Let k be a field of characteristic 6= 2, and let L be a Galois extension of k with group G. Let us denote by qL : L × L → k the trace form, defined by qL(x, y) = TrL/k(xy). Let (gx)g∈G be a normal basis of L over k. We say that this is a self–dual normal basis if qL(gx, hx) = δg,h. If the order of G is odd, then L always has a self–dual normal basis over k (cf. [1]). This is no longer true in g...

For symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the modified quantized enveloping algebra U̇(g) and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra g we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of U̇(g), where the root category is the T -orbit category ...

Abstract We introduce the concept of subalgebra spectrum, Sp ( A ), for a finite codimension in $$\mathbb {K}[x]$$ K [ x ] . The spectrum is subset underlying field. also tool, characteristic polynomial , which has as its set zeroes. ca...

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v|W ) = (d(v,w1), d(v, w2), . . . , d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The setW is a resolving set for G if distinct vertices of G have distinct representations. A resolving set of minimum ...