeda and the author in terms of the symmetrized determinant. The fact that these two elements actually coincide was shown by A. Wachi, but his observation was based on the following two non-trivial results: (i) the centrality of the first element, and (ii) the calculation of the eigenvalue of the second element. The purpose of this paper is to prove this coincidence of two central elements directly without using these (i) and (ii). Conversely this approach provides us new proofs of (i) and (ii). A similar discussion can be applied to the symplectic Lie algebras.

Energy correlations for a random matrix model of disordered bosons

Linearizing the Heisenberg equations of motion around the ground state of an interacting quantum many-body system, one gets a time-evolution generator in the positive cone of a real symplectic Lie algebra. The presence of disorder in the physical system determines a probability measure with support on this cone. The present paper analyzes a discrete family of such measures of exponential type, and does so in an attempt to capture, by a simple random matrix model, some generic statistical features of the characteristic frequencies of disordered bosonic quasiparticle systems. The level correlation functions of the said measures are shown to be those of a determinantal process, and the kernel of the process is expressed as a sum of biorthogonal polynomials. While the correlations in the bulk scaling limit are in accord with sine-kernel or Gaussian Unitary Ensemble universality, at the low-frequency end of the spectrum an unusual type of scaling behavior is found.

Skew representations of twisted Yangians

Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL N and the Yangian for $$ \mathfrak{g}\mathfrak{l}_{{N}} $$ . We prove a version of this theorem for the twisted Yangians $$ {\text{Y(}}\mathfrak{g}_{N} {\text{)}} $$associated with the orthogonal and symplectic Lie algebras $$ \mathfrak{g}_{N} = \mathfrak{o}_{N} {\text{ or }}\mathfrak{s}\mathfrak{p}_{N} $$. This gives rise to representations of the twisted Yangian $$ {\text{Y}}{\left( {\mathfrak{g}_{{N - M}} } \right)} $$ on the space of homomorphisms $$ {\text{Hom}}_{{\mathfrak{g}_{M} }} {\left( {W,V} \right)} $$, where W and V are finite-dimensional irreducible modules over $$ \mathfrak{g}_{{M}} {\text{ and }}\mathfrak{g}_{{N}} $$, respectively. In the symplectic case these representations turn out to be irreducible and we identify them by calculating the corresponding Drinfeld polynomials.We also apply the quantum Sylvester theorem to realize the twisted Yangian as a projective limit of certain centralizers in universal enveloping algebras.

Deformations of filiform Lie algebras and symplectic structures

We study symplectic structures on filiform Lie algebras, which are niplotent Lie algebras with the maximal length of the descending central sequence. Let g be a symplectic filiform Lie algebra and dim g = 2k ≥ 12. Then g is isomorphic to some ℕ-filtered deformation either of m0(2k) (defined by the structure relations [e1, e i ] = ei+1, i = 2,…, 2k − 1) or of V2k, the quotient of the positive part of the Witt algebra W+ by the ideal of elements of degree greater than 2k. We classify ℕ-filtered deformations of V n : [e i , e j ] = (j − i)ei+1 + Σl≥1c ij l e i+j+l . For dim g = n ≥ 16, the moduli space ℳn of these deformations is the weighted projective space \(\mathbb{K}P^4 \left( {n - 11,n - 10,n - 9,n - 8,n - 7} \right)\). For even n, the subspace of symplectic Lie algebras is determined by a single linear equation.

Hypersymplectic Lie algebras

We characterize real Lie algebras carrying a hypersymplectic structure as bicrossproducts of two symplectic Lie algebras endowed with a compatible flat torsion-free connection. In particular, we obtain the classification of all hypersymplectic structures on 4-dimensional Lie algebras, and we describe the associated metrics on the corresponding Lie groups.