for X,Y ∈ ḡ, m,n ∈ Z, where X(m) = X⊗t with X ∈ ḡ and m ∈ Z and (·|·) is the normalized invariant inner product of ḡ. We identify ḡ with ḡ⊗C ⊂ g. Fix the triangular decomposition ḡ = n̄− ⊕ h̄⊕ n̄+, and the Cartan subalgebra of g as h = h̄⊕CK ⊕CD. We have h = h̄ ⊕CΛ0 ⊕Cδ, where Λ0 and δ are elements dual to K and D, respectively. Let L(λ) be the irreducible highest weight representation of g of highe...

We consider pure Yang Mills theory on the four torus. A set of non-Abelian transition functions is presented which encompass all instanton sectors. It is argued that these transition functions are a convenient starting point for gauge fixing. In particular, we give an extended Abelian projection with respect to the Polyakov loop, where A0 is independent of time and in the Cartan subalgebra. In ...

(Abstract) The ordinary Schrödinger equation with minimal coupling for a nonrelativistic electron interacting with a single-mode photon field is not satisfied by the nonrelativistic limit of the exact solutions to the corresponding Dirac equation. A Schrödinger-like equation valid for arbitrary photon intensity is derived from the Dirac equation without the weak-field assumption. The ”eigenvalu...

We give a motivic proof of a character formula for depth zero supercuspidal representations of p-adic SL(2). We begin by finding the virtual Chow motives for the character values of all depth zero supercuspidal representations of p-adic SL(2), at topologically unipotent elements. Then we find the virtual Chow motives for the values of the Fourier transform of all regular elliptic orbital integr...

The analog of the principal SO(3) subalgebra of a finite dimensional simple Lie algebra can be defined for any hyperbolic Kac Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact group SO(1, 2). We exhibit the decomposition of g(A) into representations of SO(1, 2); with the exception of the adjoint SO(1, 2) algebra itself, all of these represent...

Let A ⊇ D ⊆ B be two von Neumann algebras together with a common Cartan subalgebra. Then the amalgamated free product M = A ∗D B with respect to the unique conditional expectations from A, B onto D can be considered. In our previous paper [U1], the questions of its factoriality and type classification were discussed in detail, which will be reviewed in §4. The main purpose of the paper is to gi...

The five simple exceptional complex Lie superalgbras of vector fields are described. One of them is new; the other four are explicitely described for the first time. All of the exceptional Lie superalgebras are obtained with the help of the Cartan prolongation or a generalized prolongation. The description of several of the exceptional Lie superalgebras is associated with the Lie superalgebra a...

The current article offers a mathematical toolkit for the study of waves propagating on spacetimes with nonvanishing torsion. comprises generalized versions Lichnerowicz–de Rham and Beltrami wave operators, Weitzenböck identity relating them Riemann–Cartan geometries. construction applies to any field belonging matrix representation Lie (super) algebra containing an $$\mathfrak {so} \left( \eta...

Certain phase space path integrals can be evaluated exactly using equivariant cohomology and localization in the canonical loop space. Here we extend this to a general class of models. We consider hamiltonians which are a priori arbitrary functions of the Cartan subalgebra generators of a Lie group which is defined on the phase space. We evaluate the corresponding path integral and find that it...