Clifford algebra from a bunch of commutation and anti-commutation relations
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When reading the paper by Kitaev (arXiv:0901.2686), it seems to me there is a certain kind of theorem roughly like this:
"Consider an algebra formed by $B_i$, $i=1,2...n$ with $[ B_i, B_j]_{s_{ij}} =2 phi_{ij}$, where $phi_{ij}=0$ for $ineq j$, $phi_{ii}in {pm 1}$, $s_{ij} in {pm 1}$ for $ineq j$, $s_{ii}=+1$, and $[,]_{s_{ij}}$ means the commutation or anticommutation for $s_{ij}=-1$ or $s_{ij}=1$.
If there doesn't exist any product over some of the $B_i$s that commute with all $B_i$, then one can always find some products over $B_i$ to form $gamma_{mu}$, $mu=1,2...n$, such that the $gamma_{mu}$ form a Clifford algebra
${gamma_{mu}, gamma_{nu}}=2 eta_{mu nu}$ where $ eta_{mu nu} = diag(underbrace{1,...,1}_{p},underbrace{-1,...,-1}_q)$"
Could someone please comment on whether the theorem exists/where to find it or the direction for proving it? Thanks!
linear-algebra mathematical-physics clifford-algebras
asked Dec 3 at 19:18
Yen-Ta Huang
62
add a comment |
up vote
1
down vote
favorite
When reading the paper by Kitaev (arXiv:0901.2686), it seems to me there is a certain kind of theorem roughly like this:
"Consider an algebra formed by $B_i$, $i=1,2...n$ with $[ B_i, B_j]_{s_{ij}} =2 phi_{ij}$, where $phi_{ij}=0$ for $ineq j$, $phi_{ii}in {pm 1}$, $s_{ij} in {pm 1}$ for $ineq j$, $s_{ii}=+1$, and $[,]_{s_{ij}}$ means the commutation or anticommutation for $s_{ij}=-1$ or $s_{ij}=1$.
If there doesn't exist any product over some of the $B_i$s that commute with all $B_i$, then one can always find some products over $B_i$ to form $gamma_{mu}$, $mu=1,2...n$, such that the $gamma_{mu}$ form a Clifford algebra
${gamma_{mu}, gamma_{nu}}=2 eta_{mu nu}$ where $ eta_{mu nu} = diag(underbrace{1,...,1}_{p},underbrace{-1,...,-1}_q)$"
Could someone please comment on whether the theorem exists/where to find it or the direction for proving it? Thanks!
linear-algebra mathematical-physics clifford-algebras
asked Dec 3 at 19:18
Yen-Ta Huang
62
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
When reading the paper by Kitaev (arXiv:0901.2686), it seems to me there is a certain kind of theorem roughly like this:
"Consider an algebra formed by $B_i$, $i=1,2...n$ with $[ B_i, B_j]_{s_{ij}} =2 phi_{ij}$, where $phi_{ij}=0$ for $ineq j$, $phi_{ii}in {pm 1}$, $s_{ij} in {pm 1}$ for $ineq j$, $s_{ii}=+1$, and $[,]_{s_{ij}}$ means the commutation or anticommutation for $s_{ij}=-1$ or $s_{ij}=1$.
If there doesn't exist any product over some of the $B_i$s that commute with all $B_i$, then one can always find some products over $B_i$ to form $gamma_{mu}$, $mu=1,2...n$, such that the $gamma_{mu}$ form a Clifford algebra
${gamma_{mu}, gamma_{nu}}=2 eta_{mu nu}$ where $ eta_{mu nu} = diag(underbrace{1,...,1}_{p},underbrace{-1,...,-1}_q)$"
Could someone please comment on whether the theorem exists/where to find it or the direction for proving it? Thanks!
linear-algebra mathematical-physics clifford-algebras
asked Dec 3 at 19:18
Yen-Ta Huang
62
When reading the paper by Kitaev (arXiv:0901.2686), it seems to me there is a certain kind of theorem roughly like this:
"Consider an algebra formed by $B_i$, $i=1,2...n$ with $[ B_i, B_j]_{s_{ij}} =2 phi_{ij}$, where $phi_{ij}=0$ for $ineq j$, $phi_{ii}in {pm 1}$, $s_{ij} in {pm 1}$ for $ineq j$, $s_{ii}=+1$, and $[,]_{s_{ij}}$ means the commutation or anticommutation for $s_{ij}=-1$ or $s_{ij}=1$.
If there doesn't exist any product over some of the $B_i$s that commute with all $B_i$, then one can always find some products over $B_i$ to form $gamma_{mu}$, $mu=1,2...n$, such that the $gamma_{mu}$ form a Clifford algebra
${gamma_{mu}, gamma_{nu}}=2 eta_{mu nu}$ where $ eta_{mu nu} = diag(underbrace{1,...,1}_{p},underbrace{-1,...,-1}_q)$"
Could someone please comment on whether the theorem exists/where to find it or the direction for proving it? Thanks!
linear-algebra mathematical-physics clifford-algebras
linear-algebra mathematical-physics clifford-algebras
asked Dec 3 at 19:18
Yen-Ta Huang
62
asked Dec 3 at 19:18
Yen-Ta Huang
62
edited Dec 3 at 19:29
asked Dec 3 at 19:18
Yen-Ta Huang
62
asked Dec 3 at 19:18
Yen-Ta Huang
62
asked Dec 3 at 19:18
Yen-Ta Huang
62
62
add a comment |
add a comment |
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