A semi-finite algebra associated to a subfactor planar algebra

We canonically associate to any planar algebra two type II_∞ factors M_±. The subfactors constructed previously by the authors in Guionnet et al. (2010) [6] are isomorphic to compressions of M_± to finite projections. We show that each M_± is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M₊≅M₋ is the amplification a free group factor on a finite number of generators. As an application, we show that the factors Mj constructed in Guionnet et al. (in press) [6] are isomorphic to interpolated free group factors L(F(rj)), rj=1+2δ−2j(δ−1)I, where δ² is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.     

A non-dualisable entropic algebra

We give an example of a five-element unary algebra which is entropic but does not admit a natural duality. Received September 26, 2000; accepted in final form April 4, 2001.     

A quantum octonion algebra

Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group (D) of D, we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of (D). The product in the quantum octonions is a (D)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at new ``representation theory' proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz algebra.

     

A certain functional algebra

We study the algebra of functions on the set of natural numbers with respect to the generalized convolution, generated by the generalized translation operator T_nf(k)=f(max (n, k)), n, k. With the help of the generalized Fourier transform, connected with this convolution, we establish numerous identities and recurrence relations, connecting, in particular, sums of powers of natural numbers.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 372–377, March, 1990.     

A modified Brauer algebra as centralizer algebra of the unitary group

The centralizer algebra of the action of on the real tensor powers of its natural module, , is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for and with the decomposition of into irreducible submodules is considered.

     

A specialization of prinjective Ringel-Hall algebra and the associated Lie algebra

In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives us a generalisation of Serre relations for semisimple Lie algebras. Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.     

A remarkable homogeneous Banach algebra

We give an example of a strongly homogeneous Banach algebraB withA(T)⊊B⊊C(T) on which all 1-Lipschitzian functions operate; this improves previous results of M. Zafran.     

A distinguished real Banach algebra

We present a new and elementary approach to characterize the maximal ideals and their associated multiplicative linear functionals for a classical real Banach algebra of analytic functions.     

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A ₁ is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.     

A very rigid Boolean algebra

A Boolean algebra is constructed having only those endomorphisms corresponding to prime ideals, which are present in any BA. The BA constructed is of powerc, has 2 ^c endomorphisms, and is not rigid in Bonnet’s sense.     

A quasi-Hopf algebra freeness theorem

We prove the quasi-Hopf algebra version of the Nichols-Zoeller theorem: A finite dimensional quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

     

A Monster Lie algebra?

This paper defines a remarkable Lie algebra of infinite dimension and rank, and conjectures that it may be related to the Fischer-Griess Monster group.     

一类推广的Virasoro-like李代数

构造了一类无限维李代数,它是Virasoro-like李代数的推广.研究了这类李代数的两类自同构,这两类自同构均关于映射的合成构成自同构群,一类同构于对称群S3,另一类同构于Klein交换群.得到了这类李代数一些特殊的自同态、中心.证明了这类李代数不是半单李代数.     

A crossed product of the canonical anticommutation relations algebra in the Cuntz algebra

We show that the Cuntz algebra can be represented as a crossed product of the canonical anticommutation relations algebra by an endomorphism.