Lie Algebra Of Rank Research Articles

Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank r

Given a finite dimensional Lie algebra g, let z(g) denote the center of g and let μ(g) be the minimal possible dimension for a faithful representation of g. In this paper we obtain μ(Lr,2), where Lr,k is the free k-step nilpotent Lie algebra of rank r. In particular we prove that μ(Lr,2)=⌈2r(r−1)⌉+2 for r≥4. It turns out that μ(Lr,2)∼μ(z(Lr,2))∼2dim⁡Lr,2 (as r→∞) and we present some evidence that this could be true for Lr,k for any k. This is considerably lower than the known bounds for μ(Lr,k), which are (for fixed k) polynomial in dim⁡Lr,k.

Real forms of embeddings of maximal reductive subalgebras of the complex simple Lie algebras of rank up to 8

We consider the problem of determining the noncompact real forms of maximal reductive subalgebras of complex simple Lie algebras. We briefly describe two algorithms for this purpose that are taken from the literature. We discuss applications in theoretical physics of these embeddings. The supplementary material to this paper contains the tables of embeddings that we have obtained for all real forms of the semisimple Lie algebras of rank up to 8.

Algorithmic construction of solvable rigid Lie algebras determined by generating functions

ABSTRACT We introduce the notion of saturation of a nilradical with respect to the eigenvalue spectrum of a maximal torus of derivations and show the existence of a generating function and a factor sequence that completely determine the nilradical. As an application, a cohomologically rigid series of solvable Lie algebras of rank one with saturated nilradical is obtained for any and dimensions , where the eigenvalues of the torus are given by the sequence . An algorithmic procedure to search cohomologically rigid Lie algebras is proposed, based on the imposition of certain constraints on the generating function.

A priori estimates of Toda systems, I: the Lie algebras of $\mathbf{A}_n$, $\mathbf{B}_n$, $\mathbf{C}_n$ and $\mathbf{G}_2$

It is well known that the PDE (Partial Differential Equation) system arising from the infinitesimal Plucker formulas is a particular case of the Toda system of $\mathbf{A}_n$ type. In this paper, we prove an a priori estimate of solutions of the Toda systems associated with the simple Lie algebras $\mathbf{A}_n$, $\mathbf{B}_n$, $\mathbf{C}_n$ and $\mathbf{G}_2$. Previous results in this direction have been done only for the case of Lie algebras of rank two. Our result for $n \geq 3$ is new. The proof of this fundamental theorem is to combine techniques from PDE and the monodromy theory. One of the key steps is to calculate the local mass of a sequence of blowup solutions near each blowup point. At each blowup point, a sequence of bubbling steps (via scaling) are performed, and the local mass of the present step could be computed from the previous step. We find out that this transformation of the local mass of each step is related to the action of an element in the Weyl group of the Lie algebra. The correspondence of the Pohozaev identity and the Weyl group could reduce the complicated calculation of the local mass into a simpler one.

Presentations for Subrings and Subalgebras of Finite CO-Rank

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

On generalized Melvin solutions for Lie algebras of rank 3

A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra G is considered. The gravitational model contains n 2-forms and l > n scalar fields, where n is the rank of G. The solution is governed by a set of n functions Hs (z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). The polynomials Hs(z), s = 1,…, 6, corresponding to the Lie algebra E6 are obtained. They depend upon integration constants Qs, s = 1,…, 6 . The polynomials obey symmetry and duality identities. The latter ones are used in deriving asymptotic relations for solutions at large distances which are presented in the paper. The power-law asymptotic relations for E6 - polynomials at large z are governed by integer-valued matrix v = A−1 (I + P), where A−1 is inverse Cartan matrix, I is identity matrix and P is permutation matrix, corresponding to a generator of the Z2-group of symmetry of the Dynkin diagram. The 2-form fluxes Φs are calculated, s = 1, …, 6.

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.

Projective modules over classical Lie algebras of infinite rank in the parabolic category

We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a,b,c,d. Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category O‾ over infinite rank Lie superalgebra of types a,b,c,d through the super duality is also a Koszul category.

Extended Laplace–Runge–Lentz vectors, new family of superintegrable systems and quadratic algebras

We present a useful proposition for discovering extended Laplace–Runge–Lentz vectors of certain quantum mechanical systems. We propose a new family of superintegrable systems and construct their integrals of motion. We solve these systems via separation of variables in spherical coordinates and obtain their exact energy eigenvalues and the corresponding eigenfunctions. We give the quadratic algebra relations satisfied by the integrals of motion. Remarkably these algebra relations involve the Casimir operators of certain higher rank Lie algebras in the structure constants.

Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank 4 (namely, A4, B4, C4, and D4) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in D dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions Hs(z) (s=1,…,4) of squared radial coordinate z=ρ2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1,n2,n3,n4)=(4,6,6,4),(8,14,18,10),(7,12,15,16),(6,10,6,6) for Lie algebras A4, B4, C4, and D4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4×4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A4 case) the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over 2-dimensional discs and corresponding Wilson loop factors over their boundaries.

Undecidability of equations in free Lie algebras

In this paper we prove undecidability of finite systems of equations in free Lie algebras of rank at least three over an arbitrary field. We show that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero.

On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras

Let $L$ be a free Lie algebra of rank $rgeq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$‎. ‎By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$‎, ‎we determine the dimension of $[L_2,L_2,L_1]$‎. ‎Moreover‎, ‎by this method‎, ‎we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$.

