On the transposition anti-involution in real Clifford algebras III: the automorphism group of the transposition scalar product on spinor spaces

Abstract

In Abłamowicz and Fauser [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras I: The transposition map, Linear Multilinear Alg. (to appear)] a signature ϵ = (p, q)-dependent transposition anti-involution of real Clifford algebras Cℓ _p,q for non-degenerate quadratic forms was introduced. In Abłamowicz and Fauser [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Alg. (to appear)] we showed that, depending on the value of (p − q) mod 8, the map gives rise to transposition, complex Hermitian or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [R. Abłamowicz and B. Fauser, Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple, Comput. Phys. Commun. 170 (2005), pp. 115–130]. We provide a full signature (p, q)-dependent classification of the invariance groups of this product for p + q ≤ 9. The map is identified as the ‘star’ map known [D.S. Passman, The Algebraic Structure of Group Rings, Robert E. Krieger Publishing Company, Malabar, Florida, 1985] from the theory of (twisted) group algebras where the Clifford algebra Cℓ _p,q is seen as a twisted group ring n = p + q. We discuss important subgroups of a stabilizer group G _p,q (f) of a primitive idempotent f and we relate their transversals to spinor bases in spinor spaces realized as minimal left ideals Cℓ _p,q  f.

Keywords:

• grade involution
• group ring
• indecomposable module
• involution
• minimal left ideal
• monomial order
• primitive idempotent
• reversion
• semisimple algebra
• spinor
• stabilizer
• transversal
• twisted group ring
• universal Clifford algebra

AMS Subject Classifications::

• 15A66
• 16S35
• 20C05
• 68W30

Notes

Notes

1. Recall that elements λ of 𝕂 = f Cℓ _p,q  f commute with f because each λ = fuf for some u ∈ Cℓ _p,q . The notation ∃! stands for ‘there exists a unique’.

2. Here, r _i is Radon–Hurwitz number 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 12,14.

3. In the following, for f we can pick any of the 2 ^k primitive idempotents. In examples, our default choice for f will always be the one in which all signs in the factorization (3) are plus. The default primitive idempotents for Cℓ _p,q , p + q ≤ 9, are stored in CLIFFORD 4–6, and can be retrieved by issuing the command clidata([p, q]) in a Maple worksheet.

4. See 1,2 how to use CLIFFORD.

5. Note that transversal sets, which just provide coset representatives, are not unique: for the given (left) coset aK of K, a normal subgroup of G, we may assign ℓ(aK) = a or ℓ(aK) = b as long as aK = bK or, equivalently, b ⁻¹ a ∈ K.

6. In simple Clifford algebras, the monomials s ₁ and s ₂ also satisfy: (i) and (ii) . The identity (i) (resp. (ii)) is also valid in the semisimple algebras provided β₊ ≢ 0 (resp. β₋ ≢ 0).

7. See Fulton and Harris 11 for a definition of the quaternionic unitary group U _ℍ(N). In our notation we follow loc. cit. page 100, ‘Remark on Notations’.

8. In 14, p. 233], Lounesto states correctly that ‘the element s [so that or belongs to ] can be chosen from the standard basis of Cℓ _p,q ’. In fact, one can restrict the search for s to the transversal of the stabilizer G _p,q (f) in G _p,q which has a much smaller size compared to the size 2 ^p+q of the Clifford basis.

9. An ℝ-bilinear form K : V × V → 𝕂 on a real, complex or quaternionic vector space V is non-degenerate if K(v, w) = 0 for all v implies w = 0. The form is positive definite if K(v, v) > 0 for v ≠ 0 11.

10. The choice of each of the monomials s ₁ and s ₂ is not unique of course. In general, there are 2 ^p+q /N such choices modulo the commutator subgroup {±1} of G _p,q .

11. For the definitions of U(1, 1) and Sp(2, ℂ), see 14, Chapter 18].

12. To shorten display, we set ψ_[11φ_12] = ψ₁₁φ₁₂ − ψ₁₂φ₁₁ and ψ₍₁₁φ₂₁₎ = ψ₁₁φ₂₁ + ψ₂₁φ₁₁, etc.

13. Sp(2, 2) = U(2, 2) ∩ Sp(4, ℂ) and Sp(2, 2)/{±1} ≅ SO ₊(4, 1).

14. Recall that Sp(N) = U _ℍ(N) where U _ℍ(N) is the quaternionic unitary group 11.

15. By a small abuse of notation, we write the same numerical components for ψ and ψ _g , also for φ and φ _g , although in general they may be different. In order to identify the automorphism groups, what matters is really the form of the polynomial in the numerical components but not the actual components.

16. In 14, Table 1, p. 236], Lounesto shows for short only GL(2, ℝ) whereas in 14, Table 2, p. 236], he also shows ²Sp(2, ℝ). Thus, our definition of the symplectic group Sp(N, ℝ) is the same as his.

17. The group U _p,q ℍ is defined in 11, p. 99] as the group of automorphisms of a Hermitian form of signature (p, q) on a quaternionic vector space V of dimension p + q thus having the standard form In 14, Table 1, p. 236] we find ²Sp(2, 2) and in 14, Table 2, p. 236] we find GL(2, ℍ), respectively.

18. For a general theory of group rings, see 16.

19. We could simplify notation by identifying an ordered n-tuple from (ℤ₂) ⁿ with a monomial in ℝ ^t [(ℤ₂) ⁿ ].