Abstract We show that the Kazhdan–Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure and its full subcategory $\mathcal{O}_k^{fin}$ objects semisimple Cartan subalgebra actions is a subcategory. every simple $\widehat{\mathfrak{gl...

In this paper we consider the product of two complex Cartan manifolds, the outcome being a class of product complex Cartan spaces. Then, we study the relationships between the geometric objects of a product complex Cartan space and its components, (e.g. Chern-Cartan complex nonlinear connection, Cartan tensors). By means of these, we establish the necessary and sufficient conditions under which...

The classification of the primitive transitive and effective actions of Lie groups on manifolds is a problem dating back to Lie. The classification of the infinite dimensional infinitesimal actions was originally done by Cartan [3] and was made rigorous by some joint work of Guillemin, Quillen and Sternberg [8], whose proof was further simplified by Guillemin [7] recently by using some results ...

Let S(g ) be the symmetric algebra over the complexification 9 of the real semisimple Lie algebra g. For u £ S(g ), d(u) is the corresponding differential operator on g. 3)(g) denotes the algebra generated by d(S(g )) and multiplication by polynomials on g . For any open set U C t, Diff (LO is the algebra of differential operators with C°°-coefficients on U. Let t be a Cartan subalgebra of g, f...

In this work, we study direct limits of finite dimensional basic classical simple Lie superalgebras and obtain the conjugacy classes of Cartan subalgebras under the group of automorphisms.

Recent work has shown that every 3D root system allows the construction of a correponding 4D via an `induction theorem'. In this paper, we look at icosahedral case $H_3\rightarrow H_4$ in detail and perform calculations explicitly. Clifford algebra is used to group theoretic based on versor theorem Cartan-Dieudonn\'e theorem, giving simple Pin Spin covers. Using connection with $H_3$ induction ...

This paper is concerned with the study of properties exact solution fundamental integrable G2 vertex model. The model R-matrix and respective spin chain are presented in terms basis generators Lie algebra. formulation permits us to relate number Bethe roots equations eigenvalues U(1) conserved charges from Cartan subalgebra G2. solved by a peculiar string structure which combines complex three-...