Abstract
In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras—subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras—graded central simple algebras or graded central simple algebras with involution.
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1 Introduction
In this paper, we study inner automorphisms that leave invariant fixed fundamental subspaces of the real and complex Clifford algebras. We present groups of elements that define such inner automorphisms. Some of these groups are associated with the vector subspace V of the Clifford algebra \({C}\!\ell (V,Q)\) [4, 12] over a quadratic space (V, Q) with a nondegenerate quadratic form Q: well-known Clifford group \(\Gamma \), which preserves the subspace V and is widely used in the theory of spin groups; the groups \(\Gamma ^k\), \(k=0, 1, \ldots , n\), which preserve the subspaces of fixed grades k. The others are naturally defined by the main involution (the grade involution) and the reversion. We introduce the groups \(\mathrm{P}\) (preserve subspaces of fixed parity), \(\mathrm{A}\), \(\mathrm{B}\), \(\mathrm{Q}\), and \(\mathrm{Q}^\prime \) (preserve some other fundamental subspaces determined by the reversion and the grade involution) and study their properties. As one of the anonymous reviewers noted, these subspaces can be defined in more general algebras, and some of the results of this paper can be reformulated for the more general case. Namely, the groups \(\mathrm{P}\) (see Sect. 3) can be introduced in the graded central simple algebras (GCSAs) introduced by Wall in 1964 [25]. The groups \(\mathrm{A}\), \(\mathrm{B}\), \(\mathrm{Q}\) (see Sects. 4, 5, 6) can be introduced in the graded central simple algebras with involution (GCSAsWI) introduced by Wall in 1968 [26]Footnote 1. Let us also note the paper [7] on GSCAs and GSCAsWI, including, as a special case, the Clifford algebras. Clifford algebras are widely used in different applications—in engineering, physics, robotics, computer vision, image and signal processing, etc. Groups that preserve different structures of Clifford algebras under similarity transformation may be of interest to these applications. All these groups contain spin groups as subgroups and can be considered as generalizations of Clifford, Lipschitz, and spin groups.
Let \(\mathrm{C}\) be the real Clifford algebra \({C}\! \ell (V,Q)\), \(V={\mathbb R}^{p,q}\), \(p+q=n\) (or the geometric algebra [6, 8]) or the complex Clifford algebra \({C}\! \ell ({\mathbb C}^n)\) [3, 9, 10, 12]. We denote the identity element by e and the generators of \(\mathrm{C}\) by \(e_a\), \(a=1, \ldots , n\). In the case of the real Clifford algebra, the generators \(e_a\), \(a=1, \ldots , n\) satisfy
where \(\eta =(\eta _{ab})=\mathrm{diag}(1,\ldots , 1, -1, \ldots , -1)\) is the diagonal matrix with its first p entries equal to 1 and the last q entries equal to \(-1\) on the diagonal. In the case of the complex Clifford algebra, the generators satisfy the same conditions but with the identity matrix \(\eta =I_n\) of size n. Consider the subspaces \(\mathrm{C}^k\) of grades \(k=0, 1, \ldots , n\). The elements of these subspaces are linear combinations of the basis elements \(e_{a_1 \ldots a_k}:=e_{a_1}\cdots e_{a_k}\), \(a_1< a_2< \cdots < a_k\), with multi-indices of length k. The Clifford algebra \(\mathrm{C}\) can be represented as the direct sum of the even and odd subspaces
where \({\hat{\quad }}\) is the grade involution (or the main involution). Also we consider the following four subspaces (see [16,17,18]), which are naturally determined by the grade involution \({\hat{\quad }}\) and the reversion \({\tilde{\quad }}\) :
The algebra \(\mathrm{C}\) is a \(Z_2\times Z_2\)-graded algebra w.r.t. these four subspaces and the operations of commutator \([U,V]=UV-VU\) and anticommutator \(\{U, V\}=UV+VU\):
We use the notation \(A^\times \) for the set of invertible elements of any set A. Consider the adjoint representation (inner automorphism) \(\mathrm{ad}: \mathrm{C}^{\times }\rightarrow \mathrm{Aut}\mathrm{C}\) acting on the group of all invertible elements of the Clifford algebra
as \(T\mapsto \mathrm{ad}_T\), where \( \mathrm{ad}_T U=TUT^{-1}\) for any \(U\in \mathrm{C}\). The kernel of \(\mathrm{ad}\) is
where \(\mathrm{Z}^\times \) is the group of all invertible elements of the center of \(\mathrm{C}\)
Let us consider the well-known Clifford group [3, 4, 12, 13], which consists of elements that define inner automorphisms preserving the subspace of grade 1:
where \(\mathrm{C}^{\times 1}=\{v_j\in \mathrm{C}^1: v_j^2\ne 0\}\) is the subset of all invertible elements of grade 1. The equivalence of (1.3) and (1.4) is well-known and can be proved using the Cartan–Dieudonné theorem [5, 13] or the generalization of the Pauli’s theorem [14, 15]. From (1.3) and (1.4), we can easily obtain
In this paper, we present and study groups that leave invariant different other fixed subspaces (the subspaces of fixed grades, the subspaces defined by the reversion and the grade involution) of the Clifford algebra \(\mathrm{C}\). In the case \(n=1\), the Clifford algebra \(\mathrm{C}\) is commutative, and all such groups coincide with \(\mathrm{C}^\times \). We consider the case \(n\ge 2\) below. We summarize the results for the Lie groups and Lie algebras presented in this paper in Table 1.
We use notations with multi-indices for direct sums of different subspaces:
and similar ones. The group of elements that define inner automorphisms preserving the subspace \(\mathrm{C}^{(k)\overline{lm}r}\) is denoted by \(\Gamma ^{(k)\overline{lm}r}\).
We can illustrate the considered Lie groups in the following way (see Fig. 1). Note that we have \(\Gamma \subseteq \mathrm{Q}\subseteq \mathrm{Q}^\prime \subseteq \mathrm{P}\) and \(\mathrm{Q}=\mathrm{A}\cap \mathrm{B}=\mathrm{A}\cap \mathrm{P}=\mathrm{B}\cap \mathrm{P}\). All the considered groups are subgroups of the group \(\mathrm{C}^\times \). Spin groups are subgroups of the smallest of the considered groups \(\Gamma \).
2 The Lie Groups \(\Gamma ^k\)
Let us use the following notation for the groups that preserve subspaces of fixed grades under similarity transformation
In the particular case, we get the Clifford group \(\Gamma ^1:=\Gamma \) (see (1.3)).
