Let Φ be a simply-laced root system with simple roots ∆ = {αi : i ∈ I} embedded in the real vector space V with inner product 〈·,·〉. For convenience, we may assume that the roots have been normalized so that 〈α, α〉 = 2 for all α ∈ Φ. As a second convenience, we may assume that V has been enlarged, if necessary, so that 〈·,·〉 is nondegenerate on V . Now let σ be a diagram automorphism of (Φ,∆). ...

Let X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π1(X). This group π1(X) is a profinite group which is uniquely determined (up to inner automorphisms) by the property that the category of finite, discrete sets equipped with a continuous π1(X)-action is equivalent to the category of finite étale coverings of X. Moreover, the assignment...

We’ll now turn to a topic which is a precise analog of the previous discussion of the Clifford algebra and spinor representations. By replacing the symmetric two-form (the inner product) in the earlier discussion by an antisymmetric two-form, we get a new algebra, the Heisenberg algebra. The group of automorphism of this algebra is now a symplectic group, and we again get a projective represent...

Abstract We investigate charge-parity (CP) and non-CP outer automorphisms of groups the transformation behavior group representations under them. identify situations where composite elementary states that transform in exactly same representation group, differently automorphisms. This can be instrumental discriminating from solely by their quantum numbers with respect to automorphism, i.e. withou...

1.1. Conventions used throughout. Symmetrisers and antisymmetrisers of tensors are denoted as usual using parentheses and square brackets, so, for example, C(ab) = 1 2 (Cab + Cba), C[ab] = 1 2 (Cab − Cba), and Cab = C(ab) + C[ab]. If G is a Lie group, L(G) denotes its Lie algebra. Ad(g) is the adjoint representation of G on L(G), defined as the derivative of the inner automorphism of G, g′ 7→ I...

We describe some of the geometric properties of the automorphism group Aut(F ) of Thompson’s group F . We give realizations of Aut(F ) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F ) and their metric properties. In particular, we show that the subg...

In 2003, Araujo and Jarosz showed that every bijective linear map θ : A → B between unital standard operator algebras preserving zero products in two ways is a scalar multiple of an inner automorphism. Later in 2007, Zhao and Hou showed that similar results hold if both A,B are unital standard algebras on Hilbert spaces and θ preserves range or domain orthogonality. In particular, such maps are...

The (real) GraviGUT algebra is an extension of the spin(11, 3) algebra by a 64dimensional Lie algebra, but there is some ambiguity in the literature about its definition. Recently, Lisi constructed an embedding of the GraviGUT algebra into the quaternionic real form of E8. We clarify the definition, showing that there is only one possibility, and then prove that the GraviGUT algebra cannot be e...

In this paper, we generalize the Penrose twistor theory to a Clifford algebra. This allows basic geometric forms and relationships to be expressed purely algebraically. In addition, by means of an inner automorphism of this algebra, it is possible to regard these forms and relationships as emerging from a deeper pre-space, which we are calling an implicate order. The way is then opened up for a...

Let F be a free Lie algebra of finite rank over a field K. We prove that if an ideal [Formula: see text] of the algebra [Formula: see text] contains a primitive element [Formula: see text] then the element [Formula: see text] is primitive. We also show that, in the Lie algebra [Formula: see text] there exists an element [Formula: see text] such that the ideal [Formula: see text] contains a prim...