Obtaining all structure constants of $ mathfrak{su}(N) $
$begingroup$
I want to understand certain properties of general curves $textrm{Ad}_{e^{t X}} = e^{t ad_X}$ of adjoint-representation matrices of $SU(N)$. For this purpose, I would like to have an explicit closed-form expression for all structure constants $f_{abc}$ of $mathfrak{su}(N)$ for arbitrary $N$ in the form $[T_a, T_b]=f_{abc} T_c$. Ideally, I would like to have them for the usual choice of basis ${T_a}$ with the convention that all $T_a$ are traceless, anti-Hermitian and trace-orthogonal, with standard Dynkin index $ tr(T_a T_b) = - frac{1}{2} delta_{ab} $, in which $f_{abc}$ is totally antisymmetric.
Is there a reference where such expression can be found? Otherwise, I am more or less familiar with the machinery of the Cartan classification. What would be the best way to attack the problem?
linear-algebra abstract-algebra representation-theory lie-groups lie-algebras
asked Dec 7 '18 at 0:10
SergioHCSergioHC
64
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add a comment |
$begingroup$
I want to understand certain properties of general curves $textrm{Ad}_{e^{t X}} = e^{t ad_X}$ of adjoint-representation matrices of $SU(N)$. For this purpose, I would like to have an explicit closed-form expression for all structure constants $f_{abc}$ of $mathfrak{su}(N)$ for arbitrary $N$ in the form $[T_a, T_b]=f_{abc} T_c$. Ideally, I would like to have them for the usual choice of basis ${T_a}$ with the convention that all $T_a$ are traceless, anti-Hermitian and trace-orthogonal, with standard Dynkin index $ tr(T_a T_b) = - frac{1}{2} delta_{ab} $, in which $f_{abc}$ is totally antisymmetric.
Is there a reference where such expression can be found? Otherwise, I am more or less familiar with the machinery of the Cartan classification. What would be the best way to attack the problem?
linear-algebra abstract-algebra representation-theory lie-groups lie-algebras
asked Dec 7 '18 at 0:10
SergioHCSergioHC
64
$endgroup$
$begingroup$
Well, as you know from SU(3), the antihermitean basis for the fundamental is not so clean, but it does generalize readily to SU(N). However, if you forfeit antihermiticity, you have a celebrated clock-and-shift basis with analytic trigonometric structure constants, which is the language most prefer for large N systematics... Do you want details?
$endgroup$
– Cosmas Zachos
Dec 11 '18 at 13:59
$begingroup$
Thank you very much for your comment! I have read through the article you reference, and am wondering why I had not heard about these Sylvester matrices before. The construction seems very neat and I suppose all structure constants easily follow from the equation for $(Sigma_1)^k (Sigma_3)^j$. However, these are a basis for $mathfrak{gl}(N,mathbb{C})$. Is trivial to relate the structure constants of these to those of $mathfrak{su}(N)$ generators? Yes please, I would really appreciate more details and references if you have!
$endgroup$
– SergioHC
Dec 13 '18 at 1:22
$begingroup$
Yes... late 19th century. S was a genius. The structure constants are simple sin es. Indeed, the algebra of gl(N) up to conventions amounts to that of su(N), and can fit the other ones, with suitable linear combs, as we did in that basis. The mother of all classical Lie algebras, indeed..... You might start with an easy antique talk of mine.
$endgroup$
– Cosmas Zachos
Dec 13 '18 at 4:30
add a comment |
$begingroup$
I want to understand certain properties of general curves $textrm{Ad}_{e^{t X}} = e^{t ad_X}$ of adjoint-representation matrices of $SU(N)$. For this purpose, I would like to have an explicit closed-form expression for all structure constants $f_{abc}$ of $mathfrak{su}(N)$ for arbitrary $N$ in the form $[T_a, T_b]=f_{abc} T_c$. Ideally, I would like to have them for the usual choice of basis ${T_a}$ with the convention that all $T_a$ are traceless, anti-Hermitian and trace-orthogonal, with standard Dynkin index $ tr(T_a T_b) = - frac{1}{2} delta_{ab} $, in which $f_{abc}$ is totally antisymmetric.
