Indecomposables in the composition algebra of the kronecker algebra

Let; K be the Kronecker algebra; M a K-indecomposable; C(K) the composition algebra of K. By using the triangular decomposition of C(K) we prove that [M] ? C(K) if and only if M is either a preprojective or a preinjective.     

On the algebra of quasi-shuffles

For any commutative algebra R the shuffle product on the tensor module T(R) can be deformed to a new product. It is called the quasi-shuffle algebra, or stuffle algebra, and denoted T ^q (R). We show that if R is the polynomial algebra, then T ^q (R) is free for some algebraic structure called Commutative TriDendriform (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads (As, CTD, Com) analogous to (Com, As, Lie). In the last part we give a similar interpretation of the quasi-shuffle algebra in the noncommutative setting.     

Automorphisms of the algebra ofq-difference operators

In this paper, automorphisms of the algebra ofq-difference operators, as an associative algebra for arbitraryq and as a Lie algebra forq being not a root of unity, are determined. Project supported by the NNSF of China     

Planar algebra of the subgroup-subfactor

We give an identification between the planar algebra of the subgroupsubfactor R ⋊ H ⊂ R ⋊ G and the G-invariant planar subalgebra of the planar algebra of the bipartite graph ★ _n , where n = [G: H]. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant of R ⋊ H ⊂ R ⋊ G in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor R ^G ⊂ R ^H and the G-invariant planar subalgebra of the planar algebra of the ‘flip’ of ★ _n .     

An algebra containing the two-sided convolution operators

We present an intrinsically defined algebra of operators containing the right and left invariant Calderón–Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<∞). This algebra provides an example of an algebra of singular integrals that falls outside of the classical Calderón–Zygmund theory.     

Basis for identities of the algebra of simplified insertion

We prove that all identities of the algebra of simplified insertion on countably many generators over a field of characteristic zero follow from the right-symmetric identity. We prove that the bases of the free special Jordan algebra and the special algebra of simplified insertion coincide. We construct an infinite series of relations in the algebra of simplified insertion which hold for the words of length k, k ε ℕ.     

Some categories associated with bases of the kalman algebra

We investigate relationships in the Kalman algebra, viewed as an algebra over the algebra induced by the coefficients of the characteristic polynomial of the state matrix. To this end we introduce the categories BSS \mathcal{B}\mathcal{S}\mathcal{S} and BSS^\textstrict \mathcal{B}\mathcal{S}{\mathcal{S}^{\text{strict}}} associated with relationships in some basis of the Kalman algebra and also with reconstruction of relationships from known fragments. On these categories we construct the structures of the symmetrical monoidal category induced by addition and multiplication in the Kalman algebra. We investigate the properties of some of the most important classes of morphisms, in particular, we describe the structure and the action of the automorphism group.     

The non-commutative Weil algebra

For any compact Lie group G, together with an invariant inner product on its Lie algebra ?, we define the non-commutative Weil algebra ? _G as a tensor product of the universal enveloping algebra U(?) and the Clifford algebra Cl(?). Just like the usual Weil algebra W _G =S(?^*)⊗∧?^*, ? _G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋ _G (B) for any G-differential algebra B. We construct an explicit isomorphism ?: W _G →? _G of the two Weil algebras as G-differential spaces, and prove that their multiplication maps are G-chain homotopic. This implies that the map in cohomology H _G (B)→ℋ _G (B) induced by ? is a ring isomorphism. For the trivial G-differential algebra B=ℝ, this reduces to the Duflo isomorphism S(?) ^G ≅U(?) ^G between the ring of invariant polynomials and the ring of Casimir elements. Oblatum 13-III-1999 & 27-V-1999 / Published online: 22 September 1999     

An algebra dense in its tidle algebra

We construct a closed setE in the circle such thatA(E) is dense, in but not equal to, its tilde algebra.     

On derivations of the heisenberg algebra

Derivations of the Heisenberg algebraH and some related questions are studied. The ideas and the language of formal differential geometry are used. It is proved that all derivations of this algebra are inner. The main subalgebras of the Lie algebraD(H) of all derivations ofH are distinguished, and their properties are studied. It is shown that the algebraH interpreted as a Lie algebra (with the commutator as the Lie bracket) forms a one-dimensional central extension ofD(H). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118. No. 2 pp. 163–189, February, 1999.     

Unitary representations of the generalized Virasoro current algebra

We investigate Verma modules V over the generalized Virasoro current algebrag, which is the semidirect sum of the Virasoro algebra and the central extension of a commutative algebra. It is shown that an arbitrary unitary representation with highest weight of algebrag is isomorphic to the tensor product of a unitary Fock representation ofg (or of a one-dimensional representation ofg) and a unitary representation with highest weight of the Virasoro algebra (considered as a representation of algebrag). This result is used to obtain formulas for the determinants of the matrices defining the Shapovalov form on Verma module V.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 532–538, April, 1990.     

Lorentz transforms of the invariant Dirac algebra

The Dirac equation, as a 4×4-hyperbolic system on ³, possesses an invariant algebra of global pseudodifferential operators-in the sense that conjugation with the Dirac time propagator leaves the algebra invariatn (cf. [CX]. Chapter 10). In this paper we examine the relation between the two invariant algebras att=0 and att'=0 when (t,x) and (t',x') are coordinates of Minkowsky space related by a (proper) Lorentz transform. For vanishing electromagnetic potentials these algebras are transforms of each other by the implied change of dependent and independent variables. In the general case such a space-time transform will make the potentials time dependent, hence also the algebra dependent on the initial plane.     

Full dualisability is independent of the generating algebra

Given a full duality based on a finite algebra M, we show how to create a full duality based on any other finite algebra N for which \mathbbISP(M) = \mathbbISP(N){\mathbb{ISP}({\bf M}) = \mathbb{ISP}({\bf N})}. So the full dualisability of a quasivariety is independent of the algebra chosen as the generator. We obtain this result by proving the corresponding result for multisorted full dualities.     

Irreducible representations over the diamond Lie algebra

In this paper, we use Block’s results to classify irreducible modules over the diamond Lie algebra 𝔇. As a corollary, we also give a classification of irreducible modules over the Euclidean algebra 𝔢(2).     

Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

Let M be a type I von Neumann algebra with the center Z and let LS(M) be the algebra of all locally measurable operators affiliated with M. We prove that every Z-linear derivation on LS(M) is inner. In particular, all Z-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements of LS(M). The text was submitted by the authors in English.     

On the quiver with relations of a quasitilted algebra and applications

In this paper we discuss, in terms of quiver with relations, su?cient and necessary conditions for an algebra to be a quasitilted algebra. We start with an algebra with global dimension at most two and we give a su?cient condition to be a quasitilted algebra. We show that this condition is not necessary. In the case of a strongly simply connected schurian algebra, we discuss necessary conditions, and combining both types of conditions, we are able to analyze if some given algebra is quasitilted. As an application we obtain the quiver with relations of all the tilted and cluster tilted algebras of Dynkin type E_p.     

IntegrableN-dimensional systems on the Hopf algebra andq-deformations

We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative integration of linear differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 373–390, September, 2000     

Weakly almost periodic functionals on the measure algebra

It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C*-algebra. This implies that the weakly almost periodic functionals on M(G), the measure algebra of a locally compact group G, is a C*-subalgebra of M(G)* = C ₀(G)**. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.