Closed subsets of compact-like topological spaces

<p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of<br />countably pracompact topological spaces. We construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed.</p>

MATRIX UNITS FOR THE GROUP ALGEBRA kGf = k((ℤ2 × ℤ2) ≀ Sf)

The irreducible representations of the group Gf := (ℤ2 × ℤ2) ≀ Sf are indexed by 4-partitions of f, i.e., by the set {[α]3[β]2[γ]1[δ]0|α ⊢ u3, β ⊢ u2, γ ⊢ u1, δ ⊢ u0, u0 + u1 + u2 + u3 = f}. This set is in 1 - 1 correspondence with partitions of 4f whose 4-core is empty. In this paper we construct the inequivalent irreducible representations of Gf. We also compute a complete set of seminormal matrix units for the group algebra kGf.

SLq(2) GLOBAL SYMMETRY FOR FOUR-STATE QUANTUM CHAINS

Following the method already used to obtain quantum chains with Dipper–Donkin global symmetry,14 we obtain all possible four-state quantum chains with SL q(2,C) global symmetry when q is not a root of unity. One of these Hamiltonians is written in terms of matrix units, as an example.

On the generators of elementary subgroups of general linear groups

Let R be a ring with identity and let Eij ∈ Mn(R) be the usual n X n matrix units, where n ≥ 2 and 1≤i, j≤N. Let En(R) be the subgroup of GLn(R) generated by all Tij(q where r ∈ R and i ≄ j. For each (two-sided) R-ideal q let En(R, q) be the normal subgroup of En(R) generated by Tij(q), where q ∈ q. The subgroup En(R, q) plays an important role in the theory of GLn(R). For example, Vaserˇstein has proved that, for a larger class of rings (which includes all commutative rings), every subgroup S of GLn(R), when R ∈ and n≥3, contains the subgroup En(R, q0), where q0 is the R-ideal generated by αij, rαij-αjjr (i ≄ j, r ∈ R), for all (αij) ∈ S. (See [13, Theorem 1].) In addition Vaseršstein has shown that, for the same class of rings, En(R, q) has a simple set of generators when n ≥ 3. Let Ên(R, q) be the subgroup of En(R, q) generated by Tij(r)Tij(q)Tij(−r), where r ∈ R, q ∈ q. Then Ên(R, q) = En(R, q), for all q, when R ∈ and n ≥ 3.(See [13, Lemma 8].)