Towards classification of simple finite dimensional modular Lie superalgebras

A way to construct (conjecturally all) simple finite dimensional modular Lie (super)algebras over algebraically closed fields of characteristic not 2 is offered. In characteristic 2, the method is supposed to give only simple Lie (super)algebras graded by integers and only some of the non-graded ones). The conjecture is backed up with the latest results computationally most difficult of which are obtained with the help of Grozman's software package SuperLie.Comment: 10 page

Real Simple Lie Algebras: Cartan Subalgebras, Cayley Transforms, and Classification

The differential geometry software package in Maple has the necessary tools and commands to automate the classification process for complex simple Lie algebras. The purpose of this thesis is to write the programs to complete the classification for real simple Lie algebras. This classification is difficult because the Cartan subalgebras are not all conjugate as they are in the complex case. For the process of the real classification, one must first identify a maximally noncompact Cartan subalgebra. The process of the Cayley transform is used to find this specific Cartan subalgebra. This Cartan subalgebra is used to find the simple roots for the given real simple Lie algebra. With this information, we can then create a Satake diagram. Then we match our given algebra\u27s Satake diagram to a Satake diagram of a known algebra. The programs explained in this thesis complete this process of classification

Four-level and two-qubit systems, sub-algebras, and unitary integration

Four-level systems in quantum optics, and for representing two qubits in quantum computing, are difficult to solve for general time-dependent Hamiltonians. A systematic procedure is presented which combines analytical handling of the algebraic operator aspects with simple solutions of classical, first-order differential equations. In particular, by exploiting $su(2) \oplus su(2)$ and $su(2) \oplus su(2) \oplus u(1)$ sub-algebras of the full SU(4) dynamical group of the system, the non-trivial part of the final calculation is reduced to a single Riccati (first order, quadratically nonlinear) equation, itself simply solved. Examples are provided of two-qubit problems from the recent literature, including implementation of two-qubit gates with Josephson junctions.Comment: 1 gzip file with 1 tex and 9 eps figure files. Unpack with command: gunzip RSU05.tar.g

Ternary "Quaternions" and Ternary TU(3) algebra

To construct ternary "quaternions" following Hamilton we must introduce two "imaginary "units, $q_1$ and $q_2$ with propeties $q_1^n=1$ and $q_2^m=1$. The general is enough difficult, and we consider the $m=n=3$. This case gives us the example of non-Abelian groupas was in Hamiltonian quaternion. The Hamiltonian quaternions help us to discover the $SU(2)=S^3$ group and also the group $L(2,C)$. In ternary case we found the generalization of U(3) which we called $TU(3)$ group and also we found the the SL(3,C) group. On the matrix language we are going from binary Pauly matrices to three dimensional nine matrices which are called by nonions. This way was initiated by algebraic classification of $CY_m$-spaces for all m=3,4,...where in reflexive Newton polyhedra we found the Berger graphs which gave in the corresponding Cartan matrices the longest simple roots $B_{ii}=3,4,..$ comparing with the case of binary algebras in which the Cartan diagonal element is equal 2, {\it i.e. } $A_{ii}=2$

Logarithmic W-algebras and Argyres-Douglas theories at higher rank

Families of vertex algebras associated to nilpotent elements of simply-laced Lie algebras are constructed. These algebras are close cousins of logarithmic W-algebras of Feigin and Tipunin and they are also obtained as modifications of semiclassical limits of vertex algebras appearing in the context of $S$-duality for four-dimensional gauge theories. In the case of type $A$ and principal nilpotent element the character agrees precisely with the Schur-Index formula for corresponding Argyres-Douglas theories with irregular singularities. For other nilpotent elements they are identified with Schur-indices of type IV Argyres-Douglas theories. Further, there is a conformal embedding pattern of these vertex operator algebras that nicely matches the RG-flow of Argyres-Douglas theories as discussed by Buican and Nishinaka.Comment: Comments are welcom