Is there a reference where such expression can be found? Otherwise, I am more or less familiar with the machinery of the Cartan classification. What would be the best way to attack the problem?
linear-algebra abstract-algebra representation-theory lie-groups lie-algebras
asked Dec 7 '18 at 0:10
SergioHCSergioHC
64
$endgroup$
I want to understand certain properties of general curves $textrm{Ad}_{e^{t X}} = e^{t ad_X}$ of adjoint-representation matrices of $SU(N)$. For this purpose, I would like to have an explicit closed-form expression for all structure constants $f_{abc}$ of $mathfrak{su}(N)$ for arbitrary $N$ in the form $[T_a, T_b]=f_{abc} T_c$. Ideally, I would like to have them for the usual choice of basis ${T_a}$ with the convention that all $T_a$ are traceless, anti-Hermitian and trace-orthogonal, with standard Dynkin index $ tr(T_a T_b) = - frac{1}{2} delta_{ab} $, in which $f_{abc}$ is totally antisymmetric.
Is there a reference where such expression can be found? Otherwise, I am more or less familiar with the machinery of the Cartan classification. What would be the best way to attack the problem?
linear-algebra abstract-algebra representation-theory lie-groups lie-algebras
linear-algebra abstract-algebra representation-theory lie-groups lie-algebras
asked Dec 7 '18 at 0:10
SergioHCSergioHC
64
asked Dec 7 '18 at 0:10
SergioHCSergioHC
64
edited Dec 8 '18 at 20:52
SergioHC
asked Dec 7 '18 at 0:10
SergioHCSergioHC
64
asked Dec 7 '18 at 0:10
SergioHCSergioHC
64
asked Dec 7 '18 at 0:10
SergioHCSergioHC
64
64
$begingroup$
Well, as you know from SU(3), the antihermitean basis for the fundamental is not so clean, but it does generalize readily to SU(N). However, if you forfeit antihermiticity, you have a celebrated clock-and-shift basis with analytic trigonometric structure constants, which is the language most prefer for large N systematics... Do you want details?
$endgroup$
– Cosmas Zachos
Dec 11 '18 at 13:59
$begingroup$
Thank you very much for your comment! I have read through the article you reference, and am wondering why I had not heard about these Sylvester matrices before. The construction seems very neat and I suppose all structure constants easily follow from the equation for $(Sigma_1)^k (Sigma_3)^j$. However, these are a basis for $mathfrak{gl}(N,mathbb{C})$. Is trivial to relate the structure constants of these to those of $mathfrak{su}(N)$ generators? Yes please, I would really appreciate more details and references if you have!
$endgroup$
– SergioHC
Dec 13 '18 at 1:22
$begingroup$
Yes... late 19th century. S was a genius. The structure constants are simple sin es. Indeed, the algebra of gl(N) up to conventions amounts to that of su(N), and can fit the other ones, with suitable linear combs, as we did in that basis. The mother of all classical Lie algebras, indeed..... You might start with an easy antique talk of mine.
$endgroup$
– Cosmas Zachos
Dec 13 '18 at 4:30
add a comment |
$begingroup$
Well, as you know from SU(3), the antihermitean basis for the fundamental is not so clean, but it does generalize readily to SU(N). However, if you forfeit antihermiticity, you have a celebrated clock-and-shift basis with analytic trigonometric structure constants, which is the language most prefer for large N systematics... Do you want details?
$endgroup$
– Cosmas Zachos
Dec 11 '18 at 13:59
$begingroup$
Thank you very much for your comment! I have read through the article you reference, and am wondering why I had not heard about these Sylvester matrices before. The construction seems very neat and I suppose all structure constants easily follow from the equation for $(Sigma_1)^k (Sigma_3)^j$. However, these are a basis for $mathfrak{gl}(N,mathbb{C})$. Is trivial to relate the structure constants of these to those of $mathfrak{su}(N)$ generators? Yes please, I would really appreciate more details and references if you have!
