ABSTRACT
Abstract In this chapter we present a decomposition of the subalgebra T (G) of Hom(A(G)) gen-
erated by {idA(G)} ∪ {Lt|Lt : A(G)→ A(G), t ∈ A(G)}, where G is a graph, A(G) is the Bernstein graph algebra of G, idA(G) is the identity function on A(G) and Lt is the left (= right) multiplication by t. If G is a simple connected graph, without loop and |V (G)| > 2, then we present a characterization of T (G). Key words: Baric algebras y Bernstein algebras, Bernstein graph algebra, transformation algebras 2000 MSC: 17D92
Let G = (V (G), E(G)) be a finite graph, where V (G) is the set of vertices and E(G) the set of edges of G. The edges will be denoted by, say, α = ab, where a and b are the vertices linked by α. Thus we have ab = ba, and aa is a loop in the vertex a. Suppose now F is a field of characteristic not 2, G has no isolated vertices and let N = N(G) be the vector space that is the direct sum of U and Z, where U is the F -vector space freely generated by V (G) and Z is the vector space freely generated by E(G). We introduce in N the following commutative multiplication (on the basis of U and Z)
aab = b; bab = a; other products are zero. (14.1)
For some x ∈ N , we have (x2)2 = 0. In fact, we see that if x1, x2, x3, x4 ∈ N then (x1x2)(x3x4) = 0 because both x1x2 and x3x4 are in U . So N is solvable of index 3. By applying [4, Prop. 7], we can embed N in an exceptional Bernstein algebra, by taking the linear operator τ : N −→ N defined by τ(u) = 12u, u ∈ U , and τ(z) = 0, z ∈ Z, and The Bernstein algebra just
will be referred to as the Bernstein algebra associated to the graph G or Bernstein graph algebra of G. For e = (1, 0) the elements of U are proper vectors of the linear operator Le : N(G) −→ N(G) defined by x −→ ex corresponding to the proper value 12 . The elements of Z are the proper vectors of the proper value 0 of Le. We have the following relations:
U2 = 0, UZ ⊆ U, Z2 = 0 as special cases of (19.1). Moreover the type of A(G) is (1 + |V (G)|, |E(G)|).
