The Space Groups of Two Dimensional Minkowski Space

1978 ◽  
Vol 30 (5) ◽  
pp. 1103-1120
Author(s):  
George Maxwell
Keyword(s):  
Vector Space ◽  
Minkowski Space ◽  
Bilinear Form ◽  
Orthogonal Group ◽  
Affine Space ◽  
Symmetric Bilinear Form ◽  
Two Dimensional ◽  
Space Groups ◽  
Dimensional Minkowski Space ◽  
Real Affine

LetEbe an n-dimensional real affine space,Vits vector space of translations andA(E)the affine group ofE.Suppose that (. , .) is a nondegenerate symmetric bilinear form on F of signature(n —1, 1), O(V) its orthogonal group andS(V)its group of similarities.

The two-dimensional minkowski space fermion determinant by example

1988 ◽  
Vol 211 (1-2) ◽  
pp. 107-110 ◽  
Author(s):  
D. Cangemi ◽  
M. Makowka ◽  
G. Wanders
Keyword(s):  
Minkowski Space ◽  
Two Dimensional ◽  
Dimensional Minkowski Space

On the crystallography of infinite Coxeter groups

1977 ◽  
Vol 82 (1) ◽  
pp. 13-24 ◽  
Author(s):  
George Maxwell
Keyword(s):  
Affine Space ◽  
Coxeter Groups ◽  
Root Systems ◽  
Space Groups ◽  
Coxeter Graph ◽  
Finite Dimensional ◽  
Multipartite Graphs ◽  
Real Affine ◽  
Point Groups ◽  
Complete Multipartite

Let V be the vector space of translations of a finite dimensional real affine space. The principal aim of this paper is to study (generally non-Euclidean) space groups whose point groups K are ‘linear’ Coxeter groups in the sense of Vinberg (4). This involves the investigation of lattices Λ in V left invariant by K and the calculation of cohomology groups H1(K, V/Λ) (3). The first problem is solved by generalizing classical concepts of ‘bases’ of root systems and their ‘weights’, while the second is carried out completely in the case when the Coxeter graph Γ of K contains only edges marked by 3. An important part in the calculation of H1(K, V/Λ) is then played by certain subgraphs of Γ which are complete multipartite graphs. The only subgraphs of this kind which correspond to finite Coxeter groups are of type Al× … × A1, A2, A3 or D4. This may help to explain why, in our earlier work on space groups with finite Coxeter point groups (3), (2), components of r belonging to these types played a rather mysterious exceptional role.

SUPERGRAVITY AND SUPERSTRINGS IN (2+1) DIMENSIONS

1991 ◽  
Vol 06 (29) ◽  
pp. 5215-5229 ◽  
Author(s):  
A. STERN
Keyword(s):  
Minkowski Space ◽  
Current Algebra ◽  
Central Extension ◽  
Loop Group ◽  
Unitary Representations ◽  
Two Dimensional ◽  
Induced Representations ◽  
Highest Weight ◽  
Dimensional Minkowski Space ◽  
State Construction

We study n=1 supergravity written on a two-dimensional disc (× time) in the absence of any sources. The dynamics of the boundary of the disc is equivalent to that of a superstring in (2+1)-dimensional Minkowski space. The relevant current algebra for the theory corresponds to the central extension of the super-Poincaré loop group. We find unitary representations of the current algebra by applying both a highest weight state construction and the method of induced representations.

scholarly journals On the Index of a Quadratic Form

1958 ◽  
Vol 1 (3) ◽  
pp. 180-180
Author(s):  
Jonathan Wild
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Bilinear Form ◽  
Direct Proof ◽  
Symmetric Bilinear Form ◽  
Arbitrary Field ◽  
Analytic Geometry ◽  
Fundamental Fact

Given a vector space V = {x, y, ...} over an arbitrary field. In V a symmetric bilinear form (x,y) i s given. A subspace W is called totally isotropic [t.i.] if (x,y) = 0 for every pair x W, y W.Let Vn and Vm be two t.i. subspaces of V; n < m. Lower indices always indicate dimensions. It is a well known and fundamental fact of analytic geometry that there exists a t.i. subspace Wm of V containing Vn [cf. Dieudonné: Les Groupes classiques , P. 18]. As no simple direct proof seems to be available, we propose to supply one.

Clifford and Heisenberg algebras

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Bilinear Form ◽  
Scalar Product ◽  
Heisenberg Algebra ◽  
Symmetric Bilinear Form ◽  
Rham Complex ◽  
Heisenberg Algebras ◽  
De Rham ◽  
Symplectic Vector Space

This chapter recalls various results on Clifford algebras and Heisenberg algebras. It first introduces the Clifford algebra of a vector space V equipped with a symmetric bilinear form B and then specializes the construction of the Clifford algebra to the case of V ⊕ V*. Next, the chapter argues that, if (V,ω‎) is a symplectic vector space, then the associated Heisenberg algebra is constructed and then specialized to the case of V ⊕ V*. Hereafter, the chapter considers the combination of the Clifford and Heisenberg algebras for V ⊕ V*, and constructs the complex Λ‎· (V*) ⊗ S· (V*), ̄ƌ) which is the subcomplex of polynomial forms in the de Rham complex. Finally, when V is equipped with a scalar product, this complex is related to a Witten complex over V.

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