Lemma 2.1
We have
Proof
We have \(\Gamma ^0=\mathrm{C}^\times \) in the case of arbitrary n and \(\Gamma ^n=\mathrm{C}^\times \) in the case of odd n because of the center (1.2) of \(\mathrm{C}\).
In the case of even n, we can easily verify that \(\mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)}\subseteq \Gamma ^n\) because \(e_{1\ldots n}\) commutes with all even elements and anticommutes with all odd elements. Let us prove that \(\Gamma ^n\subseteq \mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)}\). Suppose that \(T=T_0+T_1\in \Gamma ^n\), where \(T_0\in \mathrm{C}^{(0)}\) and \(T_1\in \mathrm{C}^{(1)}\). We get
for some constant \(\lambda \), which is equivalent to \((T_0-T_1)=\lambda (T_0+T_1)\), i.e. \(T_0=\lambda T_0\) and \(-T_1=\lambda T_1\). We get \(\lambda =1\), \(T_1=0\) or \(\lambda =-1\), \(T_0=0\), i.e. \(T\in \mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)}\). \(\square \)
Theorem 2.2
We haveFootnote 2
As a consequence, we obtain
Proof
Suppose \(T=W v_1 \cdots v_m\), \(W\in \mathrm{Z}^\times \), \(v_j\in \mathrm{C}^{\times 1}\), \(j=1, \ldots , m\). For \(U_k\in \mathrm{C}^{k}\), we have
Since \((v_j)^2\in \mathrm{C}^{\times 0}\), we have \((v_j)^{-1}=\lambda _j v_j\) for some nonzero constant \(\lambda _j\), \(j=1, \ldots , m\). Thus \(\widetilde{(v_j)^{-1}}=v_j^{-1}\), \(\widehat{(v_j)^{-1}}=-(v_j)^{-1}\), and we have
Using (1.1), we conclude that the elements \(U_k\) and \(v_j U_k v_j^{-1}\) are from the same subspace \(\mathrm{C}^{\overline{k}}\). The grade of element \(v_j U_k v_j^{-1}\) can take values between \(k-2\) and \(k+2\). Thus the grade equals k, and we get
and \(T U_k T^{-1}\in \mathrm{C}^k\). \(\square \)
We present new properties of the groups \(\Gamma ^k\), \(k=1, \ldots , n\) in Sect. 7 (see Lemmas 7.1, 7.2, 7.5).
3 The Lie Groups \(\mathrm{P}\)
Let us use the following notation for the groups that preserve subspaces of fixed parity under similarity transformation
Using Theorem 2.2, we get
Let us consider the following group
By Lemma 2.1, we have \(\Gamma ^n=\mathrm{P}\) for even n.
We need the following lemma to prove Theorems 3.2 and 6.3.
Lemma 3.1
We have
Proof
Let us have \(U e_{ab}=e_{ab} U\), \(\forall a<b\). Taking \(a=1\) and \(b=2\), let us represent U in the form
where the elements \(A, B, C, D\in \mathrm{C}\) do not contain \(e_1\) and \(e_2\). We get \((A+e_1B+e_2 C+e_{12}D)e_{12}=e_{12}(A+e_1B+e_2 C+e_{12}D)\). Using \(A e_{12}=e_{12} A\), \(D e_{12}=e_{12}D\), \(e_1 B e_{12}=-e_{12}e_1B\), and \(e_2C e_{12}=-e_{12}e_2C\), we obtain \(B=C=0\). Acting similarly for the other \(a<b\), we obtain \(U=ae+b e_{1\ldots n}\) for some constants a, b. \(\square \)
Theorem 3.2
The following three groups coincideFootnote 3
Proof
Let us prove that \(\mathrm{P}\subseteq \Gamma ^{(0)}\) and \(\mathrm{P}\subseteq \Gamma ^{(1)}\). Suppose that \(T\in \mathrm{Z}^\times (\mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)})\). Since \(TT^{-1}=e\), we conclude that if \(T=W T_0\in \mathrm{Z}^\times \mathrm{C}^{\times (0)}\), \(W\in \mathrm{Z}^\times \), \(T_0\in \mathrm{C}^{\times (0)}\), then \(T^{-1}=W^{-1} T_0^{-1}\in \mathrm{Z}^\times \mathrm{C}^{\times (0)}\), and if \(T=W T_1\in \mathrm{Z}^\times \mathrm{C}^{\times (1)}\), \(W\in \mathrm{Z}^\times \), \(T_1\in \mathrm{C}^{\times (1)}\), then \(T^{-1}=W^{-1} T_1^{-1}\in \mathrm{Z}^\times \mathrm{C}^{\times (1)}\). Therefore \(T \mathrm{C}^{(1)}T^{-1}\subseteq \mathrm{C}^{(1)}\) and \(T \mathrm{C}^{(0)} T^{-1}\subseteq \mathrm{C}^{(0)}\).
Let us prove that \(\Gamma ^{(1)}\subseteq \mathrm{P}\). Suppose \(T \mathrm{C}^{(1)} T^{-1}\subseteq \mathrm{C}^{(1)}\). Then we have
Multiplying both sides on the right by T and on the left by \(\widehat{T^{-1}}\), we get
thus \(\widehat{T^{-1}}T\in \mathrm{Z}^\times \), and \({\hat{T}}=W T\) for some \(W\in \mathrm{Z}^\times \). Taking grade involution, we obtain \(T={\hat{W}} {\hat{T}}\). Thus \(T={\hat{W}} W T\). If n is even, then for \(W=\lambda e\) with some constant \(\lambda \), we get \(\lambda =\pm 1\). We obtain \(T\in \mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)}\). If n is odd, then \(W=ae+be_{1\ldots n}\) for some constants a, b. Suppose we have \(T=T_0+T_1\), \(T_0\in \mathrm{C}^{(0)}\), \(T_1\in \mathrm{C}^{(1)}\). Then \(T_0-T_1=(ae+be_{1\ldots n})(T_0+T_1)\), and \(T_0=aT_0+be_{1\ldots n}T_1\). If \(b=0\), then \(a=1\), \(W=e\), and \(T\in \mathrm{C}^{(0)}\cup \mathrm{C}^{(1)}\). If \(b\ne 0\), then \(T_1=\mu e_{1 \ldots n} T_0\) for some constant \(\mu \). Thus we have \(T=T_0+T_1= (e+\mu e_{1\ldots n})T_0\in \mathrm{Z}^\times \mathrm{C}^{\times (0)}= \mathrm{Z}^\times (\mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)})\).
