Action Of Lie Algebra Research Articles

On the Mathematical Characterization of the Breaking of Hadronic Internal Symmetry

The adjoint action of a semisimple complex Lie algebra G singles out some singular directions with special properties. The directions of breaking of the hadronic internal symmetry groups, e.g., SU(3) or SU(3) × SU(3), by the electromagnetic, semileptonic, and nonleptonic weak interactions, are of this type. They are Hermitian (Q,Y) or nilpotent (Cabibbo direction) elements in the corresponding complexified Lie algebra. Their mathematical characterization is provided by the adjoint action of G. In particular, they satisfy a generalized extremal property for every G−invariant polynomial.

The actions of infinite-dimensional Lie algebras

The actions of infinite-dimensional Lie algebras

Instantons and affine algebras I: The Hilbert scheme and vertex operators

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises work of Nakajima, who considered the case when $X$ is an ALE space. In particular, this describes the combinatorial complexity of the moduli space as being precisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i\.e\., the cohomology of a determinant line bundle on the moduli space) generalising the ``insertion'' and ``deletion'' operations of conformal field theory, and indeed on any cohomology theory. In the particular case of $U(1)$-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac. Then the well known quadratic nature of $ch_2$, $$ch_2 = \frac{1}{2} c_1\cdot c_1 - c_2 $$ becomes precisely the formula for the eigenvalue of the degree operator, i\.e\. the well known quadratic behaviour of affine Lie algebras.