On generalized Melvin solutions for Lie algebras of rank 3

Generalized Melvin solutions for rank-[Formula: see text] Lie algebras [Formula: see text], [Formula: see text] and [Formula: see text] are considered. Any solution contains metric, three Abelian 2-forms and three scalar fields. It is governed by three moduli functions [Formula: see text] ([Formula: see text] and [Formula: see text] is a radial variable), obeying three differential equations with certain boundary conditions imposed. These functions are polynomials with powers [Formula: see text] for Lie algebras [Formula: see text], [Formula: see text], [Formula: see text], respectively. The solutions depend upon integration constants [Formula: see text]. The power-law asymptotic relations for polynomials at large [Formula: see text] are governed by integer-valued [Formula: see text] matrix [Formula: see text], which coincides with twice the inverse Cartan matrix [Formula: see text] for Lie algebras [Formula: see text] and [Formula: see text], while in the [Formula: see text]-case [Formula: see text], where [Formula: see text] is the identity matrix and [Formula: see text] is a permutation matrix, corresponding to a generator of the [Formula: see text]-group of symmetry of the Dynkin diagram. The duality identities for polynomials and asymptotic relations for solutions at large distances are obtained. Two-form flux integrals over a two-dimensional disc of radius [Formula: see text] and corresponding Wilson loop factors over a circle of radius [Formula: see text] are presented.

On generalized Melvin’s solutions for Lie algebras of rank 2

We consider a class of solutions in multidimensional gravity which generalize Melvin’s well-known cylindrically symmetric solution, originally describing the gravitational field of a magnetic flux tube. The solutions considered contain the metric, two Abelian 2-forms and two scalar fields, and are governed by two moduli functions H1(z) and H2(z) (z = ρ2, ρ is a radial coordinate) which have a polynomial structure and obey two differential (Toda-like) master equations with certain boundary conditions. These equations are governed by a certain matrix A which is a Cartan matrix for some Lie algebra. The models for rank-2 Lie algebras A2, C2 and G2 are considered. We study a number of physical and geometric properties of these models. In particular, duality identities are proved, which reveal a certain behavior of the solutions under the transformation ρ → 1/ρ; asymptotic relations for the solutions at large distances are obtained; 2-form flux integrals over 2-dimensional regions and the corresponding Wilson loop factors are calculated, and their convergence is demonstrated. These properties make the solutions potentially applicable in the context of some dual holographic models. The duality identities can also be understood in terms of the Z2 symmetry on vertices of the Dynkin diagram for the corresponding Lie algebra.

The Symmetric Invariants of Centralizers and Slodowy Grading II

Let 𝔤 be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field 𝕜 of characteristic zero. We identify 𝔤 with 𝔤∗ through the Killing form of 𝔤. Let (e, h, f) be an 𝔰𝔩2-triple of 𝔤. Denote by 𝔤 e the centralizer of e in 𝔤 and by $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ the algebra of symmetric invariants of 𝔤 e . We say that e is good if the nullvariety of some l homogeneous elements of $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ in (𝔤 e )∗ has codimension l. In our previous work (Charbonnel and Moreau. Math. Zeitsch. 282, n° 1-2, 273–339 2016), we showed that if e is good then $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of Charbonnel and Moreau (Math. Zeitsch. 282, n° 1-2, 273–339 2016) is true. Namely, we prove that e is good if and only if for some homogeneous generating sequence q 1, … , q l of S(𝔤) 𝔤 , the initial homogeneous components of their restrictions to e + 𝔤 f are algebraically independent over 𝕜.

Rank 2 fusion rings are complete intersections

We give a non-constructive proof that fusion rings attached to a simple complex Lie algebra of rank 2 are complete intersections.

Extensions and automorphisms of Lie algebras

Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].

A Presentation of the Free Lie Algebra M2,m,3

Let M2,m,3 be a free solvable nilpotent Lie algebra of rank 2 and nilpotency class m - 1. We show that M2,m,3 admits a minimal presentation whose set of defining relators consists of certain types of basic commutators using techniques in Grobner-Shirshov basis theory.

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ℚ-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.

The symmetric invariants of centralizers and Slodowy grading

Let $$\mathfrak {g}$$ be a finite-dimensional simple Lie algebra of rank $$\ell $$ over an algebraically closed field $$\Bbbk $$ of characteristic zero, and let e be a nilpotent element of $$\mathfrak {g}$$ . Denote by $$\mathfrak {g}^{e}$$ the centralizer of e in $$\mathfrak {g}$$ and by $$ \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} $$ the algebra of symmetric invariants of $$\mathfrak {g}^{e}$$ . We say that e is good if the nullvariety of some $$\ell $$ homogenous elements of $$ \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} $$ in $$({\mathfrak g}^{e})^{*}$$ has codimension $$\ell $$ . If e is good then $$ \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} $$ is a polynomial algebra. The main result of this paper stipulates that if for some homogenous generators of $$ \mathrm{S}({\mathfrak g}) ^{{\mathfrak g}} $$ , the initial homogenous components of their restrictions to $$e+\mathfrak {g}^{f}$$ are algebraically independent, with (e, h, f) an $$\mathfrak {sl}_2$$ -triple of $$\mathfrak {g}$$ , then e is good. As applications, we pursue the investigations of Panyushev et al. (J. Algebra 313:343–391, 2007) and we produce (new) examples of nilpotent elements that satisfy the above polynomiality condition, in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type $$\mathbf{D}_{7}$$ .