Let us prove that \(\Gamma ^{(0)}\subseteq \mathrm{P}\). Suppose \(T \mathrm{C}^{(0)}T^{-1}\subseteq \mathrm{C}^{(0)}\). Then we have
Multiplying both sides by T on the right and by \(\widehat{T^{-1}}\) on the left, we get
Using Lemma 3.1, we obtain \(\widehat{T^{-1}}T\in (\mathrm{C}^{0}\oplus \mathrm{C}^{n})^\times \), and \({\hat{T}}=W T\) for some \(W\in \mathrm{C}^{\times 0 \ n}\). In the case of odd n, the proof is analogous to the proof of the previous case because \(\mathrm{Z}=\mathrm{C}^{0\ n}\) in this case. Let us consider the case of even n. Suppose we have \(T=T_0+T_1\), \(T_0\in \mathrm{C}^{(0)}\), \(T_1\in \mathrm{C}^{(1)}\). Then \(T_0-T_1= W(T_0+T_1)\), i.e. \(T_0=W T_0\) and \(-T_1=WT_1\). We get \((W-e)T_0=0\) and \((W+e)T_1=0\). If at least one of two elements \(W+e\) and \(W-e\) is invertible, then we conclude that \(T_0=0\) or \(T_1=0\), and \(T\in \mathrm{C}^{(0)}\cup \mathrm{C}^{(1)}\). Suppose that both elements \(W+e\) and \(W-e\) are non invertible. For \(W=ae+be_{1\ldots n}\) with some constants a, b, we get \(W\pm e=(a\pm 1)e+b e_{1\ldots n}\). We obtain
We see that both elements can be non invertible only in the case \(a=0\), \(b=\pm 1\), \((e_{1\ldots n})^2=e\) or \(a=0\), \(b=\pm i\), \((e_{1\ldots n})^2=-e\) (in the case of the field \({\mathbb C}\)). We get \(W=\pm e_{1\ldots n}\) or \(W=\pm i e_{1\ldots n}\). From \({\hat{T}}=WT\), we obtain
i.e. a contradiction, because T is invertible and \(W\ne e\). \(\square \)
Note that we can prove \(\Gamma ^{(0)}=\Gamma ^{(1)}\) in the case of odd n in a simpler way. Multiplying both sides of \(T \mathrm{C}^{(0)} T^{-1}\subseteq \mathrm{C}^{(0)}\) by \(e_{1\ldots n}\in \mathrm{Z}\) and using \(\mathrm{C}^{(0)}=e_{1\ldots n}\mathrm{C}^{(1)}\), we obtain \(T \mathrm{C}^{(1)}T^{-1}\subseteq \mathrm{C}^{(1)}\). Similarly, in the opposite direction.
Note that we can also prove \(\Gamma ^{(1)}\subseteq \mathrm{P}\) in another way using the generalization of the Pauli’s theorem (see [15, 19, 23]). Using \(T \mathrm{C}^{(1)}T^{-1}\subseteq \mathrm{C}^{(1)}\), we get
The elements (3.3) satisfy \(\beta _a \beta _b+\beta _b \beta _a=2\eta _{ab}e\). By the generalized Pauli’s theorem for odd elements (see Theorems 5.1 and 5.2 in [19]), we get \(T\in \mathrm{Z}( \mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)})\) because the element T equals
for some element \(F\in \{e_{a_1\ldots a_k}\}\) up to an invertible element of the center \(\mathrm{Z}^\times \).
Note that subspaces of fixed parity are direct sums of the subspaces (1.1): \(\mathrm{C}^{(0)}=\mathrm{C}^{\overline{02}}\) and \(\mathrm{C}^{(1)}=\mathrm{C}^{\overline{13}}\). Below we study other subspaces, which are also direct sums of the subspaces (1.1): \(\mathrm{C}^{\overline{01}}\), \(\mathrm{C}^{\overline{23}}\), \(\mathrm{C}^{\overline{03}}\), and \(\mathrm{C}^{\overline{12}}\).
4 The Lie Groups \(\mathrm{A}\)
Let us use the following notation for the groups that preserve subspaces (1.1) or their direct sums under similarity transformation
Consider the following groupFootnote 4
Note that the “norm function” \(\psi (T):={\tilde{T}}T\) is widely used in the theory of spin groups (see, for example [2, 3, 10, 13]). The spin groups
are defined as normalized subgroups of the Lipschitz group
It is well-known that \(\psi : \Gamma ^\pm \rightarrow \mathrm{C}^{\times 0}\). The group \(\mathrm{A}\) contains the groups \(\Gamma \) and \(\Gamma ^\pm \) as subgroups.
Lemma 4.1
The norm function \({\tilde{T}}T\) takes values in \(\mathrm{C}^{\overline{01}}\), i.e.
As a consequence, we get
Proof
We have \(\widetilde{{\tilde{T}}T}={\tilde{T}} \tilde{{\tilde{T}}}={\tilde{T}}T\) for any \(T\in \mathrm{C}\). We conclude that the reversion does not change the expression \({\tilde{T}}T\). Using (1.1), we obtain \({\tilde{T}}T\in \mathrm{C}^{\overline{01}}\). Using (4.3), we get (4.5). \(\square \)
Theorem 4.2
The following three groups coincide
Proof
Let us prove that \(\mathrm{A}\subseteq \Gamma ^{\overline{01}}\) and \(\mathrm{A}\subseteq \Gamma ^{\overline{23}}\). Let us have \({\tilde{T}} T=W\in \mathrm{Z}^\times \). For \(U\in \mathrm{C}^{\overline{01}}\), we have
i.e. \(TUT^{-1}\in \mathrm{C}^{\overline{01}}\). For \(U\in \mathrm{C}^{\overline{23}}\), we have
i.e. \(TUT^{-1}\in \mathrm{C}^{\overline{23}}\).
Let us prove that \(\Gamma ^{\overline{01}}\subseteq \mathrm{A}\). For \(U\in \mathrm{C}^{\overline{01}}\), we have
Multiplying both sides on the right by T and on the left by \({\tilde{T}}\), we get
In particular, we conclude that \({\tilde{T}}T\) commutes with all generators \(e_a\in \mathrm{C}^{\overline{01}}\), \(a=1, \ldots , n \). Thus \({\tilde{T}}T\in \mathrm{Z}^\times \).
Let us prove that \(\Gamma ^{\overline{23}}\subseteq \mathrm{A}\). For \(U\in \mathrm{C}^{\overline{23}}\), we have
We get
If \(n\ge 3\), then we can always represent each generator \(e_a\), \(a=1,\ldots , n\) as the product of elements of grade 2 and 3. This implies that \({\tilde{T}}T\) commutes with all generators \(e_a\), \(a=1, \ldots , n \). For example, if \({\tilde{T}}T\) commutes with \(e_{123}\) and \(e_{23}\), then it commutes with \(e_1\). We obtain \({\tilde{T}}T\in \mathrm{Z}^\times \). If \(n=2\), then from \([{\tilde{T}}T, e_{12}]=0\), we obtain \({\tilde{T}}T\in \mathrm{C}^0\oplus \mathrm{C}^2\). Using \({\tilde{T}}T\in \mathrm{C}^{\overline{01}}\), we get \({\tilde{T}}T\in \mathrm{C}^{\times 0}=\mathrm{Z}^\times \). \(\square \)
5 The Lie Groups \(\mathrm{B}\)
Let us consider the following group
The norm function \(\chi (T):=\hat{{\tilde{T}}}T\) as well as the function \(\psi (T)\) (see above) is widely used in the theory of spin groups (see [2, 3, 10, 13]).
Lemma 5.1
The norm function \(\hat{{\tilde{T}}}T\) takes values in \(\mathrm{C}^{\overline{03}}\), i.e.
As a consequence, we get
In the particular cases, we have
Proof
We have \(\widehat{\widetilde{\hat{{\tilde{T}}}T}}=\hat{{\tilde{T}}} \hat{\tilde{\hat{{\tilde{T}}}}}=\hat{{\tilde{T}}}T\). We conclude that the operation \(\hat{{\tilde{\quad }}}\) does not change the expression \(\hat{{\tilde{T}}}T\). Using (1.1), we obtain \(\hat{{\tilde{T}}}T\in \mathrm{C}^{\overline{03}}\). Using (5.1), we get (5.2). In the cases \(n\le 3\), the condition \(\hat{{\tilde{T}}} T\in \mathrm{Z}^\times =\mathrm{C}^{\overline{03}}\) holds automatically. \(\square \)
Theorem 5.2
The following three groups coincide
Proof
Let us prove that \(\mathrm{B}\subseteq \Gamma ^{\overline{03}}\) and \(\mathrm{B}\subseteq \Gamma ^{\overline{12}}\). Let us have \(\hat{{\tilde{T}}} T=W\in \mathrm{Z}^\times \). For \(U\in \mathrm{C}^{\overline{03}}\), we have
i.e. \(TUT^{-1}\in \mathrm{C}^{\overline{03}}\). For \(U\in \mathrm{C}^{\overline{12}}\), we have
i.e. \(TUT^{-1}\in \mathrm{C}^{\overline{12}}\).
Let us prove that \(\Gamma ^{\overline{12}}\subseteq \mathrm{B}\). For \(U\in \mathrm{C}^{\overline{12}}\), we have
Multiplying both sides on the right by T and on the left by \(\tilde{{\hat{T}}}\), we get
In particular, we conclude that \(\tilde{{\hat{T}}}T\) commutes with all generators \(e_a\in \mathrm{C}^{\overline{12}}\), \(a=1, \ldots , n\). Thus \(\tilde{{\hat{T}}}T\in \mathrm{Z}^\times \).
Let us prove that \(\Gamma ^{\overline{03}}\subseteq \mathrm{B}\). For \(U\in \mathrm{C}^{\overline{03}}\), we have
Multiplying both sides on the right by T and on the left by \(\tilde{{\hat{T}}}\), we get
For \(n\le 3\), we get \(\Gamma ^{\overline{03}}=\mathrm{B}= \mathrm{C}^\times \). If \(n\ge 4\), then we can always represent each generator \(e_a\), \(a=1, \ldots , n\), as the product of two elements of grades 3 and 4. This implies that \(\tilde{{\hat{T}}}T\) commutes with all generators \(e_a\), \(a=1,\ldots , n\). For example, if \(\tilde{{\hat{T}}}\) commutes with \(e_{234}\) and \(e_{1234}\), then it commutes with \(e_1\). We get \(\tilde{{\hat{T}}}T\in \mathrm{Z}^\times \). \(\square \)
6 The Lie Groups \(\mathrm{Q}\) and \(\mathrm{Q}^\prime \)
Let us consider the following group
Lemma 6.1
We have
In the particular cases, we have
Note that below (see Lemma 7.4), using auxiliary statements, we also prove that \(\mathrm{Q}=\mathrm{P}=\mathrm{A}\) in the cases \(n\le 3\), and \(\mathrm{Q}\ne \mathrm{A}\) in the case \(n=4\).
Proof
The first two statements (6.2) are trivial because of the definition (6.1). Let us prove the nontrivial statement \(\mathrm{A}\cap \mathrm{B}=\mathrm{Q}\). Suppose we have element \(T\in \mathrm{C}^\times \) such that \({\tilde{T}}T=W_1\in \mathrm{Z}^\times \) and \(\hat{{\tilde{T}}} T=W_2\in \mathrm{Z}^\times \). We get \({\hat{T}}=W T\) for some \(W\in \mathrm{Z}^\times \). In the proof of Theorem 3.2, we have already shown that this implies \(T\in \mathrm{Z}^\times (\mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)})\). Thus \(\mathrm{A}\cap \mathrm{B}=\mathrm{Q}\).
For \(n\le 3\), we have \(\mathrm{C}^{\overline{0}}=\mathrm{C}^0\). Using \({\tilde{T}}T\in \mathrm{C}^{\overline{01}}\) and \(\hat{{\tilde{T}}}T\in \mathrm{C}^{\overline{03}}\), we get \(\mathrm{Q}=\mathrm{P}\) for \(n\le 3\).
In the case \(n=4\), the element
is invertible because
is a nonzero scalar. Also we have
i.e. \(T\in \mathrm{P}\), \(T\notin \mathrm{A}\), and \(T\notin \mathrm{Q}\). Thus \(\mathrm{P}\ne \mathrm{A}\) and \(\mathrm{P}\ne \mathrm{Q}\) in the case \(n=4\). \(\square \)
Let us consider the group
This group coincides with the group \(\mathrm{Q}\) in the cases \(n=1, 3{\ \mathrm mod}4\) because \(\mathrm{Z}=\mathrm{C}^0\oplus \mathrm{C}^n\) for odd n, and in the case \(n=2{\ \mathrm mod}4\) because \({\tilde{T}}T\in \mathrm{C}^{\overline{01}}\), \(\hat{{\tilde{T}}}T\in \mathrm{C}^{\overline{03}}\):
Let us consider the group \(\mathrm{Q}^\prime \) in the case \(n=0{\ \mathrm mod}4\).
Lemma 6.2
We have
where
For the groups
we have
Proof
We obtain (6.6) using the definitions of the corresponding groups.
Let us prove (6.7). For the element \(T=e+2e_{1\ldots n}\), we have
i.e. \(T\in \mathrm{Q}^\prime \) and \(T\notin \mathrm{Q}\). The element T is invertible because
is a nonzero scalar.
We have (6.8) because the condition \({\tilde{T}}T\in \mathrm{C}^{\overline{0}}=\mathrm{C}^0\oplus \mathrm{C}^4\) holds automatically in the case \(n=4\).
Let us prove (6.9). The element \(T=e_{12}+2e_{34}\in \mathrm{C}^{(0)}\) is invertible because
is a nonzero scalar. Also we have
i.e. \(T\in \mathrm{P}\) and \(T\notin \mathrm{Q}^\prime \).
Let us prove (6.12). If \(n=1, 2, 3{\ \mathrm mod}4\), then \(\mathrm{A}^\prime =\mathrm{A}\), \(\mathrm{B}^\prime =\mathrm{B}\), and \(\mathrm{Q}^\prime =\mathrm{Q}\), thus \(\mathrm{Q}^\prime =\mathrm{Q}=\mathrm{A}\cap \mathrm{B}=\mathrm{A}^\prime \cap \mathrm{B}^\prime \). Let us consider the case \(n=0{\ \mathrm mod}4\). Suppose we have an element \(T\in \mathrm{C}^\times \) such that
We get \({\hat{T}}=W T\) for some \(W\in (\mathrm{C}^{0}\oplus \mathrm{C}^n)^\times \). In the proof of Theorem 3.2, we have already shown that this implies \(T\in \mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)}\). \(\square \)
Theorem 6.3
In the cases \(n\ge 4\), we have
In the exceptional cases, we have
As a consequence, we obtain (the analogue of Theorem 2.2 for grades)
Proof
Let us prove that \(\mathrm{Q}\subseteq \Gamma ^{\overline{k}}\), \(k=0, 1, 2, 3\). Suppose \(T\in \mathrm{Z}^\times (\mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)})\) and \({\tilde{T}} T=W\in \mathrm{Z}^\times \). For an arbitrary \(U_{\overline{k}}\in \mathrm{C}^{{\overline{k}}}\), \(\overline{k}=\overline{0}, \overline{1}, \overline{2}, \overline{3}\), we have
Using (1.1), we conclude that the elements \(T U_{\overline{k}} T^{-1}\in \mathrm{C}^{{\overline{k}}}\). Thus \(\mathrm{Q}\subseteq \Gamma ^{\overline{k}}\), \(k=0, 1, 2, 3\).
Let us prove that in the cases \(n=0{\ \mathrm mod}4\), we have \(\mathrm{Q}^\prime \subseteq \Gamma ^{\overline{0}}\), \(\mathrm{Q}^\prime \subseteq \Gamma ^{\overline{2}}\). Suppose \(T\in \mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)}\) and \({\tilde{T}} T=W\in (\mathrm{C}^0\oplus \mathrm{C}^n)^\times \). For an arbitrary \(U_{\overline{k}}\in \mathrm{C}^{{\overline{k}}}\), we have
where we use that W commutes with even elements. Using (1.1), we conclude that \(T U_{\overline{k}} T^{-1}\in \mathrm{C}^{{\overline{k}}}\) for \(k=0\) and 2. Thus \(\mathrm{Q}^\prime \subseteq \Gamma ^{\overline{k}}\), \(k=0, 2\).
Let us prove that \(\Gamma ^{\overline{1}}\subseteq \mathrm{Q}\). Suppose \(T \mathrm{C}^{{\overline{1}}}T^{-1}\subseteq \mathrm{C}^{{\overline{1}}}\). From \(T e_a T^{-1}\in \mathrm{C}^{{\overline{1}}}\), \(a=1, \ldots , n\), we get
Multiplying both sides on the right by T and on the left by \({\tilde{T}}\), we get
i.e. \({\tilde{T}} T\in \mathrm{Z}^\times \), and \(\Gamma ^{\overline{1}}\subseteq \mathrm{A}\). Also we have
Multiplying both sides on the right by T and on the left by \(\hat{{\tilde{T}}}\), we get
i.e. \(\hat{{\tilde{T}}} T\in \mathrm{Z}^\times \), and \(\Gamma ^{\overline{1}}\subseteq \mathrm{B}\). Using Lemma 6.1, we obtain \(\Gamma ^{\overline{1}}\subseteq \mathrm{A}\cap \mathrm{B}=\mathrm{Q}\).
In the exceptional case \(n=3\), we have \(\Gamma ^{\overline{3}}=\mathrm{C}^\times \) because \(\mathrm{C}^{\overline{3}}=\mathrm{C}^3\in \mathrm{Z}\) in this case.
Let us prove that \(\Gamma ^{\overline{3}}\subseteq \mathrm{Q}\), \(n\ge 4\). Suppose \(T \mathrm{C}^{{\overline{3}}}T^{-1}\subseteq \mathrm{C}^{{\overline{3}}}\). We obtain
and
If \(n\ge 4\), then we can always represent each generator \(e_a\), \(a=1,\ldots , n\), as the product of 3 elements from the subspace \(\mathrm{C}^{{\overline{3}}}\). This implies that \({\tilde{T}}T\) commutes with all generators \(e_a\), \(a=1, \ldots , n \). For example, if \({\tilde{T}}T\) commutes with \(e_{123}\), \(e_{124}\), \(e_{134}\), then it is trivial that it commutes with \(e_1\). We obtain \({\tilde{T}} T\in \mathrm{Z}^\times \), i.e. \(\Gamma ^{\overline{3}}\subseteq \mathrm{A}\). Analogously, we get
and
Finally, for \(n\ge 4\), we obtain \(\hat{{\tilde{T}}}T\in \mathrm{Z}^\times \), and \(\Gamma ^{\overline{3}}\subseteq \mathrm{B}\). Thus \(\Gamma ^{\overline{3}}\subseteq \mathrm{A}\cap \mathrm{B}=\mathrm{Q}\) in the cases \(n\ge 4\).
Let us prove that \(\Gamma ^{\overline{2}}\subseteq \mathrm{Q}\) in the cases \(n=1, 2, 3{\ \mathrm mod}4\) and \(\Gamma ^{\overline{2}}\subseteq \mathrm{Q}^\prime \) in the case \(n=0{\ \mathrm mod}4\). Suppose \(T \mathrm{C}^{{\overline{2}}}T^{-1}\subseteq \mathrm{C}^{{\overline{2}}}\). We have
Therefore
This implies
because we can always (\(n\ge 2\)) represent all even basis elements of Clifford algebra as the products of elements from the subspace \(\mathrm{C}^{{\overline{2}}}\). Using Lemma 3.1, we get \({\tilde{T}} T\in \mathrm{C}^0\oplus \mathrm{C}^n\). Analogously,
Therefore
This implies
Using Lemma 3.1, we obtain \(\hat{{\tilde{T}}} T\in \mathrm{C}^0\oplus \mathrm{C}^n\). We have \({\tilde{T}}T\in \mathrm{C}^{\overline{01}}\), \(\hat{{\tilde{T}}}T\in \mathrm{C}^{\overline{03}}\). Thus in the cases \(n=1, 2, 3{\ \mathrm mod}4\), we get \({\tilde{T}} T\in \mathrm{C}^0\) and \(\hat{{\tilde{T}}} T\in \mathrm{C}^0\). We obtain the group \(\mathrm{A}\cap \mathrm{B}=\mathrm{Q}\) in this case. In the case \(n=0{\ \mathrm mod}4\), we obtain the group \(\mathrm{A}^\prime \cap \mathrm{B}^\prime =\mathrm{Q}^\prime \).
In the exceptional cases \(n=2, 3\), we have \(\Gamma ^{\overline{0}}=\Gamma ^{0}=\mathrm{C}^\times \) because \(\mathrm{C}^{\overline{0}}=\mathrm{C}^{0}\in \mathrm{Z}\) in this case.
Let us prove that \(\Gamma ^{\overline{0}}\subseteq \mathrm{Q}\) in the cases \(n=1, 2, 3{\ \mathrm mod}4\), \(n\ge 5\), and \(\Gamma ^{\overline{0}}\subseteq \mathrm{Q}^\prime \) in the cases \(n=0{\ \mathrm mod}4\), \(n\ge 4\). Suppose \(T \mathrm{C}^{{\overline{0}}}T^{-1}\subseteq \mathrm{C}^{{\overline{0}}}\). We have
Therefore
In the cases \(n\ge 5\), this implies
because we can always represent each basis element of grade 2 as the product of elements from the subspace \(\mathrm{C}^{{\overline{0}}}\). For example, if \({\tilde{T}}T\) commutes with \(e_{1345}\) and \(e_{2345}\), then it commutes with \(e_{12}\). Using Lemma 3.1, we get
If \(n=4\), then \(\mathrm{C}^{{\overline{0}}}=\mathrm{C}^0\oplus \mathrm{C}^4\) and, from (6.16), we get \({\tilde{T}}T\in \mathrm{C}^{(0)}=\mathrm{C}^{024}\). Using \({\tilde{T}}T\in \mathrm{C}^{\overline{0}}\), we get \({\tilde{T}}T\in \mathrm{C}^{04}\), i.e. (6.17) again. Analogously, we obtain
Therefore
In the cases \(n\ge 5\), this implies
Using Lemma 3.1, we get
In the case \(n=4\), from (6.18), we get again (6.19). Finally, in the cases \(n=1, 2, 3{\ \mathrm mod}4\), for \(n\ge 4\), using \({\tilde{T}}T\in \mathrm{C}^{\overline{01}}\), \(\hat{{\tilde{T}}}T\in \mathrm{C}^{\overline{03}}\), we get from (6.17) and (6.19) the conditions \({\tilde{T}}T\in \mathrm{C}^0\) and \(\hat{{\tilde{T}}} T\in \mathrm{C}^0\). We obtain the group \(\mathrm{A}\cap \mathrm{B}=\mathrm{Q}\) in this case. In the cases \(n=0{\ \mathrm mod}4\), \(n\ge 4\), from the conditions (6.17) and (6.19) we obtain the group \(\mathrm{A}^\prime \cap \mathrm{B}^\prime =\mathrm{Q}^\prime \). \(\square \)
Let us give one example for the case \(n=4\), \(p=1\), \(q=3\). The element \(T=e+e_{1234}\in \Gamma ^{\overline{2}}=\mathrm{Q}^\prime \) and \(T\notin \Gamma ^{\overline{1}}=\mathrm{Q}\). We have \((e_{1234})^2=-e\) and \(T^{-1}=\frac{1}{2}(e-e_{1234})\). Thus
and for an arbitrary \(U\in \mathrm{C}^{\overline{2}}\)
We get \(\mathrm{Q}=\Gamma ^{\overline{1}}\ne \Gamma ^{\overline{2}}=\mathrm{Q}^\prime \) in this case.
7 The Relations Between Groups
In this section, we prove some new properties of the groups \(\Gamma ^k\), \(\mathrm{Q}\), \(\mathrm{Q}^\prime \) and \(\mathrm{A}\). In particular, we show that these groups are closely related in the cases of small dimensions.
Lemma 7.1
We have
Below (see Theorem 7.3 and Lemma 7.5) we prove that we have equality in (7.1), (7.2), and (7.3) in the cases \(n\le 5\).
Proof
Let us have \(TU_kT^{-1}\subseteq \mathrm{C}^k\) for any \(U_k\in \mathrm{C}^k\) with some fixed k. We get
Multiplying both sides by T on the right and by \({\tilde{T}}\) on the left, we get
If k is odd and \(k\ne n\), then we can always represent each generator \(e_a\), \(a=1, \ldots , n\) as the product of elements of grade k. We obtain
i.e. \({\tilde{T}}T\in \mathrm{Z}^\times \).
If k is even and \(k\ne n\), \(k\ne 0\), then we can always represent each basis element of grade 2 as the product of elements of grade k. We obtain
Using Lemma 3.1, we get \({\tilde{T}}T\in \mathrm{C}^0\oplus \mathrm{C}^n\), which is equivalent to \({\tilde{T}}T\in \mathrm{C}^0\) in the cases \(n=1, 2, 3{\ \mathrm mod}4\).
Also we have
Multiplying both sides by T on the right and by \(\hat{{\tilde{T}}}\) on the left, we get
If k is odd and \(k\ne n\), then we obtain \(\hat{{\tilde{T}}}T\in \mathrm{Z}^\times \). If k is even and \(k\ne n\), \(k\ne 0\), then we obtain \(\hat{{\tilde{T}}}T\in \mathrm{C}^0\oplus \mathrm{C}^n\), which is equivalent to \(\hat{{\tilde{T}}}T\in \mathrm{C}^0\) in the cases \(n=1, 2, 3{\ \mathrm mod}4\).
