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The Freudenthal–Tits magic square m (A 1 , A 2) for A = R , C , H , O of semi-simple Lie algebras can be extended by including the sextonions S . A series of non-reductive Lie algebras naturally appear in the new row associated with the sextonions, which we will call the intermediate exceptional series, with the largest one as the intermediate Lie algebra E 7 + 1 / 2 constructed by Landsberg–Manivel. We study various aspects of the intermediate vertex operator (super)algebras associated with the intermediate exceptional series, including rationality, coset constructions, irreducible modules, (super)characters and modular linear differential equations. For all g I belonging to the intermediate exceptional series, the intermediate VOA L 1 (g I) has characters of irreducible modules coinciding with those of the simple rational C 2 -cofinite W-algebra W - h ∨ / 6 (g , f θ) studied by Kawasetsu, with g belonging to the Cvitanović–Deligne exceptional series. We propose some new intermediate VOA L k (g I) with integer level k and investigate their properties. For example, for the intermediate Lie algebra D 6 + 1 / 2 between D 6 and E 7 in the subexceptional series and also in Vogel’s projective plane, we find that the intermediate VOA L 2 (D 6 + 1 / 2) has a simple current extension to a SVOA with four irreducible Neveu–Schwarz modules, and the supercharacters can be realized by a fermionic Hecke operator on the N = 1 Virasoro minimal model L (c 16 , 2 , 0) . We also provide some (super) coset constructions such as L 2 (E 7) / L 2 (D 6 + 1 / 2) and L 1 (D 6 + 1 / 2) ⊗ 2 / L 2 (D 6 + 1 / 2) . In the end, we find that the theta blocks associated with the intermediate exceptional series produce some new holomorphic Jacobi forms of critical weight and lattice index. [ABSTRACT FROM AUTHOR]

E 7 + 1 / 2 is an intermediate Lie algebra filling a hole between E 7 and E 8 in the Deligne–Cvitanović exceptional series. It was found independently by Mathur, Muhki, Sen in the classification of 2d RCFTs via modular linear differential equations (MLDE) and by Deligne, Cohen, de Man in representation theory. In this paper we propose some new vertex operator algebras (VOA) associated with E 7 + 1 / 2 and give some useful information at small levels. We conjecture that the affine VOA (E 7 + 1 / 2) k is rational if and only if the level k is at most 5, and provide some evidence from the viewpoint of MLDE. We propose a conjectural Weyl dimension formula for infinitely many irreducible representations of E 7 + 1 / 2 , which generates almost all irreducible representations of E 7 + 1 / 2 with level k ≤ 4 . More concretely, we propose the affine VOA E 7 + 1 / 2 at level 2 and the rank-two instanton VOA associated with E 7 + 1 / 2 . We compute the VOA characters and provide some coset constructions. These generalize the previous works of Kawasetsu for affine VOA E 7 + 1 / 2 at level 1 and of Arakawa–Kawasetsu at level - 5 . We then predict the conformal weights of affine VOA E 7 + 1 / 2 at level 3, 4, 5. [ABSTRACT FROM AUTHOR]

We consider a family of Riemannian manifolds M such that for each unit speed geodesic γ of M there exists a distinguished bijective correspondence L between infinitesimal translations along γ and infinitesimal rotations around it. The simplest examples are R 3 , S 3 and hyperbolic 3-space, with L defined in terms of the cross product. More generally, M is a connected compact semisimple Lie group, or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M. Given λ ∈ R , we define a left invariant distribution on G accounting for infinitesimal roto-translations of M of pitch λ . We give conditions for the controllability of the associated control system on G and find explicitly all the geodesics of the natural sub-Riemannian structure. We also study a similar system on R 7 ⋊ S O 7 involving the octonionic cross product. In an appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of “small rotations”. [ABSTRACT FROM AUTHOR]

The Lie algebra e 7 1 2 comes from P. Deligne's work on the exceptional series of Lie groups. Using the triality algebra, J. Landsberg and L. Manivel construct this Lie algebra in 2006. In this paper, we study the structure of e 7 1 2 following B. Gross and N. Wallach's work on the highest root and Heisenberg parabolic subalgebra. The process of removing the lowest root from the extended Dynkin diagram of E 8 will contribute to the components of e 7 1 2 . We also study the branching rule of e 7 1 2 to s l (3 , C) and g 2 . We use the computer algebra system SageMath to carry out the branching rule calculations. Then we calculate the Dynkin indices for e 7 1 2 . We find that the number 24 behaves like the 'dual Coxeter number' of e 7 1 2 . [ABSTRACT FROM AUTHOR]

We present a binary code for spinors and Clifford multiplication using non-negative integers and their binary expressions, which can be easily implemented in computer programs for explicit calculations. As applications, we present explicit descriptions of the triality automorphism of Spin(8), explicit representations of the Lie algebras 𝔰𝔭𝔦𝔶 (8), 𝔰𝔭𝔦𝔶 (7) and 𝔤2, etc.

