Split Lie Algebras Research Articles

The compact exceptional Lie algebra 𝔤^{𝔠}₂ as a twisted ring group

A new highly symmetrical model of the compact Lie algebra g 2 c \mathfrak {g}^c_2 is provided as a twisted ring group for the group Z 2 3 \mathbb {Z}_2^3 and the ring R ⊕ R \mathbb {R}\oplus \mathbb {R} . The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in s u ( 2 ) \mathfrak {su}(2) and of the Gell-Mann matrices in s u ( 3 ) \mathfrak {su}(3) . As a bonus, the split Lie algebra g 2 \mathfrak {g}_2 is also seen as a twisted ring group.

Takiff algebras, Toda systems, and jet transformations

We study a family of completely integrable systems attached to Takiff algebras gN, extending Toda systems for split simple Lie algebras g. With respect to Darboux coordinates on coadjoint orbits O, the potentials of the hamiltonians are products of polynomial and exponential functions. Explicit general solutions for their equations of motion are obtained using differential operators called jet transformations. The classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, quantizing the Poisson algebra of functions on O. These results are illustrated with concrete examples, including a 3-body problem based on sl(2) and an extension of soliton solutions for A∞ to associated Takiff algebras.

The Toda flow on Hessenberg elements of real, split simple Lie algebras

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits.

Faithfulness of generalised Verma modules for Iwasawa algebras

We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type A2. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.

Classification of four-rebit states

We classify states of four rebits, that is, we classify the orbits of the group Gˆ(R)=SL(2,R)4 in the space (R2)⊗4. This is the real analogon of the well-known SLOCC operations in quantum information theory. By constructing the Gˆ(R)-module (R2)⊗4 via a Z/2Z-grading of the simple split real Lie algebra of type D4, the orbits are divided into three groups: semisimple, nilpotent and mixed. The nilpotent orbits have been classified in Dietrich et al. (2017) [26], yielding applications in theoretical physics (extremal black holes in the STU model of N=2,D=4 supergravity, see Ruggeri and Trigiante (2017) [51]). Here we focus on the semisimple and mixed orbits which we classify with recently developed methods based on Galois cohomology, see Borovoi et al. (2021) [8,9]. These orbits are relevant to the classification of non-extremal (or extremal over-rotating) and two-center extremal black hole solutions in the STU model.

Non-group gradings on simple Lie algebras

A set grading on the split simple Lie algebra of type D13, that cannot be realized as a group-grading, is constructed by splitting the set of positive roots into a disjoint union of pairs of orthogonal roots, following a pattern provided by the lines of the projective plane over GF(3). This answers in the negative [3, Question 1.11].Similar non-group gradings are obtained for types Dn with n≡1(mod12), by substituting the lines in the projective plane by blocks of suitable Steiner systems.

Split Lie–Rinehart algebras

We introduce the class of split Lie–Rinehart algebras as the natural extension of the one of split Lie algebras. We show that if [Formula: see text] is a tight split Lie–Rinehart algebra over an associative and commutative algebra [Formula: see text] then [Formula: see text] and [Formula: see text] decompose as the orthogonal direct sums [Formula: see text] and [Formula: see text], where any [Formula: see text] is a nonzero ideal of [Formula: see text], any [Formula: see text] is a nonzero ideal of [Formula: see text], and both decompositions satisfy that for any [Formula: see text], there exists a unique [Formula: see text] such that [Formula: see text]. Furthermore, any [Formula: see text] is a split Lie–Rinehart algebra over [Formula: see text]. Also, under mild conditions, it is shown that the above decompositions of [Formula: see text] and [Formula: see text] are by means of the family of their, respective, simple ideals.

Split Lie algebras of order 3

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form [Formula: see text] with [Formula: see text] a linear subspace of [Formula: see text] and any [Formula: see text] a well-described (split) ideal of [Formula: see text] satisfying [Formula: see text], with [Formula: see text], if [Formula: see text]. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that [Formula: see text] is the direct sum of the family of its (split) simple ideals) is stated.

Projections of Semisimple Lie Algebras

It is proved that the property of being a semisimple algebra is preserved under projections (lattice isomorphisms) for locally finite-dimensional Lie algebras over a perfect field of characteristic not equal to 2 and 3, except for the projection of a three-dimensional simple nonsplit algebra. Over fields with the same restrictions, we give a lattice characterization of a three-dimensional simple split Lie algebra and a direct product of a one-dimensional algebra and a three-dimensional simple nonsplit one.

On the Classification of Lie Bialgebras by Cohomological Means

We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms of faithfully flat cohomology over an arbitrary ring over \mathbb{Q} , and solve it for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero. We consider the classification up to isomorphism, as opposed to equivalence, and treat split and non-split Lie algebras alike. We moreover give a new interpretation of scalar multiples of Lie bialgebras hitherto studied using twisted Belavin-Drinfeld cohomology.

Split regular Hom-Leibniz color 3-algebras

We introduce and describe the class of split regular $Hom$-Leibniz color $3$-algebras as the natural extension of the class of split Lie algebras, split Leibniz algebras, split Lie $3$-algebras, split Lie triple systems, split Leibniz $3$-algebras, and some other algebras. More precisely, we show that any of such split regular $Hom$-Leibniz color $3$-algebras $T$ is of the form ${T}={\mathcal U} +\sum\limits_{j}I_{j}$, with $\mathcal U$ a subspace of the $0$-root space ${T}_0$, and $I_{j}$ an ideal of $T$ satisfying {for} $j\neq k:$ \[[{ T},I_j,I_k]+[I_j,{ T},I_k]+[I_j,I_k,T]=0.\] Moreover, if $T$ is of maximal length, we characterize the simplicity of $T$ in terms of a connectivity property in its set of non-zero roots.