Finally, in the cases \(n=1, 2, 3{\ \mathrm mod}4\), we obtain the group \(\mathrm{Q}=\mathrm{A}\cap \mathrm{B}\). In the case \(n=0{\ \mathrm mod}4\), we obtain the group \(\mathrm{Q}=\mathrm{A}\cap \mathrm{B}\) for odd k and the group \(\mathrm{Q}^\prime =\mathrm{A}^\prime \cap \mathrm{B}^\prime \) for even k. \(\square \)
Using previous Lemma, we conclude that \(\Gamma ^k\subseteq \mathrm{P}\) because \(\mathrm{Q}\subseteq \mathrm{P}\), \(\mathrm{Q}^\prime \subseteq \mathrm{P}\) (see Lemmas 6.1 and 6.2). Thus we can rewrite the definitions of the groups \(\Gamma ^k\) (2.1) in the form
Lemma 7.2
We have
Proof
If n is odd, then \(T \mathrm{C}^k T^{-1}\subseteq \mathrm{C}^k\) is equivalent to \(T \mathrm{C}^{n-k} T^{-1}\subseteq \mathrm{C}^{n-k}\), \(k=1, \ldots , n-1\), because we can multiply both sides of this conditions by \(e_{1\ldots n}\in \mathrm{Z}\) and use \(\mathrm{C}^k e_{1\ldots n}=\mathrm{C}^{n-k}\).
If n is even, then \(T \mathrm{C}^k T^{-1}\subseteq \mathrm{C}^k\) is also equivalent to \(T \mathrm{C}^{n-k}T^{-1}\subseteq \mathrm{C}^{n-k}\), \(k=1, \ldots , n-1\). We can multiply both sides of this conditions by \(e_{1\ldots n}\) and use that \(T\in \mathrm{C}^{\times (0)}\cup \mathrm{C}^{\times (1)}\) (see the note before this Lemma), use that \(e_{1\ldots n}\) commutes with all even elements and anticommutes with all odd elements, and \(\mathrm{C}^k e_{1\ldots n}=\mathrm{C}^{n-k}\). \(\square \)
Theorem 7.3
We have
Proof
In the cases \(n\le 4\), we have
because \(\mathrm{C}^{\overline{1}}=\mathrm{C}^1\). In the case \(n=5\), we have for \(T\in \mathrm{Q}\)
Suppose that
with some constant \(\lambda \). Then
where we use the property \(\langle AB\rangle _0=\langle BA \rangle _0\) of the scalar part operation [10] (it is the projection onto the subspace \(\mathrm{C}^0\))Footnote 5, and \(e_{1\ldots 5}\in \mathrm{Z}\).
In the case \(n=p=6\), \(q=0\), consider the element
We have
and
but \(e_{23456}\in \mathrm{C}^{\overline{1}}\). Thus \(T\in \mathrm{Q}\), \(T\notin \Gamma \), and \(\mathrm{Q}\ne \Gamma \) in this case. \(\square \)
Note that the following analogue of the statement of the previous theorem is known: in the cases \(n\le 5\), if \(T\in \mathrm{C}^{(0)}\) and \({\tilde{T}}T=\pm e\), then \(T\mathrm{C}^1T^{-1}\subseteq \mathrm{C}^1\). That is why the spin group in the cases \(n\le 5\) (see, for example, [13]) can be defined as
Lemma 7.4
We have
Proof
We have \(\mathrm{Q}=\mathrm{P}\) for \(n\le 3\) by Lemma 6.1. Let us prove that \(\mathrm{Q}=\mathrm{A}\) for \(n\le 3\). If \(T\in \mathrm{A}\), then \(T\mathrm{C}^{\overline{01}}T^{-1}\subseteq \mathrm{C}^{\overline{01}}\), in particular,
Using the property \(\langle TUT^{-1} \rangle _0=\langle U \rangle _0\) of the scalar part operation, we get
Thus, \(\mathrm{A}\subseteq \Gamma \) in the cases \(n\le 3\). We have \(\Gamma =\mathrm{Q}\) for \(n\le 5\), and \(\mathrm{Q}\subseteq \mathrm{A}\). Thus we get \(\mathrm{A}=\mathrm{Q}\) for \(n\le 3\).
In the case \(n=4\), for the element
we have
Thus \(S\in \mathrm{A}\), \(S\notin \mathrm{Q}\), and \(\mathrm{A}\ne \mathrm{Q}\) in the case \(n=4\). \(\square \)
Lemma 7.5
We have the following relations in the cases of small dimensions
Proof
We have \(\Gamma ^1=\mathrm{Q}\) for \(n\le 5\). If we have \(\Gamma ^1\subseteq \Gamma ^k\subseteq \mathrm{Q}\) for some fixed k (see Lemma 7.1), then we can conclude that \(\Gamma ^1=\Gamma ^k=\mathrm{Q}\) for this k. In the case \(n=4\), we get \(\Gamma ^2=\Gamma ^{\overline{2}}=\mathrm{Q}^\prime =\mathrm{P}\) from Theorem 6.3. \(\square \)
Note that \(\Gamma ^1 \ne \Gamma ^3\) in the case \(n=6\). This fact follows from Theorem 6.3 (we have \(\Gamma ^3=\Gamma ^{\overline{3}}=\mathrm{Q}\) in the case \(n=6\)) and Theorem 7.3 (we have \(\Gamma \ne \mathrm{Q}\) in the case \(n=6\)). For example, the element \(T=e+2e_{123456}\) is invertible because
is a nonzero scalar. We have
for an arbitrary \(U\in \mathrm{C}^3\) because \(e_{123456}\) anticommutes with all odd elements. Also we have
Thus \(T\notin \Gamma \) and \(T\in \Gamma ^3\).
8 The Corresponding Lie Algebras
It is well-known (see, for example, [3, 13]) that the Lie algebra of the Lie group \(\Gamma \) is the following subspace with respect to the operation of commutator
with dimension
Theorem 8.1
The Lie groups \(\mathrm{P}\), \(\mathrm{A}\), \(\mathrm{B}\), \(\mathrm{Q}\), and \(\mathrm{Q}^\prime \) have the corresponding Lie algebrasFootnote 6\({\mathfrak {p}}\), \({\mathfrak {a}}\), \({\mathfrak {b}}\), \({\mathfrak {q}}\), and \({\mathfrak {q}}^\prime \) illustrated in Table 1 with corresponding dimensions. Also we have
In the cases \(n=0{\ \mathrm mod}4\), we have
Note that the statements (8.1) are analogues of the corresponding statements (6.2), (6.3), (6.4), (7.1), (7.2), (7.4), (7.5), (7.6) for the Lie groups. The statements (8.2) are analogues of the corresponding statements (6.6), (6.7), (6.8), (6.9), (7.3) for the Lie groups.
Proof
To prove these statements, we use the standard facts about the relation between an arbitrary Lie group and the corresponding Lie algebra.
To calculate dimensions, we use (see, for example, [22])Footnote 7
All statements (8.1), (8.2) are easily verified using definitions of the corresponding Lie algebras. \(\square \)
9 The Cases of Small Dimensions \(n\le 5\)
Let us write down all the Lie groups considered in this paper and the corresponding Lie algebras for the cases of small dimensions \(n\le 5\).
