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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]

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.

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]

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]

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]

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]