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in sl 2 corresponds to the KdV hierarchy, and the triple (f , 0 , e θ) , where f is the sum of negative simple root vectors and e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy. [ABSTRACT FROM AUTHOR]

We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$, the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

In this paper we face the study of the representations of the exceptional Lie superalgebra E(5,10). We recall the construction of generalized Verma modules and give a combinatorial description of the restriction to s l 5 of the Verma module induced by the trivial representation. We use this description to classify morphisms between Verma modules of degree one, two and three proving in these cases a conjecture given by Rudakov (8). A key tool is the notion of dual morphism between Verma modules. [ABSTRACT FROM AUTHOR]

The space of invariant affine connections on every 3-Sasakian homogeneous manifold of dimension at least seven is described. In particular, the subspace of invariant affine metric connections and the subclass with skew torsion are also determined. To this aim, an explicit construction of all 3-Sasakian homogeneous manifolds is exhibited. It is shown that the 3-Sasakian homogeneous manifolds which admit nontrivial Einstein with skew torsion invariant affine connections are those of dimension seven, that is, S 7 , R P 7 and the Aloff–Wallach space W 1 , 1 7 . On S 7 and R P 7 , the set of such connections is bijective to two copies of the conformal linear transformation group of the Euclidean space, while it is strictly bigger on W 1 , 1 7 . The set of invariant connections with skew torsion whose Ricci tensor satisfies that its eigenspaces are the canonical vertical and horizontal distributions, is fully described on 3-Sasakian homogeneous manifolds. An affine connection satisfying these conditions is distinguished, by parallelizing all the Reeb vector fields associated with the 3-Sasakian structure, which is also Einstein with skew torsion on the 7-dimensional examples. The invariant metric affine connections on 3-Sasakian homogeneous manifolds with parallel skew torsion have been found. Finally, some results have been adapted to the non-homogeneous setting. [ABSTRACT FROM AUTHOR]

We consider a complex simple Lie algebra $${\mathfrak{g}}$$ , with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct $${\mathfrak{g}}$$ -invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra $${{\rm U}(\mathfrak{g})}$$ . This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters.

We classify the simple modules of the exceptional algebraic supergroups over an algebraically closed field of prime characteristic. [ABSTRACT FROM AUTHOR]

By using the idea of Wakimoto’s free field, we construct a class of representations for the Lie superalgebra D(2, 1; α) on the tensor product of a polynomial algebra and an exterior algebra involving one parameter λ. Then we obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter λ satisfies (λ+m)(λ−1+ααm)≠0 for any m ∈ ℤ+. [ABSTRACT FROM AUTHOR]

These notes have been prepared for the Workshop on “(Non)-existence of complex structures on S 6 ”, celebrated in Marburg in March, 2017. The material is not intended to be original. It contains a survey about the smallest of the exceptional Lie groups: G 2 , its definition and different characterizations as well as its relationship to the spheres S 6 and S 7 . With the exception of the summary of the Killing–Cartan classification, this survey is self-contained, and all the proofs are provided. Although these proofs are well-known, they are scattered, some of them are difficult to find, and others require stronger background, while we will usually stick to linear algebra arguments. The approach is algebraical, working at the Lie algebra level most often. We analyze the complex Lie algebra (and group) of type G 2 as well as the two real Lie algebras of type G 2 , the split and the compact one. The octonion algebra will play its role, but it is not the starting point. Also, both the 3-forms approach and the spinorial approach are viewed and connected. Special emphasis is put on relating all the viewpoints by providing precise models. [ABSTRACT FROM AUTHOR]

