Direct Sum Of Irreducible Modules Research Articles

Module structure of the center of the universal central extension of a genus zero Krichever–Novikov algebra

We describe how the center of the universal central extension of the genus zero Krichever–Novikov current algebra decomposes as a direct sum of irreducible modules for automorphism group of the coordinate ring of this algebra.

Lie algebras with S3- or S4-action and generalized Malcev algebras

Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In the case of S3-symmetry, the Lie algebras are coordinatized by some non-associative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary product, and include both the Malcev algebras and the Jordan triple systems.

Decomposition of the lattice vertex operator algebra [formula omitted

Motivated by the work of Dong et al. [Associative subalgebras of Griess algebra and related topics, in: J. Ferrar, K. Harada (Eds.), Proc. Conf. Monster and Lie Algebras, de Gruyter, Berlin, 1998], we study a decomposition of the lattice vertex operator algebra V 2 A l as a direct sum of irreducible modules of a certain tensor product of Virasoro vertex operator algebras and a parafermion algebra W l+1 (2 l/( l+3)). We find that the vertex operator algebra V 2 A l contains a subalgebra isomorphic to a parafermion algebra W l+1 (2 l/( l+3)) of central charge 2 l/( l+3). A complete decomposition of the vertex operator algebra V 2 A l as a direct sum of irreducible modules of W=L(c 1,0)⊗L(c 2,0)⊗⋯⊗L(c l,0)⊗W l+1(2l/(l+3)) , where c i , i=1,…, l, is given by the discrete series c i =1−6/( i+2)( i+3), is also obtained.

Boundary states, extended symmetry algebra, and module structure for certain rational torus models

The massless bosonic field compactified on the circle of rational R2 is reexamined in the presense of boundaries. A particular class of models corresponding to R2=1/2k is distinguished by demanding the existence of a consistent set of Neumann boundary states. The boundary states are constructed explicitly for these models and the fusion rules are derived from them. These are the ones prescribed by the Verlinde formula from the S-matrix of the theory. In addition, the extended symmetry algebra of these theories is constructed, which is responsible for the rationality of these theories. Finally, the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra.

Representations of the Lie Ring sl2( Z ) over the Ring of Integers

Let O denote the category of finite-dimensional torsion-free sl2(Z)-modules and let S stand for the Lie ring sl2(Z). Thus, V ∈ O if, for v ∈ V and n ∈ Z, we have nv = 0⇒ n = 0 or v = 0; moreover, dimV = dimQ V ⊗ Q < ∞. An S-module V ∈ O is said to be irreducible if the S ⊗Z Q-module V ⊗Z Q is irreducible over sl2(Q). Fix a standard basis {e, h, f} for S such that eh = 2e, ef = h, and fh = −2f . Given a module V ∈ O, denote by Vd the submodule that is generated by the set {v ∈ V | (∃i ∈ Z) vh = iv}. An S-module V is said to be diagonal if V = Vd and semisimple if V decomposes into direct sum of irreducible modules. Clearly, each diagonal S-module V has a unique maximal semisimple submodule, which is denoted by Vs. The next proposition provides a complete description for semisimple S-modules.

A note on multiplicity-free tensor representations

A given tensor power of a complex vector space is a direct sum of irreducible modules for the general linear group on that vector space. Each irreducible module has finite, but in general quite large, multiplicity. We describe a reduced version of the tensor power in which each multiplicity is one.