Related papers: The ring of Weyl invariant $E_8$ Jacobi forms
We study the ring of Weyl invariant $E_8$ weak Jacobi forms. Wang recently proved that the ring is not a polynomial algebra. We consider a polynomial algebra which contains the ring as a subset and clarify the necessary and sufficient…
In 1992 Wirthm\"{u}ller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a…
We investigate the Jacobi forms for the root system $E_8$ invariant under the Weyl group. This type of Jacobi forms has significance in Frobenius manifolds, Gromov--Witten theory and string theory. In 1992, Wirthm\"{u}ller proved that the…
We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n…
The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an…
We prove the polynomiality of the bigraded ring $J_{*,*}^{w, W}(F_4)$ of weak Jacobi forms for the root system $F_4$ which are invariant with respect to the corresponding Weyl group. This work is a continuation of the joint article with…
We study a ring of weak Jacobi forms indexed by integral lattices of rank two. We find an explicit finite set of generators of this ring and give a dimension formula for weak Jacobi forms of rank two lattice index.
We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of $(0,2)$-model in string…
In this paper, we prove that the ring of polynomial invariants of the Weyl group for an indecomposable and indefinite Kac-Moody Lie algebra is generated by invariant symmetric bilinear form or is trivial depending on $A$ is symmetrizable or…
$D_4$ triality invariants are modular forms as well as polynomial invariants for a fiber product of the modular group and the Weyl group of type $F_4$. We show that the ring of $D_4$ triality invariants satisfying a certain cusp condition…
In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by…
In the paper we study two types of relations: a one is between the elliptic genus of Calabi-Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac-Moody…
We construct a tower of arithmetic generators of the bigraded polynomial ring J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This tower…
In this paper we study Jacobi forms associated with the Leech lattice $\Lambda$ which are invariant under the Conway group $\mathrm{Co}_0$. We determine and construct generators of modules of both weak and holomorphic Jacobi forms of…
We show that the higher-order Weyl algebras over a field of characteristic zero, which are formally rigid as associative algebras, can be formally deformed in a nontrivial way as hom-associative algebras. We also show that these…
This paper introduces and systematically studies Weyl-type, Witt-type, and non-associative algebras defined over expolynomial rings -- commutative rings generated by exponential functions $e^{\alpha x}$, exponentials of exponentials $e^{\pm…
We give an explicit affine algebraic variety whose coordinate ring is isomorphic (as an algebra with the action of the Weyl group) with the equivariant cohomology of some Springer fibers.
We show that the convolution algebra $K^G(\mathcal{B} \times \mathcal{B})$ is isomorphic to the Based ring of the lowest two-sided cell of the extended affine Weyl group associated to $G$, where $G$ is a connected reductive algebraic group…
An explicit form of the generators of quantum and ordinary semisimple algebras for an arbitrary finite-dimensional representation is found. The generators corresponding to the simple roots are obtained in terms of a solution of a system of…
A minimal system of homogeneous generating elements of the algebra of covariants for the binary form of degree 8 is calculated.