Let g be a simple finite-dimensional Lie algebra and ĝκ, where κ is an invariant inner product on g, the corresponding affine Kac-Moody algebra. Consider the vacuum module Vκ over ĝκ (see Section 2 for the precise definitions). According to the results of [FF, Fr], the algebra of endomorphisms of Vκ is trivial, i.e., isomorphic to C, unless κ = κc, the critical value. The algebra Endĝκc Vκc is ...

The Lie algebra W = DerA is called the Witt algebra. It consists of “vector fields” f∂, f ∈ A. In particular, dimF W = dimF A = p. As any Lie algebra of derivations of a commutative algebra over F, W has a canonical structure of a restricted Lie algebra. Recall that a restricted Lie algebra is a Lie algebra over F with an additional unary (in general, non-linear) operation g 7→ g satisfying the...

We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice Our interpretation seems computationally more direct than the one based on G odel s Dialectica interpretation Interestingly this interpretation uses a re nement of the realizibility semantics of the absurdity proposition which is not interpreted as the empty type here We also s...

in this paper, we introduce a notion of property (t) for a c∗-dynamical system (a, g, α) consisting of a unital c∗-algebra a,a locally compact group g, and an action α on a. as a result,we show that if a has strong property (t) and g has kazhdan’sproperty (t), then the triple (a, g, α) has property (t).

in this paper we initiate the study of real group algebras and investigate some of its aspects.let l1 (g) be a group algebra of a locally compact group g,τ :g →g be a group homeomorphismsuch that τ 2 =τοτ = 1, the identity map, and lp (g,τ ) = { f ∈ lp (g) : fοτ = f } ( p ≥ 1) . in thispaper, among other results, we clarify the structure of lp (g,τ ) and characterize amenability ofl1 (g,τ ) and...

we construct a family of two parameter quantum grou-ps‎ ‎$u_{r,s}(mathfrak{g})$ associated with a generalized kac-moody‎ ‎algebra corresponding to symmetrizable admissible borcherds cartan‎ ‎matrix‎. ‎we also construct the $textbf{a}$-form $u_{textbf{a}}$ and‎ ‎the classical limit of $u_{r,s}(mathfrak{g})$‎. ‎furthermore‎, ‎we‎ ‎display the equitable presentation for a subalgebra‎ ‎$u_{r,...

Gödel’s famous incompleteness theorems (there are two of them) concern the ability of a formal system to state and derive all true statements, and only true statements, in some fixed domain; and concern the ability of logic to determine if a formal system has that property. They were developed in the early 1930s. Very loosely, the first theorem says that in any “sufficiently rich” formal proof ...

An MV-pair is a pair (B, G) where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain conditions. Let ∼G be the equivalence relation on B naturally associated with G. We prove that for every MV-pair (B, G), the effect algebra B/ ∼G is an MV-effect algebra. Moreover, for every MV-effect algebra M there is an MV-pair (B, G) such that M is isomorphic to B/ ∼G.

This paper investigates (modal) extensions of Heyting-Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We rst develop matrix as well as Kripke style semantics for those logics. Then, by extending the Godel-embedding of intuitionistic logic into S4, it is shown that all (modal) extensions of Heyting...