Universal central extensions of direct limits of Lie superalgebras

We show that the universal central extension of a direct limit of perfect Lie superalgebras L_i is (isomorphic to) the direct limit of the universal central extensions of L_i. As an application we describe the universal central extensions of some infinite rank Lie superalgebras

An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G --> H. Unlike the group of inner automorphisms of G itself, the group of such extended systems of automorphisms is always isomorphic to G. A similar characterization holds for inner automorphisms of an associative algebra R over a field K; here the group of functorial systems of automorphisms is isomorphic to the group of units of R modulo units of K. If one substitutes "endomorphism" for "automorphism" in these considerations, then in the group case, the only additional example is the trivial endomorphism; but in the K-algebra case, a construction unfamiliar to ring theorists, but known to functional analysts, also arises. Systems of endomorphisms with the same functoriality property are examined in some other categories; other uses of the phrase "inner endomorphism" in the literature, some overlapping the one introduced here, are noted; the concept of an inner {\em derivation} of an associative or Lie algebra is looked at from the same point of view, and the dual concept of a "co-inner" endomorphism is briefly examined. Several questions are posed.Comment: 20 pages. To appear, Publicacions Mathem\`{a}tiques. The 1-1-ness result in the appendix has been greatly strengthened, an "Overview" has been added at the beginning, and numerous small rewordings have been made throughou

Representation theory for subfactors, λ\lambda-lattices and C*-tensor categories

We develop a representation theory for $\lambda$-lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $\lambda$-lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with property (T) standard invariant and that are not constructed from property (T) groups.Comment: v3: minor changes, final version to appear in Communications in Mathematical Physics. v2: improved exposition; permanence of property (T) under quotients adde

The Group of Quasisymmetric Homeomorphisms of the Circle and Quantization of the Universal Teichm\"uller Space

In the first part of the paper we describe the complex geometry of the universal Teichm\"uller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo M\"obius transformations may be treated as a smooth part of $\mathcal T$. In the second part we consider the quantization of universal Teichm\"uller space $\mathcal T$. We explain first how to quantize the smooth part $\mathcal S$ by embedding it into a Hilbert-Schmidt Siegel disc. This quantization method, however, does not apply to the whole universal Teichm\"uller space $\mathcal T$, for its quantization we use an approach, due to Connes

Regulators of canonical extensions are torsion: the smooth divisor case

In this paper, we prove a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $>1$) are torsion, of a flat bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of Deligne's canonical extension of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion

Mutant knots and intersection graphs

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy and well known. We discuss relationship between our results and certain Lie algebra weight systems.Comment: 13 pages, many figure