A Note on Strongly Split Lie Algebras

Split Lie algebras are possibly the most known examples of graded Lie algebras. Since an important category in the class of graded algebras is the category of strongly graded algebras, we introduce, in a natural way, the category of strongly split Lie algebras 𝔏 and show that if 𝔏 is centerless, then 𝔏 is the direct sum of split ideals each of which is a split-simple strongly split Lie algebra.

Split 3-Leibniz algebras

We introduce and describe the class of split 3-Leibniz algebras as the natural extension of the class of split Lie algebras, split Leibniz algebras, split Lie triple systems and split 3-Lie algebras. More precisely, we show that any of such split 3-Leibniz algebras T is of the form T=U+∑jIj, with U a subspace of the 0-root space T0, and Ij an ideal of T satisfying [T,Ij,Ik]+[Ij,T,Ik]+[Ij,Ik,T]=0 for j≠k. Moreover, if T is of maximal length, we characterize the simplicity of T in terms of a connectivity property in its set of non-zero roots.

Belavin\u2013Drinfeld solutions of the Yang\u2013Baxter equation: Galois cohomology considerations

We relate the Belavin–Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field mathbb {K} of characteristic 0 to the standard non-abelian Galois cohomology H^1(mathbb {K}, mathbf{H}) for a suitable algebraic mathbb {K}-group mathbf{H}. The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras.

The structure of split regular BiHom-Lie algebras

We introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. We show that an arbitrary split regular BiHom-Lie algebra L is of the form L = U + ∑ j I j with U a linear subspace of a fixed maximal abelian subalgebra H and any I j a well described (split) ideal of L , satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals.

Positive Energy Representations for Locally Finite Split Lie Algebras

Let $\mathfrak g$ be a locally finite split simple complex Lie algebra of type $A_J$, $B_J$, $C_J$ or $D_J$ and $\mathfrak h \subseteq \mathfrak g$ be a splitting Cartan subalgebra. Fix $D \in \mathrm{der}(\mathfrak g)$ with $\mathfrak h \subseteq \ker D$ (a diagonal derivation). Then every unitary highest weight representation $(\rho_\lambda, V^\lambda)$ of $\mathfrak g$ extends to a representation $\tilde\rho_\lambda$ of the semidirect product $\mathfrak g \rtimes \mathbb C D$ and we say that $\tilde\rho_\lambda$ is a positive energy representation if the spectrum of $-i\tilde\rho_\lambda(D)$ is bounded from below. In the present note we characterise all pairs $(\lambda,D)$ with $\lambda$ bounded for which this is the case. If $U_1(\mathcal H)$ is the unitary group of Schatten class $1$ on an infinite dimensional real, complex or quaternionic Hilbert space and $\lambda$ is bounded, then we accordingly obtain a characterisation of those highest weight representations $\pi_\lambda$ satisfying the positive energy condition with respect to the continuous $\mathbb R$-action induced by $D$. In this context the representation $\pi_\lambda$ is norm continuous and our results imply the remarkable result that, for positive energy representations, adding a suitable inner derivation to $D$, we can achieve that the minimal eigenvalue of $\tilde\rho_\lambda(D)$ is $0$ (minimal energy condition). The corresponding pairs $(\lambda,D)$ satisfying the minimal energy condition are rather easy to describe explicitly.

Some special features of Cayley algebras, and [formula omitted], in low characteristics

Some features of Cayley algebras (or algebras of octonions) and their Lie algebras of derivations over fields of low characteristic are presented. More specifically, over fields of characteristic 7, explicit embeddings of any twisted form of the Witt algebra into the simple split Lie algebra of type G2 are given. Over fields of characteristic 3, even though the Lie algebra of derivations of a Cayley algebra is not simple, it is shown that still two Cayley algebras are isomorphic if and only if their Lie algebras of derivations are isomorphic. Finally, over fields of characteristic 2, it is shown that the Lie algebra of derivations of any Cayley algebra does not depend on the Cayley algebra as it is always isomorphic to the projective special linear Lie algebra of degree four. The twisted forms of this latter algebra are described too. Any such form is, up to isomorphism, the second derived power of the Lie algebra of skew elements relative to a symplectic involution in a central simple associative algebra of degree six.

On the structure of split involutive Lie algebras

On the structure of split involutive Lie algebras

Verma modules over p-adic Arens–Michael envelopes of reductive Lie algebras

Let K be a locally compact p-adic field, g a split reductive Lie algebra over K and U(g) its universal enveloping algebra. We investigate the category Cg of coadmissible modules over the p-adic Arens–Michael envelope Uˆ(g) of U(g). Let p⊆g be a parabolic subalgebra. The main result gives a canonical equivalence between the classical parabolic BGG category of g relative to p and a certain explicitly given highest weight subcategory of Cg. This completely clarifies the “Verma module theory” over Uˆ(g).

$$SK_1$$ SK 1 and Lie algebras

We investigate the vanishing of the group $$SK_1(\Lambda (G))$$ for the Iwasawa algebra $$\Lambda (G)$$ of a pro- $$p$$ $$p$$ -adic Lie group $$G$$ (with $$p \ne 2$$ ). We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras.