A Note on Centralizers and Twisted Centralizers in Clifford Algebras
E. R. Filimoshina D. S. Shirokov

































A Note on Centralizers and Twisted Centralizers in Clifford Algebras]
 A Note on Centralizers and Twisted Centralizers in Clifford Algebras

E. Filimoshina]Ekaterina Filimoshina

HSE University

Moscow 101000

Russia

filimoshinaek@gmail.com



D. Shirokov]Dmitry Shirokov

HSE University

Moscow 101000

Russia


and


Institute for Information Transmission Problems of the Russian Academy of Sciences 

Moscow 127051 

Russia
dm.shirokov@gmail.com

15A66, 11E88





Last Revised:
 April 29, 2024


This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in computer science, physics, and engineering.



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    April 29, 2024
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§ INTRODUCTION









In this work, we consider degenerate and non-degenerate real and complex Clifford (geometric) algebras _p,q,r of arbitrary dimension and signature  (in
the case of any complex Clifford algebra, we can take q = 0). 
Degenerate Clifford algebras have  applications in physics <cit.>, geometry <cit.>, computer vision and image processing <cit.>, motion capture and robotics <cit.>, neural networks and machine learning <cit.>, etc.


Several recent works on Clifford algebras use the notion of centralizers and twisted centralizers in _p,q,r  <cit.>.
We call a centralizer of a set in _p,q,r a subset of all elements of _p,q,r that commute with all elements of this set. 
A twisted centralizer of a set in _p,q,r is a subset of such multivectors that their projections onto the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces commute and anticommute respectively with all elements of this set (see details in Section <ref>). 
Centralizers and twisted centralizers of some particular sets in _p,q,r are used in literature for various purposes. 
For example, the recent paper <cit.> finds an explicit form of the twisted centralizer of the grade-1 subspace in _p,q,r and applies it in the construction of Clifford group equivariant neural networks. 
The work <cit.> uses the explicit form of the same twisted centralizer when considering degenerate spin groups. 
The book <cit.> finds an explicit form of the centralizer of the even subspace in the case of the non-degenerate Clifford algebra _p,q,0.
In the papers <cit.>, the centralizers and twisted centralizers of the even subspace and the grade-1 subspace are employed in consideration of Lie groups preserving the even and odd subspaces under the adjoint and twisted adjoint representations in the non-degenerate Clifford algebras _p,q,0.
The works <cit.> find an explicit form of these centralizers in the case of arbitrary _p,q,r.

In light of appearance of centralizers and twisted centralizers in _p,q,r in the recent papers,
we decided to investigate them in this note.
We concentrate on the centralizers and twisted centralizers of the subspaces of fixed grades ^k_p,q,r, k=0,1,…,n, the subspaces ^m_p,q,r, m=0,1,2,3, determined by the grade involution and the reversion, and their direct sums. In particular, we consider the centralizers and twisted centralizers of the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces. We find an explicit form of these centralizers and twisted centralizers in the case of arbitrary k=0,1,…,n and m=0,1,2,3. 
We study the relations between the considered centralizers and the twisted centralizers.
This paper also considers the centralizers and twisted centralizers in the particular cases of the non-degenerate Clifford algebra _p,q,0 and the Grassmann algebra _0,0,n. Theorems <ref>, <ref>, <ref> and Lemmas <ref>, <ref>, <ref> are new.


The paper is structured as follows. 
Section <ref> introduces all the necessary notation related to _p,q,r. 
Section <ref> provides an explicit form of the centralizers and twisted centralizers of the subspaces ^k_p,q,r, k=0,1,…,n and considers the relations between them.
In Section <ref>, we write out all the considered centralizers and twisted centralizers in the particular cases _p,q,0 and _0,0,n and in the case of small k≤4.
Section <ref> provides an explicit form of the centralizers and twisted centralizers of the subspaces ^m_p,q,r, m=0,1,2,3, and their direct sums, in particular, the even and odd subspaces.
The conclusions follow in Section <ref>.







§ DEGENERATE AND NON-DEGENERATE CLIFFORD ALGEBRAS _P,Q,R


Let us consider the Clifford (geometric) algebra <cit.> (V)=_p,q,r, p+q+r=n≥1, over a vector space V with a symmetric bilinear form, where V can be real ^p,q,r or complex  ^p+q,0,r. We use  to denote the field of real numbers  in the first case and the field of complex numbers  in the second case. In this work, we consider both the case of the non-degenerate Clifford algebras _p,q,0 and the case of the degenerate Clifford algebras _p,q,r, r≠0.
We use Λ_r to denote the subalgebra _0,0,r, which is the Grassmann (exterior) algebra <cit.>.
The identity element is denoted by e, the generators are denoted by e_a, a=1,…,n. The generators satisfy the following conditions:

    e_a e_b + e_b e_a = 2 η_abe,    ∀ a,b=1,…,n,

where η=(η_ab) is the diagonal matrix with p times +1, q times -1, and r times 0 on the diagonal in the real case (^p,q,r) and p+q times +1 and r times 0 on the diagonal in the complex case (^p+q,0,r).

Let us consider the subspaces ^k_p,q,r of fixed grades k=0,…,n. Their elements are linear combinations of the basis elements e_A=e_a_1… a_k:=e_a_1⋯ e_a_k, a_1<⋯<a_k, labeled by ordered multi-indices A of length k, where 0≤ k≤ n. The multi-index with zero length k=0 corresponds to the identity element e. The grade-0 subspace is denoted by ^0 without the lower indices p,q,r, since it does not depend on the Clifford algebra's signature. We have ^k_p,q,r={0} for k<0 and k>n.
Let us use the following notation:

    ^≥ k_p,q,r   :=   ^k_p,q,r⊕^k+1_p,q,r⊕⋯⊕^n_p,q,r,
    
    ^≤ k_p,q,r   :=   ^0⊕^1_p,q,r⊕⋯⊕^k_p,q,r

for 0≤ k≤ n. For example, ^≥0_p,q,r=^≤ n_p,q,r=_p,q,r and ^≥ n_p,q,r=^n_p,q,r. 





Consider such conjugation operations as grade involution and reversion. The grade involute of an element  U∈_p,q,r is denoted by U and the reversion is denoted by U. These operations satisfy

    UV=UV,    UV=VU,    ∀ U, V∈_p,q,r.


The grade involution defines the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces:

    ^(k)_p,q,r = {U∈_p,q,r:  U=(-1)^kU}=⊕_j=k2^j_p,q,r,   k=0,1.

We can represent any element U∈_p,q,r as a sum 

    U=⟨ U⟩_(0)+⟨ U⟩_(1),    ⟨ U⟩_(0)∈^(0)_p,q,r,  ⟨ U⟩_(1)∈^(1)_p,q,r.

We use the angle brackets ⟨·⟩_(l) to denote the operation of projection of multivectors onto the subspaces ^(l)_p,q,r, l=0,1.
For an arbitrary subset ⊆̋_p,q,r, we have

    ⟨⟩̋_(0) := ∩̋^(0)_p,q,r,    ⟨⟩̋_(1) := ∩̋^(1)_p,q,r.




The grade involution and the reversion define four subspaces ^0_p,q,r,  ^1_p,q,r,  ^2_p,q,r, and  ^3_p,q,r (they are called the subspaces of quaternion types 0, 1, 2, and 3 respectively in the papers <cit.>):

    ^k_p,q,r={U∈_p,q,r: U=(-1)^k U,  U=(-1)^k(k-1)/2 U},  k=0, 1, 2, 3.

Note that the Clifford algebra _p,q,r can be represented as a direct sum of the subspaces ^k_p,q,r, k=0, 1, 2, 3, and viewed as ℤ_2×ℤ_2-graded algebra with respect to the commutator and anticommutator <cit.>.
To denote the direct sum of different subspaces, we use the upper multi-index and omit the direct sum sign. For instance, ^(1)24_p,q,r:=^(1)_p,q,r⊕^2_p,q,r⊕^4_p,q,r.










§ CENTRALIZERS AND TWISTED CENTRALIZERS OF THE SUBSPACES OF FIXED GRADES



Consider
the subset  ^m_p,q,r of all elements of _p,q,r commuting with all elements of the grade-m subspace for some fixed m:

    ^m_p,q,r:={X∈_p,q,r:   X V = V X,  ∀ V∈^m_p,q,r}.

Note that  ^m_p,q,r=_p,q,r for m<0 and m>n.
We call the subset ^m_p,q,r the centralizer (see, for example, <cit.>) of the subspace ^m_p,q,r in _p,q,r.

The center of the Clifford algebra _p,q,r is also the centralizer but of the entire Clifford algebra _p,q,r.
We denote the center of the degenerate and non-degenerate Clifford algebra _p,q,r by _p,q,r. It is well known (see, for example, <cit.>) that

    _p,q,r=
    {[ ^(0)_r⊕^n_p,q,r,                ,;          ^(0)_r,                . ].



Similarly, consider 
the set ^m_p,q,r:

    ^m_p,q,r:={X∈_p,q,r:  X V = V X,  ∀ V∈^m_p,q,r}.

Note that ^m_p,q,r=_p,q,r for m<0 and m>n.
We call the set ^m_p,q,r the twisted centralizer of the subspace ^m_p,q,r in _p,q,r. 
The particular case ^1_p,q,r is considered in the papers <cit.>.


Note that the projections ⟨^m_p,q,r⟩_(0) and ⟨^m_p,q,r⟩_(0) of ^m_p,q,r and ^m_p,q,r respectively onto the even subspace ^(0)_p,q,r  (<ref>) coincide by definition:

    ⟨^m_p,q,r⟩_(0)=⟨^m_p,q,r⟩_(0),    ∀ m=0,1,…,n.

In the case m=0, we have

    ^0_p,q,r=_p,q,r,    ^0_p,q,r={X∈_p,q,r:  X=X}=^(0)_p,q,r.

In Theorem <ref>, we find explicit forms of the centralizers ^m_p,q,r and the twisted centralizers  ^m_p,q,r of the subspaces of fixed grades for an arbitrary m=1,…,n. 







To prove Theorem <ref>, let us prove auxiliary Lemmas <ref>, <ref>, and <ref>.
In Lemmas <ref> and <ref>, we use that any non-zero X∈_p,q,r has the following decomposition over a basis:

    X=X_1+⋯+X_k,   X_i=α_ie_A_i,  α_i∈^×,   i=1,…,k,

where A_i is an ordered multi-index, A_i≠ A_j for i≠ j, and each X_i is non-zero, i.e. α_i≠0.



For any even m, we have

    ⟨^m_p,q,r⟩_(1)⊆⟨^m+1_p,q,r⟩_(1).



In the case of even  m<0 or m≥ n, we have ^m_p,q,r=^m+1_p,q,r=_p,q,r. For m=0, we have ⟨^0_p,q,r⟩_(1)=⟨^(0)_p,q,r⟩_(1)={0}, where we use (<ref>). 
Let us consider the case 0<m<n.
Consider a non-zero element X∈^m_p,q,r∩^(1)_p,q,r, where m is even, and its 
 decomposition over a basis  (<ref>).
 For any fixed e_a_1… a_m∈^m_p,q,r,  each summand X_i, i=1,…,k, contains at least one such e_x_i that x_i∈{a_1,…, a_m}, because otherwise we have X_i e_a_1… a_m=e_a_1… a_mX_i, so 
Xe_a_1… a_m≠ e_a_1… a_mX,
and we get a contradiction. 
Therefore, for any e_a_1… a_m+1∈^m+1_p,q,r,

    Xe_a_1… a_m+1=±X_1e_a_1…x̌_1… a_m+1e_x_1±⋯±X_ke_a_1…x̌_k… a_m+1e_x_k,

where X_i contains e_x_i, i=1,…,k, the sign  depends on the parity of the corresponding permutation, and the checkmarks indicate that the corresponding indices are missing. From (<ref>), we obtain

    Xe_a_1… a_m+1=
    ± e_a_1…x̌_1… a_m+1X_1 e_x_1±⋯± e_a_1…x̌_k… a_m+1X_k e_x_k
    
    =± e_a_1…x̌_1… a_m+1e_x_1 X_1±⋯± e_a_1…x̌_k… a_m+1e_x_kX_k=e_a_1… a_m+1 X,

where all the signs preceding the terms remain the same in (<ref>)–(<ref>),
since X_i∈^m_p,q,r∩^(1)_p,q,r and X_i contains e_x_i, i=1,…,k. This completes the proof.



For any odd m, we have

    ⟨^m_p,q,r⟩_(1)⊆⟨^m+1_p,q,r⟩_(1).



In the case m<0 or m≥ n, we have ^m_p,q,r=^m+1_p,q,r=_p,q,r.
Let us consider the case 0<m<n.
Consider a non-zero X∈^m_p,q,r∩^(1)_p,q,r, where m is odd, and its decomposition (<ref>).   
For any fixed e_a_1… a_m∈^m_p,q,r, each summand X_i, i=1,…,k, contains at least one such e_x_i that x_i∈{a_1,… a_m}, because otherwise X_i e_a_1… a_m=e_a_1… a_mX_i, so 
X e_a_1… a_m≠ e_a_1… a_m X,
and we get a contradiction. 
Hence, for any e_a_1… a_m+1∈^m+1_p,q,r, we have

    X e_a_1… a_m+1=±X_1e_a_1…x̌_1… a_m+1e_x_1±⋯±X_ke_a_1…x̌_k… a_m+1e_x_k,

where X_i contains e_x_i, i=1,…,k, the sign  depends on the parity of the corresponding permutation, and the checkmarks indicate that the corresponding indices are missing. From (<ref>), we obtain

    Xe_a_1… a_m+1=
    ± e_a_1…x̌_1… a_m+1X_1 e_x_1±⋯± e_a_1…x̌_k… a_m+1X_k e_x_k
    
    =± e_a_1…x̌_1… a_m+1e_x_1 X_1±⋯± e_a_1…x̌_k… a_m+1e_x_kX_k=e_a_1… a_m+1 X,

where all the signs preceding the terms in (<ref>)–(<ref>) remain the same,
since X_i∈^m_p,q,r∩^(1)_p,q,r
and X_i contains e_x_i, i=1,…,k. This completes the proof.




Note that the more general statement than (<ref>) holds true:

    ^m_p,q,r⊆^m+1_p,q,r,    ,

which follows from Theorem <ref> below and is provided in the formula (<ref>) of  Remark <ref>. 
Note that the statement (<ref>) can not be generalized in a similar way. For any even m, we have

    ⟨^m_p,q,r⟩_(0)⊆⟨^m+1_p,q,r⟩_(0),  ,
    
       (⟨^m_p,q,r⟩_(0)∖^n_p,q,r)⊆⟨^m+1_p,q,r⟩_(0),  ,

which follows from Theorem <ref> below as well.






For any M∈^m_p,q,r, K∈^k_p,q,r,  and L∈Λ^n-m_r, we have

    (KL)M=
    {[ M(KL)     ;; M(KL)       ].



     Suppose m,k=02. We have 
     
    (KL)M=(LM)K=(ML)K=M(KL),

     where we use that LM∈^n_p,q,r commutes with any even element, LM=ML, and LK=KL, since m and k are even respectively. 
     If m,k,n=12, then we again have (<ref>), since L is even and LM∈^n_p,q,r⊂_p,q,r is odd.

     Consider the case m=12 and k=02. 
     If n is odd, then L is even. We get (<ref>) again and obtain (KL)M=M(KL), since both K and L are even.
     If n is even, then L is odd and
     
    (KL)M=(LM)K=(ML)K=M(KL)=M(KL).

     Finally, suppose m,n=02 and k=12. We obtain 
     
    (KL)M=(LM)K=(ML)K=M(KL)=M(KL),

     since L is even, ML∈^n_p,q,r is even, and it anticommutes with all odd elements, including K.






    Note that 
    
    Λ^k_r ^m_p,q,r⊆^k+m_p,q,r,   k≥ 1;    Λ^0_r ^m_p,q,r =  ^m_p,q,r.

    Moreover, if at least one of k and m is even, then 
    
    X V = V X,    ∀ X∈Λ^k_r,  ∀ V∈^m_p,q,r.

    If both k and m are odd, then
    
    X V = V X,    ∀ X∈Λ^k_r,  ∀ V∈^m_p,q,r.


We use Remark <ref> in the proof of Theorem <ref>.
In Theorem <ref>, we find the centralizers and twisted centralizers for any m=1,…,n in the case r≠ n. The case of the Grassmann algebra _0,0,n is written out separately in Remark <ref>  for the sake of brevity in the theorem statement.

Consider the case r≠ n.

    
  * For an arbitrary even m, where n≥ m≥ 2, the centralizer has the form
    
    ^m_p,q,r=Λ^≤ n-m-1_r⊕⊕_k=12^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=02^m-2^k_p,q,0Λ^≥ n-m_r ⊕^n_p,q,r,

    and the twisted centralizer is equal to
    
    ^m_p,q,r=⟨Λ_r^≤ n-m-1⊕⊕_k=12^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=02^m-2^k_p,q,0Λ^≥ n-m_r ⊕^n_p,q,r⟩_(0)
                      ⊕⟨⊕_k=02^m-2^k_p,q,0Λ^≥ n-(m-1)_r
            ⊕⊕_k=12^m-1^k_p,q,0Λ^≥ n-m_r ⟩_(1).


    
  * For an arbitrary odd m, where n≥ m ≥ 1, we have
    
    ^m_p,q,r=Λ_r^≤ n-m-1⊕⊕_k=12^m-2^k_p,q,0Λ_r^≥ n-(m-1)⊕⊕_k=02^m-1^k_p,q,0Λ_r^≥ n-m

    and
     
    ^m_p,q,r = ⟨Λ_r^≤ n-m-1⊕⊕_k=12^m-2^k_p,q,0Λ^≥ n-(m-1)_r ⊕⊕_k=02^m-1^k_p,q,0Λ^≥ n-m_r ⟩_(0)
                  ⊕⟨⊕_k=02^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=12^m-2^k_p,q,0Λ^≥ n-m_r⊕^n_p,q,r⟩_(1).









Let us prove (<ref>). 
Namely, we prove that for any X∈_p,q,r and even m, where n≥ m≥ 2, the condition X e_a_1… a_m = e_a_1… a_m X for any basis element e_a_1… a_m∈^m_p,q,r is equivalent to the condition 

    X∈Λ^≤ n-m-1_r⊕⊕_k=12^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=02^m-2^k_p,q,0Λ^≥ n-m_r ⊕^n_p,q,r.

