Clifford Algebras and Graphs

I show how to associate a Clifford algebra to a graph. I describe the structure of these Clifford graph algebras and provide many examples and pictures. I describe which graphs correspond to isomorphic Clifford algebras and also discuss other related sets of graphs. This construction can be used to build models of representations of simplylaced compact Lie groups. 1 Clifford Algebras Let A be a unital algebra over C, with n generators e1, e2, . . . , en and relations ei = −1 for any i, and eiej = −ejei, for i 6= j. A is a classical Clifford algebra. As a vector space A has dimension 2 and is generated by monomials ei1ei2 . . . eik , where i1 < i2 < . . . < ik. The monomials are in one-to-one correspondence with the subsets of the set {1, 2, . . . n} or with binary strings of length n. Suppose α is a binary string of length n. We associate with this string the monomial eα = ei1ei2 . . . eik , where 1 ≤ i1 < i2 < . . . < ik ≤ n and i1, i2, . . ., ik are positons of ones in the string α. We associate 1 with the string of all zeroes. If β is a binary string too, then eαeβ = ±eγ , where γ = αXORβ, and XOR is the standard parity (xoring) operation on binary strings. Let us look at the center of this algebra — the subalgebra of elements that commute with all elements. Are there central elements not in C1? Every monomial either commutes or anticommutes with generators ei. From this we can deduce that the center is spanned by monomials. Suppose a monomial c