Representations of Bihom-Lie Algebras

A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.

Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

Brauer–Clifford Group of Lie–Rinehart Algebras

In this paper we define the notion of Brauer–Clifford group for [Formula: see text]-Azumaya algebras when [Formula: see text] is a commutative algebra and[Formula: see text] is a [Formula: see text]-Lie algebra over a commutative ring [Formula: see text]. This is the situation that arises in applications having connections to differential geometry. This Brauer–Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.

A sandwich in thin lie algebras

A thin Lie algebra is a Lie algebra $L$ , graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$ , and such that each non-zero ideal of $L$ lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of $L$ (that is, the next diamond past $L_1$ ) occurs in degree $k$ . We prove that if $k>5$ , then $[Lyy]=0$ for some non-zero element $y$ of $L_1$ . In characteristic different from two this means $y$ is a sandwich element of $L$ . We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.

Real dimension of the Lie algebra of S-skew-Hermitian matrices

Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $\mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $\mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.

Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

The holographic c-theorem and infinite-dimensional Lie algebras

We discuss a non-dynamical theory of gravity in three dimensions which is based on an infinite-dimensional Lie algebra that is closely related to an infinite-dimensional extended AdS algebra. We find an intriguing connection between on the one hand higher-derivative gravity theories that are consistent with the holographic c-theorem and on the other hand truncations of this infinite-dimensional Lie algebra that violate the Lie algebra structure. We show that in three dimensions different truncations reproduce, up to terms that do not contribute to the c-theorem, Chern-Simons-like gravity models describing extended 3D massive gravity theories. Performing the same procedure with similar truncations in dimensions larger than or equal to four reproduces higher derivative gravity models that are known in the literature to be consistent with the c-theorem but do not have an obvious connection to massive gravity like in three dimensions.

Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

In this note, we study in a finite dimensional Lie algebra $${\mathfrak g}$$ g the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone $$C_x$$ C x . Assuming that $${\mathfrak g}$$ g is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying $$[h,x]=x$$ [ h , x ] = x for which $$C_x$$ C x pointed. Given x, we show that such elements h can be constructed in such a way that $$\mathop {\mathrm{ad}}\nolimits h$$ ad h defines a 5-grading, and characterize the cases where we even get a 3-grading.

Clebsh–Gordan coefficients for the algebra 𝔤𝔩₃ and hypergeometric functions

The Clebsh–Gordan coefficients for the Lie algebra g l 3 \mathfrak {gl}_3 in the Gelfand–Tsetlin base are calculated. In contrast to previous papers, the result is given as an explicit formula. To obtain the result, a realization of a representation in the space of functions on the group G L 3 GL_3 is used. The keystone fact that allows one to carry the calculation of Clebsh–Gordan coefficients is the theorem that says that functions corresponding to the Gelfand–Tsetlin base vectors can be expressed in terms of generalized hypergeometric functions.