Lie Algebra Sl Research Articles

Special function models of indecomposable sl(2) representations: the Laguerre case

In this paper, we point out connections between certain types of indecomposable representations of sl(2) and generalizations of well-known orthogonal polynomials. Those representations take the form of infinite dimensional chains of weight or generalised weight spaces, for which the Cartan generator acts in a diagonal way or via Jordan blocks. The other generators of the Lie algebras sl(2) act as raising and lowering operators but are now allowed to relate the different chains as well. In addition, we construct generating functions, we calculate the action of the Casimir invariant and present relations to systems of non-homogeneous second-order coupled differential equations. We present different properties as higher-order linear differential equations for building blocks taking the form of one variable polynomials. We also present insight into the zeroes and recurrence relations.

Specht property for the graded identities of the pair (M2(D),sl2(D))

Let D be a Noetherian infinite integral domain, denote by M2(D) and by sl2(D) the 2×2 matrix algebra and the Lie algebra of the traceless matrices in M2(D), respectively. In this paper we study the weak polynomial identities for the natural grading by the cyclic group Z2 of order 2 on M2(D) and on sl2(D). We describe a finite basis of the graded polynomial identities for the pair (M2(D),sl2(D)). Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for (M2(D),sl2(D)), is finitely generated. The polynomial identities for M2(D) are known if D is any field of characteristic different from 2. The identities for the Lie algebra sl2(D) are known when D is an infinite field. The identities for the pair we consider were first described by Razmyslov when D is a field of characteristic 0, and afterwards by the second author when D is an infinite field. The graded identities for the pair (M2(D),gl2(D)) were also described, by Krasilnikov and the second author.In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets.

Notes on emergent conformal symmetry for black holes

We examine the motion of the massless scalar field and nearly bound null geodesics in the near-ring region of a black hole, which may possess either acceleration or a gravitomagnetic mass. Around such black holes, the photon ring deviates from the equatorial plane. In the large angular momentum limit, we demonstrate that the massless scalar field exhibits an emergent conformal symmetry in this near-ring region. Additionally, in the nearly bound limit, we observe the emergence of a conformal symmetry for the null geodesics that constitute the photon ring in the black hole image. These findings suggest that the hidden conformal symmetry, associated with the Lie algebra 𝔰𝔩(2, ℝ), persists even for black holes lacking north-south reflection symmetry, thereby broadening the foundation of photon ring holography. Finally, we show that the conformal symmetry also emerges for nearly bound timelike geodesics and scalar fields in proximity to the particle ring, and with specific mass around a Schwarzschild black hole.

Пространства постоянной кривизны, уравнение Лиувилля и контракции алгебр Ли

The Liouville equation is equivalent to the two-dimensional
 Laplace equation and is reduced to it by the Lie-Bäcklund
 transformation, which, from the point of view of group theory,
 can be interpreted from using the Inönü-Wigner contraction.
 We analyze the construction of solutions to the Liouville
 equation to represent zero curvature using the contraction of
 the Lie algebra sl(2) and use nilpotent generators for this.

Zero product determined n-th Schrödinger algebra

The n-th Schrödinger algebra schn:=sl2⋉hn is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn. Let K be an algebraically closed field of characteristic zero. A Lie algebra L is zero product determined over K, if for every K-linear space V and every bilinear map ϕ:L×L→V with the property that ϕ(x,y)=0 whenever [x,y]=0, then there exists a linear map f:[L,L]→V such that ϕ(x,y)=f([x,y]) for all x,y∈L. This article shows that the n-th Schrödinger algebra schn is zero product determined. Applying this result, product zero derivations and two-sided commutativity-preserving maps on schn are determined. Furthermore, quasi-derivations, linear anti-commuting maps, quasi-automorphisms and strong commutativity-preserving maps of schn are all obtained.

Quasi-Whittaker modules for the n-th Schrödinger algebra

The n-th Schrödinger algebra schn defined in [14] is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn, which generalizes the Schrödinger algebra sl2⋉h1. Let ϕ:hn→C be a nonzero Lie algebra homomorphism. A schn-module V is called quasi-Whittaker of type ϕ if V=U(schn)v, where U(schn) is the universal enveloping algebra of schn, v is a nonzero vector such that xv=ϕ(x)v for any x∈hn. In this paper, we prove that a simple schn-module V is a quasi-Whittaker module if and only if V is a locally finite hn-module. Then we classify the simple quasi-Whittaker modules of ϕ, according to the rank of ϕ. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of ϕ.

Coquaternions, Metric Invariants of Biologic Systems and Malignant Transformations

Different hypotheses of carcinogenesis have been proposed based on local genetic factors and physiologic mechanisms. It is assumed that changes in the metric invariants of a biologic system (BS) determine the general mechanisms of cancer development. Numerous pieces of data demonstrate the existence of three invariant feedback patterns of BS: negative feedback (NFB), positive feedback (PFB) and reciprocal links (RL). These base patterns represent basis elements of a Lie algebra sl(2,R) and an imaginary part of coquaternion. Considering coquaternion as a model of a functional core of a BS, in this work a new geometric approach has been introduced. Based on this approach, conditions of the system are identified with the points of three families of hypersurfaces in R42: hyperboloids of one sheet, hyperboloids of two sheets and double cones. The obtained results also demonstrated the correspondence of an indefinite metric of coquaternion quadratic form with negative and positive entropy contributions of the base elements to the energy level of the system. From that, it can be further concluded that the anabolic states of the system will correspond to the points of a hyperboloid of one sheet, whereas catabolic conditions correspond to the points of a hyperboloid of two sheets. Equilibrium states will lie in a double cone. Physiologically anabolic and catabolic states dominate intermittently oscillating around the equilibrium. Deterioration of base elements increases positive entropy and causes domination of catabolic states, which is the main metabolic determinant of cancer. Based on these observations and the geometric representation of a BS’s behavior, it was shown that conditions related to cancer metabolic malfunction will have a tendency to remain inside the double cone.

Towards an [formula omitted] action on the annular Khovanov spectrum

Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra sl2, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of sl2 are lifted to maps of spectra. In particular, it follows that the sl2 action on homology commutes with the action of the Steenrod algebra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.

On simple modules of the n-th Schrödinger algebra

The n -th Schrödinger algebra sch n : = sl 2 ⋉ h n is the semi-direct product of the Lie algebra sl 2 with the n -th Heisenberg Lie algebra h n . We give a classification of simple Harish-Chandra modules and simple Whittaker modules over the n -th Schrödinger algebra. We determine the centres of some algebras that are related to the universal enveloping algebra of the n -th Schrödinger algebra.

Characteristic polynomials and finitely dimensional representations of [formula omitted

In this paper, we obtain a general formula for the characteristic polynomial of a finitely dimensional representation of Lie algebra sl(2,C) and the form for these characteristic polynomials, and prove there is one to one correspondence between representations and their characteristic polynomials. Moreover we define a monoid structure on these characteristic polynomials.

A Closed Structure Constant Formula for the Universal Enveloping Algebra of the Lie Algebra sl2

Gourley and Kennedy gave a recursive formula for the structure constants of the universal enveloping algebra of sl2. Using Kostant's formula we give a closed formula for these structure constants.

Modified graded Hennings invariants from unrolled quantum groups and modified integral

The second author constructed a topological ribbon Hopf algebra from the unrolled quantum group associated with the super Lie algebra sl ( 2 | 1 ) . We generalize this fact to the context of unrolled quantum groups and construct the associated topological ribbon Hopf algebras. Then we use such an algebra, the discrete Fourier transforms , a symmetrized graded integral and a modified trace to define a modified graded Hennings invariant of 3-manifolds endowed with a cohomology class and which contains a ribbon graph. Finally, we use the notion of a modified integral to extend this invariant to manifolds without ribbon graphs inside and show that it recovers the invariant of [6] .

Jordan–Kronecker Invariants of Semidirect Sums of the Form sl(n) + (ℝn)k and gl(n) + (ℝn)k

We calculate Jordan–Kronecker invariants for semidirect sums of Lie algebras sl(n) and gl(n) with k copies of ℝn with respect to their standard representation for cases where k > n or n is a multiple of k.

A family of simple modules over the Rueda's algebras

The Rueda's algebras R(f,ς) (simply R) are a class of algebras similar to the universal enveloping algebra of sl2. We study the category H of R-modules whose objects are free of rank 1 when restricted to R0=C[H]. We classify the isomorphism classes of objects in H and determine the simplicity of these modules. As a result, we also give an explicit description of submodule structures and obtain new simple non-weight modules over R. In particular, we recover some results about U(h)-free modules over the Lie algebra sl2 obtained by J. Nilsson and over the Lie superalgebra osp(1|2) obtained by Y. Cai and K. Zhao.

Blocks and characters of G(3)-modules of non-integral weights

We classify blocks in the BGG category O of modules of non-integral weights for the exceptional Lie superalgebra G(3). We compute the characters for tilting modules of non-integral weights in O. Reduction methods are established to connect non-integral blocks of G(3) with blocks of the special linear Lie algebra sl(2), the exceptional Lie algebra G2, the general linear Lie superalgebras gl(1|1), gl(2|1) and the ortho-symplectic Lie superalgebra osp(3|2).

A Dixmier theorem for Poisson enveloping algebras

We consider a skew-symmetric n-ary bracket on the polynomial algebra K[x1,…,xn,xn+1] (n≥2) over a field K of characteristic zero defined by {a1,…,an}=Jac(a1,…,an,C), where C is a fixed element of K[x1,…,xn,xn+1] and Jac is the Jacobian. If n=2 then this bracket is a Poisson bracket and if n≥3 then it is an n-Lie-Poisson bracket on K[x1,…,xn,xn+1]. We describe the center of the corresponding n-Lie-Poisson algebra and show that the quotient algebra K[x1,…,xn,xn+1]/(C−λ), where (C−λ) is the ideal generated by C−λ, 0≠λ∈K, is a simple central n-Lie-Poisson algebra if C is a homogeneous polynomial that is not a proper power of any nonzero polynomial. This construction includes the quotients P(sl2(K))/(C−λ) of the Poisson enveloping algebra P(sl2(K)) of the simple Lie algebra sl2(K), where C is the standard Casimir element of sl2(K) in P(sl2(K)). It is also proven that the quotients P(M)/(CM−λ) of the Poisson enveloping algebra P(M) of the exceptional simple seven dimensional Malcev algebra M are central simple, where CM is the standard Casimir element of M in P(M).

Lie algebras of matrix difference differential operators and special matrix functions

We discuss certain models of irreducible representations of Lie algebra sl(2,C) from the special matrix functions point of view. These models are constructed in terms of matrix differential operators and matrix difference differential operators and are connected through a matrix integral transformation. In the process, we find new matrix function identities involving one and two variable special matrix functions.

Second cohomology space of $\mathfrak{sl}(2)$ acting on the space of bilinear bidifferential operators

We consider the sl(2)-module structure on the spaces of bilinear bidifferential operators acting on the spaces of weighted densities. We compute the second cohomology group of the Lie algebra sl(2) with coefficients in the space of bilinear bidifferential operators that act on tensor densities Dλ,ν,µ.

The quantum group SLq⋆(2) and quantum algebra Uq(sl2⋆) based on a new associative multiplication on 2 × 2 matrices

We present classical groups SL⋆(2) and SU⋆(2) as well as classical Lie algebra sl2⋆(C) associated with a new associative multiplication on 2 × 2 matrices. The idea of the new multiplication is generalized to the action of a 2 × 2 square matrix on a 2 × 1 column one. The coordinate Hopf algebra O(SLq⋆(2)) is introduced as a q-generalization of Hopf algebra O(SL⋆(2)), and it is shown that the coordinate algebra corresponding to the quantum plane Cq2 is a SLq⋆(2)-left-covariant algebra. Furthermore, the quantized universal enveloping algebra Uq(sl2⋆) with the Hopf structure as a dual of O(SLq⋆(2)) is introduced. For each of the Hopf algebras O(SLq⋆(2)) and Uq(sl2⋆), we associate two different real forms with two inequivalent families of *-involutions, with O(SUq⋆(2)) and Uq(su2⋆) as one of the real forms. It is shown that the Hopf algebra pairing is a dual pairing of two Hopf *-algebras O(SUq⋆(2)) and Uq(su2⋆).

On 3-dimensional complex Hom-Lie algebras

We study and classify the 3-dimensional Hom-Lie algebras over C. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space g. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps g→g that turn g into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps g→g are not homomorphisms for their products are exhibited. Examples also arise of non-isomorphic families of Hom-Lie algebras which share, however, a fixed Lie-algebra product on g. In particular, this is the case for the complex simple Lie algebra sl2(C). Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on g.