On the transposition anti-involution in real Clifford algebras II: stabilizer groups of primitive idempotents

Abstract

In the first article [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras I: The transposition map, Linear Multilinear Algebra, to appear] we showed that real Clifford algebras Cℓ(V, Q) possess a unique transposition anti-involution . The map reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of associated matrix of that element in the left regular representation of the algebra. In this article we show that, depending on the value of (p − q) mod 8, where ϵ = (p, q) is the signature of Q, the anti-involution gives rise to transposition, Hermitian complex and Hermitian quaternionic conjugation of representation matrices in spinor representations. We realize spinors in minimal left ideals S = Cℓ _p,q  f generated by a primitive idempotent f. The map allows us to define a dual spinor space S*, and a new spinor norm on S, which is different, in general, from two spinor norms known to exist. We study a transitive action of generalized Salingaros' multiplicative vee groups G _p,q on complete sets of mutually annihilating primitive idempotents. Using the normal stabilizer subgroup G _p,q (f), we construct left transversals, spinor bases and maps between spinor spaces for different orthogonal idempotents f _i summing up to 1. We classify the stabilizer groups according to the signature in simple and semisimple cases.

Keywords:

• correlation
• dual space
• exterior algebra
• grade involution
• group action
• group ring
• indecomposable module
• involution
• left regular representation
• minimal left ideal
• monomial order
• primitive idempotent
• reversion
• simple algebra
• semisimple algebra
• spinor
• stabilizer
• transversal
• universal Clifford algebra

AMS Subject Classifications:

• 11E88
• 15A66
• 16S34
• 20C05
• 68W30

Acknowledgements

B. Fauser would like to thank the Emmy-Noether Zentrum for Algebra at the University of Erlangen for their hospitality during his stay at the Zentrum in 2008/2009.

Notes

1. As explained in 3, this is not the linear dual Clifford algebra, which is not employed in this article, but the Clifford algebra over the space V ^♭ ≅ V* for the same quadratic form Q.

2. It may interest the curious reader that this map is in fact the antipode on the Hopf algebra over the group algebra k[G].

3. The Radon–Hurwitz number is defined by recursion as r _i+8 = r _i  + 4 and these initial values: r ₀ = 0, r ₁ = 1, r ₂ = r ₃ = 2, r ₄ = r ₅ = r ₆ = r ₇ = 3. (see 5,8).

4. Two idempotents f ₁ and f ₂ are mutually annihilating when f ₁ f ₂ = f ₂ f ₁ = 0. Such idempotents are also called orthogonal. A decomposition of the unity 1 into a sum of mutually annihilating primitive idempotents is called a primitive idempotent decomposition.

5. See 10, Chap. 9] for a discussion of quaternionic linear spaces, and 10, Chap. 12] for double fields and their anti-involutions.

6. In a semisimple Clifford algebra, we have N = 2 ^k−1. Semisimple algebras are considered in Section 3.

7. We will see shortly that the monomial set ℳ is a (left) transversal for G _p,q (f) in G _p,q .

8. From now on, we always sort the monomial set ℳ by InvLex order so that m ₁ = 1 and f ₁ = f.

9. See [8, Sect. 17.2] for the definition of Pin(p, q).

10. The commutator subgroup G′ of a group G is G′ = [G, G] = ⟨[x, y]|x, y ∈ G⟩. That is, it is a subgroup of G generated by all commutators [x, y] = xyx ⁻¹y⁻¹ for x, y ∈ G. In general, G′ ◃ G and G/G′ is Abelian 11, Prop. 5.57]. We have since any two monomials in G _p,q either commute or anti-commute.

11. The coset representatives are precomputed for all Clifford algebras Cℓ _p,q , n = p + q ≤ 9, in CLIFFORD 2. They are also shown in 1.

12. Later, when we will be computing matrices of Clifford elements in spinor representation, we will always take the ordered basis 𝒮₁ = [m ₁ f ₁, … , m _N f ₁].

13. In fact, it is enough to pick m _j from the left transversal ℳ.

14. In 4 we provide a complete classification of all groups

15. We remark here that the product always coincides with β₊ in Euclidean signatures (p, 0) and with β₋ in anti-Euclidean signatures (0, q).

16. Decomposition (49) is also valid in semi-simple Clifford algebras when p − q = 1 mod 4.

17. Recall, that û denotes the grade involution of u ∈ Cℓ _p,q . Then, , J ₊ J ₋ = J ₋ J ₊ = 0 and .