If \(n=1\), then the Clifford algebra \(\mathrm{C}\) is a commutative algebra and all considered Lie groups coincide with \(\mathrm{C}^\times \). The corresponding Lie algebra is \(\mathrm{C}\) considered w.r.t. the operation of commutator.
If \(n=2\), then we have two different groups
with the corresponding Lie algebras \(\mathrm{C}\) and \(\mathrm{C}^{02}\) respectively.
If \(n=3\), then we have two different groups
with the corresponding Lie algebras \(\mathrm{C}\) and \(\mathrm{C}^{023}\) respectively.
If \(n=4\), then we have five different groups
with the corresponding Lie algebras \(\mathrm{C}\), \(\mathrm{C}^{024}\), \(\mathrm{C}^{02}\), \(\mathrm{C}^{012}\), and \(\mathrm{C}^{023}\) respectively.
If \(n=5\), then we have five different groups
with the corresponding Lie algebras \(\mathrm{C}\), \(\mathrm{C}^{0245}\), \(\mathrm{C}^{025}\), \(\mathrm{C}^{012}\), and \(\mathrm{C}^{0235}\) respectively.
The relations between all the considered Lie groups are illustrated in Fig. 1.
10 Conclusions
In this paper, we present an explicit form of the groups of elements that define inner automorphisms preserving different naturally defined subspaces (subspaces defined by the reversion and the grade involution and subspaces of fixed grades) of the real or complex Clifford algebra \(\mathrm{C}\)
We present the corresponding Lie algebras. The relations between Lie groups (and Lie algebras) are illustrated in Table 1 and Fig. 1. One of this groups (the Clifford group \(\Gamma \), which preserves the subspace of grade 1) is well-known and is widely used in the theory of spin groups. The other groups \(\mathrm{A}, \mathrm{B}, \mathrm{Q}, \mathrm{Q}^\prime \) are related to the norm functions \(\psi (T)={\tilde{T}}T\) and \(\chi (T)=\hat{{\tilde{T}}}T\) and can be also used in different applications.
Note that the spin groups are defined as normalized subgroups (using the norm functions \(\psi \) and \(\chi \)) of the Lipschitz group
which is a subgroup of the Clifford group \(\Gamma \) (we have \(\Gamma ^\pm \subseteq \Gamma \) and \(\Gamma =\mathrm{Z}^\times \Gamma ^\pm \)). We can consider normalized subgroups (using the same norm functions \(\psi \) and \(\chi \)) of the groups \(\mathrm{P}\), \(\mathrm{A}\), \(\mathrm{B}\), \(\mathrm{Q}\), \(\mathrm{Q}^\prime \). These normalized subgroups can be interpreted as generalizations of the spin groups and can be used in different applications of Clifford algebras. In the papers [20,21,22], we have already considered some of these groups (see the groups \(\mathrm{G}^{23}_{p,q}\), \(\mathrm{G}^{12}_{p,q}\), \(\mathrm{G}^2_{p,q}\) in [22]) and found isomorphisms between these groups and classical matrix Lie groups. Some of these groups are related to automorphism groups of the scalar products on the spinor spaces (see [1, 3, 10, 12]).
All the groups considered in this paper contain spin groups as subgroups. The real Clifford algebras are isomorphic to the matrix algebras over \({\mathbb R}\), \({\mathbb R}\oplus {\mathbb R}\), \({\mathbb C}\), \({\mathbb H}\), or \({\mathbb H}\oplus {\mathbb H}\) depending on \(p-q{\ \mathrm mod}8\) and the complex Clifford algebras are isomorphic to the matrix algebras over \({\mathbb C}\) or \({\mathbb C}\oplus {\mathbb C}\) depending on \(n{\ \mathrm mod}2\). In the opinion of the author, the structure of naturally defined fundamental subspaces (subspaces of fixed grades and subspaces defined by the reversion and the grade involution) favourably distinguishes Clifford algebras from the corresponding matrix algebras, when we use them for applications. Groups that preserve different subspaces of Clifford algebra under similarity transformation may be of interest for different applications of Clifford algebras—in physics, engineering, robotics, computer vision, image and signal processing (see about applications of various groups in physics using Clifford algebra formalism, for example, in [11, 24]).
Notes
The area of interest of the author of this paper is mostly the applications of some specific algebras (as Clifford algebras) in physics, in particular, in the field theory. Therefore, let us present results only for the particular case of Clifford algebras in this paper. One can reformulate the statements for the more general cases of GCSAs or GCSAsWI, if the need arises for some purpose. We accompanied the statements of this paper with footnotes, including those received from the reviewer, about their possible reformulation for the more general case.
According to the reviewer, this fact is well-known. We present this statement for the sake of completeness. The new results for the groups \(\Gamma ^k\) are presented in Sect. 7.
As one of the anonymous reviewers of this paper noted, this statement can be reformulated for the more general case of the graded central simple algebras (GCSAs).
As one of the anonymous reviewers of this paper noted, the groups \(\mathrm{A}, \mathrm{B}, \mathrm{Q}\) can be defined in the more general case of the graded central simple algebras with involution (GCSAsWI) and the corresponding statements can be reformulated for this more general case.
This operation coincides with the trace of the corresponding matrix representation up to the scalar, which is the dimension of this representation (see [13]).
We present Lie algebras for all Lie groups considered in this paper for the sake of completeness. Some of them are known.
As one of the anonymous reviewers noted, the dimensions of the four subspaces \(\mathrm{C}^{\overline{k}}\), \(k=0, 1, 2, 3\) in the case of more general algebras (GCSAsWI) are also known from other works.
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Acknowledgements
The study presented in this paper was stimulated by discussions with Prof. A. Odzijewicz and other participants during the International Conference on Mathematical Methods in Physics (Morocco, Marrakesh, 2019). The author is grateful to the organizers and the participants of this conference. The author is grateful to the anonymous reviewers (especially to the second reviewer, who pointed out the feasibility of some of the results of this paper not only for the Clifford algebras but for more general algebras, as well as for other important comments) for their careful reading of the paper and helpful comments on how to improve the presentation. This work is supported by the grant of the President of the Russian Federation (Project MK-404.2020.1).
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This article is part of the Topical Collection on ICMMP, April 1–6, 2019, Marrakech, Morocco, edited by Zouhair Mouayn.
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Shirokov, D. On Inner Automorphisms Preserving Fixed Subspaces of Clifford Algebras. Adv. Appl. Clifford Algebras 31, 30 (2021). https://doi.org/10.1007/s00006-021-01135-6
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DOI: https://doi.org/10.1007/s00006-021-01135-6