For any fixed a_1,…, a_m, we can always represent X as a sum of 2^m summands:

    X=Y  + e_a_1Y_a_1 + ⋯ + e_a_mY_a_m + e_a_1 a_2 Y_a_1 a_2 + ⋯ + e_a_1… a_mY_a_1… a_m,

where Y, Y_a_1, …, Y_a_1… a_m∈_p,q,r do not contain e_a_1,…, e_a_m. We get

    X e_a_1… a_m=(Y + ⋯ + e_a_1… a_mY_a_1… a_m) e_a_1… a_m
          = e_a_1… a_m( ∑^m_k=02 e_a_i_1… a_i_kY_a_i_1… a_i_k
        -∑^m-1_k=12 e_a_i_1… a_i_kY_a_i_1… a_i_k),

where a_i_1,…,a_i_k∈{a_1,…,a_m},
a_i_1<⋯<a_i_k, the elements 
e_A and Y_A with the multi-indices of zero length are the identity element e and Y respectively, and the minus sign precedes summands with e_a_i_1… a_i_k∈^(1)_p,q,r. 
We get that the condition X e_a_1… a_m=e_a_1… a_mX is equivalent to

    2e_a_1… a_m(∑^m-1_k=12 e_a_i_1… a_i_kY_a_i_1… a_i_k)=0.

The equation (<ref>) is equivalent to the system of 2^m-1 equations:

    (e_a_1)^2 e_a_2… a_mY_a_1=0,  …,    (e_a_m)^2 e_a_1… a_m-1Y_a_m=0,
       (e_a_1)^2 (e_a_2)^2 (e_a_3)^2 e_a_4… a_mY_a_1 a_2 a_3=0, …, (e_a_2)^2⋯(e_a_m)^2e_a_1Y_a_2… a_m=0.

Using (e_a_1)^2 e_a_2… a_mY_a_1=0 (<ref>), we get that if (e_a_1)^2≠0, i.e. a_1∈{1,…, p+q}, then Y_a_1=0, since Y_a_1 does not contain e_a_2,…,e_a_m. On the other hand, if each summand of X  either contains only the non-invertible generators, or contains  at least 1 invertible generator and at the same time does not contain less than m-1 generators, then the equation (e_a_1)^2 e_a_2… a_mY_a_1=0 is satisfied. Therefore, (e_a_1)^2 e_a_2… a_mY_a_1=0 is satisfied if and only if  X has no summands containing at least 1 invertible generator and at the same time not containing m-1 or more of any generators. 
Similarly, for any other odd k<m, using (e_a_1)^2 (e_a_2)^2 … (e_a_k)^2 e_a_k+1… a_mY_a_1 a_2 … a_k=0 (<ref>), we get Y_a_1 a_2 … a_k=0 if a_1,…, a_k∈{1,…,p+q}. 
Moreover, the equation (e_a_1)^2 (e_a_2)^2 … (e_a_k)^2 e_a_k+1… a_mY_a_1 a_2 … a_k=0 is satisfied if and only if X has no summands containing at least k invertible generators and at the same time not containing m-k or more of any generators for any odd k≤ m. 
This implies that for 

    X∈_p,q,r=Λ_r⊕^1_p,q,0Λ_r⊕⋯⊕^p+q_p,q,0Λ_r,

we finally obtain

    X   ∈   Λ_r⊕^1_p,q,0Λ^≥ n-(m-1)_r⊕^2_p,q,0Λ^≥ n-m_r⊕^3_p,q,0Λ^≥ n-(m-1)_r 
          ⊕⋯⊕^m-3_p,q,0Λ^≥ n-(m-1)_r⊕^m-2_p,q,0Λ^≥ n-m_r⊕^n_p,q,r,

since for any fixed odd k and even k+1, we have the following condition on d for the subspaces ^k_p,q,0Λ^d_r and ^k+1_p,q,0Λ^d_r respectively: the number of not contained generators should be less than m-k, i.e. n-(k+d)<m-k, so d≥ n-(m-1) for ^k_p,q,0Λ^d_r, and n-(k+1+d)<m-k, thus, d≥ n-m for ^k+1_p,q,0Λ^d_r.
This completes the proof.





Let us prove (<ref>). Namely, let us prove that for any X∈_p,q,r and odd m, where n≥ m≥ 1, the condition  Xe_a_1… a_m=e_a_1 … a_mX for any basis element e_a_1 … a_m∈^m_p,q,r is equivalent to the condition

    X∈Λ_r^≤ n-m-1⊕⊕_k=12^m-2^k_p,q,0Λ_r^≥ n-(m-1)⊕⊕_k=02^m-1^k_p,q,0Λ_r^≥ n-m.

For any fixed a_1,…,a_m, we can represent X as a sum of 2^m summands (<ref>), where Y,…,Y_a_1… a_m∈_p,q,r do not contain e_a_1,…,e_a_m.
We obtain

    Xe_a_1… a_m=(⟨ X⟩_(0)-⟨ X⟩_(1))e_a_1… a_m
          = e_a_1… a_m(∑^m-1_k=02 e_a_i_1… a_i_kY_a_i_1… a_i_k
        -∑^m_k=12 e_a_i_1… a_i_kY_a_i_1… a_i_k),

where  a_i_1,…,a_i_k∈{a_1,…,a_m}, a_i_1<⋯<a_i_k, the elements 
e_A and Y_A with the multi-indices of zero length are the identity element e and Y respectively, and the minus sign precedes summands with e_a_i_1… a_i_k∈^(1)_p,q,r. 
We get that the equality Xe_a_1… a_m=e_a_1… a_mX is equivalent to the formula

    2e_a_1… a_m(∑^m_k=12 e_a_i_1… a_i_kY_a_i_1… a_i_k)=0.

Similar to how it is done for the formula (<ref>) above, from the formula (<ref>), we get that it
 is equivalent to the condition that X has no summands containing at least k invertible generators and at the same time not containing m-k or more of any generators for any odd k≤ m. So, for X∈Λ_r⊕^1_p,q,0Λ_r⊕⋯⊕^p+q_p,q,0Λ_r, we get

    X   ∈   Λ_r⊕^1_p,q,0Λ^≥ n-(m-1)_r⊕^2_p,q,0Λ^≥ n-m_r⊕^3_p,q,0Λ^≥ n-(m-1)_r 
          ⊕⋯⊕^m-2_p,q,0Λ^≥ n-(m-1)_r⊕^m-1_p,q,0Λ^≥ n-m_r,

since, similarly to the proof of (<ref>) above, for any odd k, for ^k_p,q,0Λ^d_r, we have the condition n-(k+d)<m-k, i.e. d≥ n-(m-1), and for any ^k+1_p,q,0Λ^d_r, we get n-(k+1+d)<m-k, thus, d≥ n-m. This completes the proof.









Now we prove (<ref>). Suppose m is even and n≥ m≥ 2. Since ⟨^m_p,q,r⟩_(0)=⟨^m_p,q,r⟩_(0) (<ref>), we only need to prove 

    ⟨^m_p,q,r⟩_(1)=⟨⊕_k=02^m-2^k_p,q,0Λ^≥ n-(m-1)_r
            ⊕⊕_k=12^m-1^k_p,q,0Λ^≥ n-m_r ⟩_(1).

First, we prove that the right set is a subset of the left one in (<ref>). 
We have ^k_p,q,0Λ^n-m_r⊆^m_p,q,r for any odd k, where m-1≥ k≥ 1, and even n by Lemma <ref>. 
Let us prove ^k_p,q,0Λ^≥ n-(m-1)_r⊆^m_p,q,r for any even k=0,…,m-2.
Note that n-(m-1)≥ 1; hence, we have Λ^≥ n-(m-1)_r^m_p,q,r⊆^≥ n-(m-1)+m_p,q,r=^n+1_p,q,r={0} and, similarly, ^m_p,q,rΛ^≥ n-(m-1)_r={0} by Remark <ref>. Therefore, for any   k=0,…,n, 

    (^k_p,q,0Λ^≥ n-(m-1)_r) ^m_p,q,r= ^m_p,q,r (^k_p,q,0Λ^≥ n-(m-1)_r)={0}.

Thus, 
^k_p,q,0Λ^≥ n-(m-1)_r⊆^m_p,q,r. 
Let us prove that the left set is a subset of the right one in (<ref>). 
Using ⟨^m_p,q,r⟩_(1)⊆⟨^m+1_p,q,r⟩_(1) for any even m by Lemma <ref> and applying (<ref>) proved above,  we get

    ⟨^m_p,q,r⟩_(1)⊆⟨Λ^≤ n-(m+1)_r⊕⊕_k=02^m^k_p,q,0Λ_r^≥ n-(m+1)⊕⊕_k=12^m-1^k_p,q,0Λ_r^≥ n-m⟩_(1).

Now let us show that the inclusion above implies the inclusion of the left set in the right one in (<ref>).
The projection of ⟨^m_p,q,r⟩_(1) onto the subspace ⟨Λ^≤ n-(m+1)_r⟩_(1) equals zero, since for any odd basis element X∈Λ^≤ n-(m+1)_r there exists such an even basis element V∈^m_p,q,r that XV≠0 and XV=VX. For example, in the case n=r=4 and m=2, for X=e_1, we have V=e_23, and e_1e_23=e_23e_1≠ e_23e_1.
The projection of ⟨^m_p,q,r⟩_(1) onto ⟨^k_p,q,0Λ^n-m_r⟩_(1) equals zero for any even k, m and odd n by Lemma <ref>. The projection of ⟨^m_p,q,r⟩_(1) onto ⟨^k_p,q,0Λ^n-m-1_r⟩_(1) equals zero for any even k, where m≥ k>0, since for any basis elements K=e_a_1… a_k∈^k_p,q,0⊂^(0)_p,q,r and L∈Λ^n-m-1_r⊆^(1)_p,q,r, there exists such an even grade-m element M∈ e_a_1… a_k^m-k_p,q,r that LM≠0, LM=ML, and KM=MK, so we get (KL)M=M(KL)≠ M(KL), where we use Remark <ref>. For example, if n=6, p=k=2, and r=m=4, for K=e_12∈^2_2,0,0 and L=e_3∈Λ^1_4, there exists M=e_1245∈ e_12^2_2,0,4, such that (e_12e_3)e_1245=e_1245(e_12e_3)≠ e_1245(e_12e_3). Finally, the projection of ^m_p,q,r onto ^m_p,q,0Λ^≥ n-(m-1)_r={0} equals zero as well. Thus, we obtain (<ref>), and the proof is completed.



Finally, let us prove (<ref>). Suppose m is odd and n≥ m≥ 1. We have ⟨^m_p,q,r⟩_(0)=⟨^m_p,q,r⟩_(0) (<ref>), so we only need to prove 

    ⟨^m_p,q,r⟩_(1)=⟨⊕_k=02^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=12^m-2^k_p,q,0Λ^≥ n-m_r⊕^n_p,q,r⟩_(1).

We obtain that the right set is a subset of the left one in (<ref>), using ^m_p,q,rΛ^≥ n-(m-1)_r={0}, Lemma <ref>, and ⟨^n_p,q,r⟩_(1)⊂_p,q,r.
Let us prove that the left set is a subset of the right one in (<ref>). Using ⟨^m_p,q,r⟩_(1)⊆⟨^m+1_p,q,r⟩_(1) by Lemma <ref> and applying (<ref>) proved above, we get

    ⟨^m_p,q,r⟩_(1)   ⊆   ⟨Λ^≤ n-m-2_r⊕⊕_k=02^m-1^k_p,q,0Λ^≥ n-m-1_r
        
          ⊕⊕_k=12^m-2^k_p,q,0Λ^≥ n-m_r ⊕^n_p,q,r⟩_(1).

The projection of ⟨^m_p,q,r⟩_(1) onto the subspace ⟨Λ^≤ n-m-2_r⟩_(1) equals zero because for any odd X∈Λ^≤ n-m-2_r there exists such an odd V∈^m_p,q,r that XV≠0 and XV=-VX. For example, if n=r=4 and m=1, for X=e_1, we have V=e_2, and e_1e_2=-e_2e_1. The projection of  ⟨^m_p,q,r⟩_(1) onto ^k_p,q,0Λ^n-m_r for any even  k and odd m, n is zero by Lemma <ref>. The projection of ⟨^m_p,q,r⟩_(1) onto ⟨^k_p,q,0Λ^n-m-1_r⟩_(1) for any even k≤ m-1 equals zero because for any basis elements K=e_a_1… a_k∈^k_p,q,0⊆^(0)_p,q,r and L∈Λ^n-m-1_r⊆^(1)_p,q,r, there exists such an odd grade-m element M∈ e_a_1… a_k^m-k_p,q,r that LM≠0, LM=ML, and KM=MK, therefore, (KL)M=KML=M(KL)≠ M(KL).  For example, if n=5, p=k=2, and r=m=3, for K=e_12∈^2_2,0,0 and L=e_3∈Λ^1_3, we can take M=e_124∈ e_12^1_2,0,3 and get (e_12e_3)e_124=-e_124(e_12e_3). Finally, ^m-1_p,q,0Λ^≥ n-(m-1)_r=^n_p,q,r. Thus, we obtain  (<ref>), and the proof is completed.










    Note that Theorem  <ref> implies

    ^m_p,q,r⊆^m+2_p,q,r,  ^m_p,q,r⊆^m+2_p,q,r,     m=1,…, n-2;
       ^m_p,q,r⊆^m+1_p,q,r,    ^m_p,q,r⊆^m+1_p,q,r,    ;
       ^m_p,q,r⊆^m+2_p,q,r,    .

Using (<ref>)–(<ref>), we get

    ^m_p,q,r⊆^4_p,q,r,    ^m_p,q,r⊆^4_p,q,r,     m=1,2,3.

If r≤ n-(m+1), then
    
    ^m_p,q,r=^1_p,q,r,  ^m_p,q,r=^1_p,q,r,  ;
            
          ^m_p,q,r=^2_p,q,r,  ^m_p,q,r=^2_p,q,r,  ,   m≠0.

We use these relations to prove Theorems <ref> and <ref>.







  




§ PARTICULAR CASES OF CENTRALIZERS AND TWISTED CENTRALIZERS


In this section, we consider the centralizers and twisted centralizers in the particular cases that are important for applications. In Remarks <ref> and <ref> below, we explicitly write out ^m_p,q,r and ^m_p,q,r, m=0,1,…,n, in the cases of the non-degenerate Clifford algebra _p,q,0 and the Grassmann algebra _0,0,n respectively. Note that in these special cases, the centralizers and twisted centralizers have a much simpler form than in the general case of arbitrary _p,q,r (Theorem <ref>).


 In the particular case of the non-degenerate algebra _p,q,0, we get from Theorem <ref>

    ^m_p,q,0=
    {[                _p,q,0, m=0;   m=n     m,n=12;;            ^(0)_p,q,0,        m=n     m,n=02;;             ^0n_p,q,0,        m≠ 0,n     m=02;                              m≠ n     m,n=12;;                    ^0,         m=12     n=02; ].

and

    ^m_p,q,0={[                   _p,q,0,            m=n,   m,n=02;;               ^(0)_p,q,0,    m=0;   m=n     m,n=12;;                ^0n_p,q,0,        m≠ 0,n     m,n=02;;                       ^0,             m≠ n     m=12;                           m≠ 0,    m=02,      n=12. ].






 In the particular case of the Grassmann algebra _0,0,n=Λ_n, we obtain from Theorem <ref>

    ^0_0,0,n=Λ_n,    ^0_0,0,n=Λ^(0)_n;
    
       ^m_0,0,n=Λ_n,  ^m_0,0,n=Λ^(0)_n ⊕⟨Λ^≥ n-m+1_n⟩_(1),   m=02,   m≠0;
    
       ^m_0,0,n=Λ^(0)_n⊕⟨Λ^≥ n-m+1_n⟩_(1),  ^m_0,0,n=Λ_n,   m=12.











In Remark <ref> below, we explicitly write out the particular cases of Theorem <ref> and Remark <ref> in the case of small k≤ 4. We use these centralizers and twisted centralizers in Theorems <ref> and <ref>.
Note that some of these centralizers and twisted centralizers are considered, for instance, in the papers <cit.>.
The cases ^2_p,q,r (<ref>) and ^1_p,q,r (<ref>) are proved in detail, for example, in <cit.>. The other cases are presented for the first time. 


We have:

    ^1_p,q,r=_p,q,r={[ Λ^(0)_r⊕^n_p,q,r,                 ,;          Λ^(0)_r,                 , ].
          ^2_p,q,r=
    {[ Λ_r⊕^n_p,q,r,         r≠ n,;          Λ_r,          r=n, ].
          ^3_p,q,r={[          Λ^(0)_r⊕Λ^n-2_r⊕^1_p,q,0(Λ^n-3_r⊕Λ^n-2_r);                         ⊕^2_p,q,0Λ^n-3_r⊕^n_p,q,r,                                                  ,;                                                   ; Λ^(0)_r⊕Λ^n-1_r⊕^1_p,q,0Λ^≥ n-2_r⊕^2_p,q,0Λ^n-2_r,                                                  , ].
    
          ^4_p,q,r={[ Λ_r⊕^1_p,q,0(Λ^n-3_r⊕Λ^n-2_r)⊕^2_p,q,0(Λ^n-4_r⊕Λ^n-3_r)⊕^n_p,q,r,                                                             r≠ n,;                                                              Λ_r,                                                              r=n, ].

and:

    ^1_p,q,r=Λ_r,
          ^2_p,q,r=
        {[            Λ^(0)_r⊕Λ^n_r⊕^1_p,q,0Λ^n-1_r,                                         ,; Λ^(0)_r⊕Λ^n-1_r⊕^1_p,q,0Λ^n-2_r⊕^n_p,q,r,                                 ,   r≠ n,;                          Λ^(0)_r⊕Λ^n-1_r,                                  ,   r=n, ].
        
          ^3_p,q,r=Λ_r⊕^1_p,q,0Λ^≥ n-2_r⊕^2_p,q,0Λ^≥ n-3_r,
          ^4_p,q,r=
        {[ Λ^(0)_r ⊕Λ^n-2_r⊕Λ^n_r⊕^1_p,q,0Λ^≥ n-3_r;      ⊕^2_p,q,0Λ^≥ n-3_r⊕^3_p,q,0Λ^n-3_r,                                        ,;                                         ; Λ^(0)_r ⊕Λ^n-3_r⊕Λ^n-1_r⊕^3_p,q,0Λ^n-4_r;      ⊕^2_p,q,0(Λ^n-4_r⊕Λ^n-3_r)⊕^n_p,q,r;    ⊕^1_p,q,0(Λ^n-4_r ⊕Λ^n-3_r ⊕Λ^n-2_r),                                ,   r≠ n,;                                         ;                Λ^(0)_r ⊕Λ^n-3_r⊕Λ^n-1_r,                                 ,   r=n. ].






In Remark <ref>, we consider how the centralizers and the twisted centralizers are related to the kernels of the adjoint and twisted adjoint representations.

    Note that 

    ^1×_p,q,r=(),    ^1×_p,q,r=()

and 

    ()⊆^m×_p,q,r,    ()=Λ^(0)×_r⊆^m×_p,q,r,      m=0,…,n,

where (), (), and () are the kernels of the adjoint representation  and the twisted adjoint  representations  and  respectively. 
The adjoint representation :^×_p,q,r→(_p,q,r) acts on the group of all invertible elements as T↦_T, where 

    _T(U)=TU T^-1,     U∈_p,q,r,     T∈^×_p,q,r.

