Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid II1 factor M containing an “exotic” maximal abelian subalgebra A: as an A,A-bimodule, L(M) is neither coarse nor discrete. Thus we show that there exist II1 factors with such property but without Cartan subalgebras. It also follows from Voiculescu’s free entropy results that M is not an inter...

We construct irreducible modules Vα, α ∈ C over W3 algebra with c = −2 in terms of a free bosonic field. We prove that these modules exhaust all the irreducible modules of W3 algebra with c = −2. Highest weights of modules Vα, α ∈ C with respect to the full (two-dimensional) Cartan subalgebra of W3 algebra are ( 1 2α(α − 1), 16α(α − 1)(2α − 1) ) . They are parametrized by points (t, w) on a rat...

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 ...

We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact non-elementary convergence and lattices in non-compact semi-simple Lie but excludes inner amenable groups. show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions groups this have at most one weakly compact Cartan subalgebra, up to u...

In a previous work [AS2] we showed how to attach to a pointed Hopf algebra A with coradical kΓ, a braided strictly graded Hopf algebra R in the category Γ Γ YD of Yetter-Drinfeld modules over Γ. In this paper, we consider a further invariant of A, namely the subalgebra R of R generated by the space V of primitive elements. Algebras of this kind are known since the pioneering work of Nichols. It...

We construct a class of non-weight modules over the twisted N=2 superconformal algebra L. Let h=CL0⊕CG0 be Cartan subalgebra L, and let t=CL0 even part L0¯. These L when restricted to h are free rank 1 or t 2. provide necessary sufficient conditions for these simple, give two such L-modules isomorphic. also compute action an automorphism on them. Moreover, based weighting functor introduced in ...

Examples of such algebras are subspaces of associative algebras of characteristic p^O which are closed under the Lie multiplication [ab] =ab — ba and under pih powers. Then one may take a[pl =ap. It is known that every restricted Lie algebra is isomorphic to one of this type. For this reason we may simplify our notation in the sequel and write a" for alp]. We call the mapping a—>ap the p-operat...

We describe four types of inner involutions of the Cartan-Weyl basis providing (for |q| = 1 and q real) three types of real quantum Lie algebras: Uq(O(3, 2)) (quantum D = 4 anti-de-Sitter), Uq(O(4, 1)) (quantum D = 4 de-Sitter) and Uq(O(5)). We give also two types of inner involutions of the Cartan-Chevalley basis of Uq(Sp(4;C)) which can not be extended to inner involutions of the Cartan-Weyl ...

Examples of such algebras are subspaces of associative algebras of characteristic p^O which are closed under the Lie multiplication [ab] =ab — ba and under pih powers. Then one may take a[pl =ap. It is known that every restricted Lie algebra is isomorphic to one of this type. For this reason we may simplify our notation in the sequel and write a" for alp]. We call the mapping a—>ap the p-operat...

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification g = (H ⊗ U(g), [ , ]gH ) of g is defined by the bracket [ z ⊗ X , w ⊗ Y ] gH = (z · w) ⊗ (XY ) − (w · z) ⊗ (Y X) , for z, w ∈ H and the basis vectors X and Y of U(g). Let SH be the ( non-commutative) algebra of H-valued smooth mappings over S and let Sg = SH ⊗ U(g). ...