The Adapted Ordering Method for the Representation Theory of Lie Algebras and Superalgebras and their Generalizations

The Adapted Ordering Method for Lie Algebras and Superalgebras and their Generalizations

In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras in two dimensions. It allows: to determine maximal dimensions for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this article we present the Adapted Ordering Method for general Lie algebras and superalgebras, and their generalizations, provided they can be triangulated. We also review briefly the results obtained for the Virasoro algebra and…

The Adapted Ordering Method in Representation Theory

In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras. This method, which proves to be very powerful, can be applied to most algebras and superalgebras, however. It allows: to determine maximal dimensions for a given type of singular vector space, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this article we present the Adapted Ordering Method for general algebras and superalgebras which admit a triangulation and review briefly the res…

Differential operator realizations of superalgebras and free field representations of corresponding current algebras

Based on the particular orderings introduced for the positive roots of finite dimensional basic Lie superalgebras, we construct the explicit differential operator representations of the $osp(2r|2n)$ and $osp(2r+1|2n)$ superalgebras and the explicit free field realizations of the corresponding current superalgebras $osp(2r|2n)_k$ and $osp(2r+1|2n)_k$ at an arbitrary level $k$. The free field representations of the corresponding energy-momentum tensors and screening currents of the first kind are also presented.

Functional Representations for Fock Superalgebras

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite dimensional superspaces, and construct superanalogs of the classical function spaces with a reproducing kernel -- including the Bargmann-Fock representation -- and of the Wiener-Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein-Uhlenbeck semigroup on the Fock space.

Formulation of Supersymmetry on a Lattice as a Representation of a Deformed Superalgebra

The lattice superalgebra of the link approach is shown to satisfy a Hopf algebraic supersymmetry where the difference operator is introduced as a momentum operator. The breakdown of the Leibniz rule for the lattice difference operator is accommodated as a coproduct operation of (quasi)triangular Hopf algebra and the associated field theory is consistently defined as a braided quantum field theory. Algebraic formulation of path integral is perturbatively defined and Ward-Takahashi identity can be derived on the lattice. The claimed inconsistency of the link approach leading to the ordering ambi…

$\frac{\mathbb{Z}}{2}$-分次亚纯开弦顶点代数及其 $\mathbb{Z}_2$-扭曲模的费米子构造，I

Fermionic construction of the $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra and its $\mathbb{Z}_2$-twisted module, I

本文定义了$\frac{\mathbb{Z}}{2}$-分次亚纯开弦顶点代数，基于非反交换Fock空间构造了自由费米子的非交换推广，并利用正规序和Wick定理证明了弱结合性及关联函数的闭合公式。