The twisted adjoint representation  has been introduced in a particular case by Atiyah, Bott, and Shapiro in <cit.>. The representation :^×_p,q,r→(_p,q,r) acts on ^×_p,q,r as T↦_T with

    _T(U)=TU T^-1,     U∈_p,q,r,     T∈^×_p,q,r.

The representation :^×_p,q,r→(_p,q,r) acts on ^×_p,q,r  as T↦_T with 

    _T(U)=T⟨ U⟩_(0) T^-1+T⟨ U⟩_(1) T^-1,  ∀ U∈_p,q,r,   T∈^×_p,q,r.

See the details about , , , and their kernels, for example, in <cit.>.












§ CENTRALIZERS AND TWISTED CENTRALIZERS OF THE SUBSPACES DETERMINED BY THE GRADE INVOLUTION AND THE REVERSION


This section finds explicit forms of the centralizers and twisted centralizers of the subspaces ^m_p,q,r (<ref>), m=0,1,2,3, determined by the grade involution and the reversion  and their direct sums. In particular, we consider the centralizers and twisted centralizers of the even ^(0)_p,q,r=^02_p,q,r and odd ^(1)_p,q,r=^13_p,q,r subspaces.


Let us consider the centralizers ^m_p,q,r and twisted centralizers ^m_p,q,r of the subspaces ^m_p,q,r (<ref>), m=0,1,2,3, in _p,q,r:

    ^m_p,q,r   :=   {X∈_p,q,r:   X V = V X,  ∀ V∈^m_p,q,r},   m=0,1,2,3,
    ^m_p,q,r   :=   {X∈_p,q,r:  X V = V X,  ∀ V∈^m_p,q,r},   m=0,1,2,3.

In Theorem <ref>, we prove that ^m_p,q,r and ^m_p,q,r coincide with some of the centralizers ^m_p,q,r and the twisted centralizers ^m_p,q,r  of the subspaces of fixed grades, which are considered in Sections <ref> and <ref>. 



We have

    ^m_p,q,r=^m_p,q,r,  ^m_p,q,r=^m_p,q,r,   m=1,2,3;
    
       ^0_p,q,r=^4_p,q,r,  ^0_p,q,r=⟨^4_p,q,r⟩_(0).


The centralizers ^m_p,q,r and twisted centralizers  ^m_p,q,r, m=1,2,3,4, are written out explicitly in Remark <ref> for the readers' convenience. In the formula (<ref>), we have

    ⟨^4_p,q,r⟩_(0)={[          Λ^(0)_r⊕^1_p,q,0Λ^n-2_r⊕^2_p,q,0Λ^n-3_r,                                                  ; Λ^(0)_r⊕^1_p,q,0Λ^n-3_r⊕^2_p,q,0Λ^n-4_r⊕^n_p,q,r,                                                 . ].



The inclusions ^m_p,q,r⊆^m_p,q,r,  ^m_p,q,r⊆^m_p,q,r, m=1,2,3,  and ^0_p,q,r⊆^4_p,q,r follow from ^k_p,q,r⊆^k_p,q,r, k=1,2,3, and ^4_p,q,r⊂^0_p,q,r respectively. We get ^0_p,q,r⊆^4_p,q,r∩^0_p,q,r=^4_p,q,r∩^(0)_p,q,r, using ^04_p,q,r⊆^0_p,q,r and ^0_p,q,r=^(0)_p,q,r (<ref>). 


Let us prove ^m_p,q,r⊆^m_p,q,r and ^m_p,q,r⊆^m_p,q,r, m=1,2,3. Any basis element of ^k_p,q,r, k=1,2,3,  can be represented as a product of one basis element of ^k_p,q,r and basis elements of ^4_p,q,r. Since ^m_p,q,r⊆^4_p,q,r and ^m_p,q,r⊆^4_p,q,r by the statement (<ref>) of Remark <ref>, we get ^m_p,q,r⊆^m_p,q,r and ^m_p,q,r⊆^m_p,q,r.

We obtain ^4_p,q,r⊆^0_p,q,r and ^4_p,q,r∩^(0)_p,q,r⊆^0_p,q,r because ^4_p,q,r⊆^0_p,q,r and ^4_p,q,r∩^(0)_p,q,r⊆^(0)_p,q,r=^0_p,q,r (<ref>) respectively and any basis element of ^0_p,q,r∖^0 can be represented as a product of basis elements of ^4_p,q,r. 





Let us denote by ^km_p,q,r and ^km_p,q,r, k,m=0,1,2,3, the centralizers and the twisted centralizers respectively of the direct sums of the subspaces ^m_p,q,r (<ref>) in _p,q,r:

    ^km_p,q,r   :=   ^k_p,q,r∩^m_p,q,r={X∈_p,q,r:   X V = V X,  ∀ V∈^km_p,q,r},
    
    ^km_p,q,r   :=   ^k_p,q,r∩^m_p,q,r={X∈_p,q,r:  X V = V X,  ∀ V∈^km_p,q,r}.

Note that ^02_p,q,r, ^02_p,q,r, ^13_p,q,r,  and ^13_p,q,r are the centralizers and the twisted centralizers of the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces respectively.
In Theorem <ref>, we find explicit forms of ^km_p,q,r and ^km_p,q,r, k,m=0,1,2,3.









We have

    ^01_p,q,r=^12_p,q,r=^13_p,q,r=_p,q,r,  ^23_p,q,r=^2_p,q,r∩^3_p,q,r,
       ^02_p,q,r=^2_p,q,r,  ^03_p,q,r=^3_p,q,r,
       ^12_p,q,r=^1_p,q,r∩^2_p,q,r,  ^23_p,q,r=^2_p,q,r∩^3_p,q,r,  ^13_p,q,r=^1_p,q,r, 
       ^01_p,q,r=⟨^1_p,q,r⟩_(0),  ^02_p,q,r=⟨^2_p,q,r⟩_(0),  ^03_p,q,r=⟨^3_p,q,r⟩_(0).


The centralizers ^2_p,q,r, ^3_p,q,r, _p,q,r=^1_p,q,r and the twisted centralizer ^1_p,q,r are written out explicitly in Remark <ref>.
In the formulas (<ref>)–(<ref>), we have

    ^2_p,q,r∩^3_p,q,r=
         {[                        Λ^(0)_r⊕Λ^n-2_r⊕^n_p,q,r,                                                ;; Λ^(0)_r⊕Λ^n-1_r⊕^1_p,q,0Λ^n-1_r⊕^2_p,q,0Λ^n-2_r,                                                ; ].
         