Let $\mathfrak {g}$ be a simple complex Lie algebra and let $\mathfrak {t} \subset \mathfrak {g}$ be a toral subalgebra of $\mathfrak {g}$ . As a $\mathfrak {t}$ -module $\mathfrak {g}$ decomposes as where $\mathfrak {s} \subset \mathfrak {g}$ is the reductive part of a parabolic subalgebra of $\mathfrak {g}$ and $\mathcal {R}$ is the Kostant root system associated to $\mathfrak {t}$ . When $\mathfrak {t}$ is a Cartan subalgebra of $\mathfrak {g}$ the decomposition above is nothing but the root decomposition of $\mathfrak {g}$ with respect to $\mathfrak {t}$ ; in general the properties of $\mathcal {R}$ resemble the properties of usual root systems. In this note we study the following problem: 'Given a subset $\mathcal {S} \subset \mathcal {R}$ , is there a parabolic subalgebra $\mathfrak {p}$ of $\mathfrak {g}$ containing $\mathcal {M} = \oplus _{\nu \in \mathcal {S}} \mathfrak {g}^{\nu }$ and whose reductive part equals $\mathfrak {s}$ ?'. Our main results is that, for a classical simple Lie algebra $\mathfrak {g}$ and a saturated $\mathcal {S} \subset \mathcal {R}$ , the condition $(\text {Sym}^{\cdot }(\mathcal {M}))^{\mathfrak {s}} = \mathbb {C}$ is necessary and sufficient for the existence of such a $\mathfrak {p}$ . In contrast, we show that this statement is no longer true for the exceptional Lie algebras F,E,E, and E. Finally, we discuss the problem in the case when $\mathcal {S}$ is not saturated. [ABSTRACT FROM AUTHOR]

By exploiting suitably constrained Zorn matrices, we present a new construction of the algebra of sextonions (over the algebraically closed field $$\mathbb {C}$$ ). This allows for an explicit construction, in terms of Jordan pairs, of the non-semisimple Lie algebra $$\mathbf {e}_{\mathbf{7} \frac{\mathbf{1}}{\mathbf{2}}}$$ , intermediate between $$\mathbf {e_7}$$ and $$\mathbf {e_8}$$ , as well as of all Lie algebras occurring in the sextonionic row and column of the extended Freudenthal Magic Square. [ABSTRACT FROM AUTHOR]

For any grading by an abelian group G on the exceptional simple Lie algebra $\mathcal {L}$ of type E or E over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple $\mathcal {L}$ -modules, as well as necessary and sufficient conditions for a finite-dimensional $\mathcal {L}$ -module to admit a G-grading compatible with the given G-grading on $\mathcal {L}$ . [ABSTRACT FROM AUTHOR]

In this paper we give an elementery construction of the Lie algebras of type E6(K) for fields K of characteristic two. [ABSTRACT FROM AUTHOR]

We recall the construction of triality automorphism of $$\mathfrak {so}(8)$$ given by E. Cartan and we give a matrix representation for the real form $$\mathfrak {so}(4,4)$$ . We compute the induced results on the characteristic classes. Paralelly we study the triality automorphism of the singularity $$D_4$$ (in Arnolds classification of smooth functions) and its miniversal deformation. The similarity with Lie theory leads us to a definition of $$G_2$$ singularity. [ABSTRACT FROM AUTHOR]

In this paper, we study the representations for the filiform Lie superalgebras L m , n , a particular class of nilpotent Lie superalgebras. We determine the minimal dimension of a faithful module over L m , n using the theory of linear algebra. In addition, using the method of Feingold and Frenkel (1985), we construct some finite and infinite dimensional modules over L m , n on the Grassmann algebra and the mixed Clifford–Weyl algebra. [ABSTRACT FROM AUTHOR]

We find a new representation of the simple Lie algebra of typeE7on the polynomial algebra in 27 variables. Using this representation and Shen's idea of mixed product, we construct a new functor fromE6-ModtoE7-Mod. A condition for the functor to map a finite-dimensional irreducibleE6-module to an infinite-dimensional irreducibleE7-module is obtained. Our general framework also gives a direct polynomial extension from irreducibleE6-modules to irreducibleE7-modules, which can be used to derive Gel'fand–Zetlin bases forE7from those forE6that can be obtained from those forD5according to our earlier work. [ABSTRACT FROM AUTHOR]

By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E into a sum of irreducible submodules. This essentially gives a partial differential equation proof of a combinatorial identity on the dimensions of certain irreducible modules of E. We also determine two three-parameter families of irreducible submodules in the solution space of Cartan's well-known fourth-order E-invariant partial differential equation. [ABSTRACT FROM AUTHOR]

Some forms of Lie algebras of typesE6,E7, andE8are constructed using the exterior cube of a rank 9 finitely generated projective module. [ABSTRACT FROM AUTHOR]