       ^1_p,q,r∩^2_p,q,r =
          {[   Λ^(0)_r⊕Λ^n_r,                ;; Λ^(0)_r⊕Λ^n-1_r,                ; ].

    
    ^2_p,q,r∩^3_p,q,r={[                     Λ^(0)_r⊕Λ^n_r⊕^1_p,q,0Λ^n-1_r,                                                  ;; Λ^(0)_r⊕Λ^n-1_r⊕^1_p,q,0Λ^≥ n-2_r⊕^2_p,q,0Λ^n-2_r,                                                  ; ].
         
       ⟨^1_p,q,r⟩_(0)=Λ^(0)_r,  ⟨^2_p,q,r⟩_(0) =
         {[          Λ^(0)_r,              ;  ;; Λ^(0)_r⊕^n_p,q,r,                 ; ].
        
       ⟨^3_p,q,r⟩_(0) = 
        {[ Λ^(0)_r⊕^1_p,q,0Λ^n-2_r⊕^2_p,q,0Λ^n-3_r,                                        ;; Λ^(0)_r⊕^1_p,q,0Λ^n-1_r⊕^2_p,q,0Λ^n-2_r,                                        . ].



 First, let us prove (<ref>) and (<ref>).
    We get 
 
    ^23_p,q,r=^2_p,q,r∩^3_p,q,r=^2_p,q,r∩^3_p,q,r

by Theorem <ref>.
 For k=1,2,3, we obtain
 
    ^0k_p,q,r=^0_p,q,r∩^k_p,q,r=^4_p,q,r∩^k_p,q,r=^k_p,q,r,

 using  ^k_p,q,r=^k_p,q,r,  ^0_p,q,r=^4_p,q,r by Theorem <ref> and ^k_p,q,r⊆^4_p,q,r by Remark <ref>. Since ^1_p,q,r=_p,q,r by Remark <ref>, we get ^01_p,q,r=_p,q,r. Similarly, for l=2,3, we obtain
 
    ^1l_p,q,r = ^1_p,q,r∩^l_p,q,r=^1_p,q,r∩^l_p,q,r=_p,q,r∩^l_p,q,r=_p,q,r,

 where we use _p,q,r⊆^2_p,q,r and _p,q,r⊆^3_p,q,r by the formula (<ref>) of Remark <ref>.

Let us prove (<ref>). We get 

    ^12_p,q,r=^1_p,q,r∩^2_p,q,r=^1_p,q,r∩^2_p,q,r,   ^23_p,q,r=^2_p,q,r∩^3_p,q,r=^2_p,q,r∩^3_p,q,r,

 and ^13_p,q,r=^1_p,q,r∩^3_p,q,r=^1_p,q,r∩^3_p,q,r=^1_p,q,r, using Theorem  <ref> and ^1_p,q,r⊆^3_p,q,r by Remark <ref>.
    Now we prove (<ref>).  For k=1,2,3, we get
    
    ^0k_p,q,r   =   ^0_p,q,r∩^k_p,q,r=^4_p,q,r∩^(0)_p,q,r∩^k_p,q,r
       =   ^k_p,q,r∩^(0)_p,q,r=^k_p,q,r∩^(0)_p,q,r=⟨^k_p,q,r⟩_(0),
 
using Theorem  <ref>, ^k_p,q,r⊆^4_p,q,r by Remark <ref>, and (<ref>).










The equalities for the centralizer ^02_p,q,r and the twisted centralizer ^02_p,q,r of the even subspace presented in the formulas (<ref>) and (<ref>) are proved in Lemma 3.2 <cit.> in the case of the degenerate and non-degenerate algebras _p,q,r. In the non-degenerate case  _p,q,0, the set ^02_p,q,0 is considered, for example, in <cit.> and the set ^02_p,q,0 is considered in <cit.>. The other equalities in the formulas 
(<ref>)–(<ref>) are presented for the first time.




















§ CONCLUSIONS



In this work, we consider the centralizers and  twisted centralizers in degenerate and non-degenerate Clifford algebras _p,q,r. 
In Theorems <ref>, <ref>, and <ref>, we find an explicit form of the centralizers
and the twisted centralizers 

    ^k_p,q,r,  ^m_p,q,r,  ^ml_p,q,r,    ^k_p,q,r,  ^m_p,q,r,  ^ml_p,q,r

of the subspaces of fixed grades ^k_p,q,r, k=0,1,…,n, the subspaces ^m_p,q,r, m=0,1,2,3, determined by the grade involution and the reversion, and their direct sums ^ml_p,q,r, m,l=0,1,2,3. 
In particular, we consider the centralizers and twisted centralizers of the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces. 
The relations between ^k_p,q,r and ^k_p,q,r for different k are considered in Remark <ref>. 
We also consider the relation between their projections ⟨^k_p,q,r⟩_(1) and ⟨^k_p,q,r⟩_(1) onto the odd subspace ^(1)_p,q,r in Lemmas <ref>, <ref> and Remark <ref>. 

In the particular cases of the non-degenerate Clifford algebras _p,q,0 and the Grassmann algebras _0,0,n, the considered centralizers and the twisted centralizers have a simpler form than in the general case of arbitrary _p,q,r (see Remarks <ref> and <ref> respectively).
In the particular case of small k, the centralizers ^k_p,q,r and the twisted centralizers ^k_p,q,r have simple form as well and are written out in Remark <ref> for k≤ 4.

In the further research, we are going to use the explicit forms of the centralizers and the twisted centralizers presented in Theorems <ref>, <ref>, and <ref> to define and study several families of Lie
groups in _p,q,r. These groups preserve the subspaces ^m_p,q,r and their direct sums under the adjoint and twisted adjoint representations. These  Lie groups can be considered as generalizations of Clifford and Lipschitz groups and are important for the theory of spin groups.
We hope that the explicit forms of centralizers and twisted centralizers can be useful for applications of Clifford algebras in physics <cit.>, computer science, in particular, for neural networks and machine learning <cit.>, image processing <cit.>, and in other areas.





























§ ACKNOWLEDGEMENTS


The results of this paper were reported at the 13th International Conference on Clifford Algebras and Their Applications in Mathematical Physics, Holon, Israel, June 2023. The authors are grateful to the organizers and the participants of this conference for fruitful discussions.


The publication was prepared within the framework of the Academic Fund Program at HSE University (grant 24-00-001 Clifford algebras and applications).



Data availability Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.



Declarations

Conflict of interest The authors declare that they have no conflict of interest.
  

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