$endgroup$
– SergioHC
Dec 13 '18 at 1:22
$begingroup$
Yes... late 19th century. S was a genius. The structure constants are simple sin es. Indeed, the algebra of gl(N) up to conventions amounts to that of su(N), and can fit the other ones, with suitable linear combs, as we did in that basis. The mother of all classical Lie algebras, indeed..... You might start with an easy antique talk of mine.
$endgroup$
– Cosmas Zachos
Dec 13 '18 at 4:30
$begingroup$
Well, as you know from SU(3), the antihermitean basis for the fundamental is not so clean, but it does generalize readily to SU(N). However, if you forfeit antihermiticity, you have a celebrated clock-and-shift basis with analytic trigonometric structure constants, which is the language most prefer for large N systematics... Do you want details?
$endgroup$
– Cosmas Zachos
Dec 11 '18 at 13:59
$begingroup$
Well, as you know from SU(3), the antihermitean basis for the fundamental is not so clean, but it does generalize readily to SU(N). However, if you forfeit antihermiticity, you have a celebrated clock-and-shift basis with analytic trigonometric structure constants, which is the language most prefer for large N systematics... Do you want details?
$endgroup$
– Cosmas Zachos
Dec 11 '18 at 13:59
$begingroup$
Thank you very much for your comment! I have read through the article you reference, and am wondering why I had not heard about these Sylvester matrices before. The construction seems very neat and I suppose all structure constants easily follow from the equation for $(Sigma_1)^k (Sigma_3)^j$. However, these are a basis for $mathfrak{gl}(N,mathbb{C})$. Is trivial to relate the structure constants of these to those of $mathfrak{su}(N)$ generators? Yes please, I would really appreciate more details and references if you have!
$endgroup$
– SergioHC
Dec 13 '18 at 1:22
$begingroup$
Thank you very much for your comment! I have read through the article you reference, and am wondering why I had not heard about these Sylvester matrices before. The construction seems very neat and I suppose all structure constants easily follow from the equation for $(Sigma_1)^k (Sigma_3)^j$. However, these are a basis for $mathfrak{gl}(N,mathbb{C})$. Is trivial to relate the structure constants of these to those of $mathfrak{su}(N)$ generators? Yes please, I would really appreciate more details and references if you have!
$endgroup$
– SergioHC
Dec 13 '18 at 1:22
$begingroup$
Yes... late 19th century. S was a genius. The structure constants are simple sin es. Indeed, the algebra of gl(N) up to conventions amounts to that of su(N), and can fit the other ones, with suitable linear combs, as we did in that basis. The mother of all classical Lie algebras, indeed..... You might start with an easy antique talk of mine.
$endgroup$
– Cosmas Zachos
Dec 13 '18 at 4:30
$begingroup$
Yes... late 19th century. S was a genius. The structure constants are simple sin es. Indeed, the algebra of gl(N) up to conventions amounts to that of su(N), and can fit the other ones, with suitable linear combs, as we did in that basis. The mother of all classical Lie algebras, indeed..... You might start with an easy antique talk of mine.
$endgroup$
– Cosmas Zachos
Dec 13 '18 at 4:30
add a comment |
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$begingroup$
Well, as you know from SU(3), the antihermitean basis for the fundamental is not so clean, but it does generalize readily to SU(N). However, if you forfeit antihermiticity, you have a celebrated clock-and-shift basis with analytic trigonometric structure constants, which is the language most prefer for large N systematics... Do you want details?
$endgroup$
– Cosmas Zachos
Dec 11 '18 at 13:59
$begingroup$
Thank you very much for your comment! I have read through the article you reference, and am wondering why I had not heard about these Sylvester matrices before. The construction seems very neat and I suppose all structure constants easily follow from the equation for $(Sigma_1)^k (Sigma_3)^j$. However, these are a basis for $mathfrak{gl}(N,mathbb{C})$. Is trivial to relate the structure constants of these to those of $mathfrak{su}(N)$ generators? Yes please, I would really appreciate more details and references if you have!
$endgroup$
– SergioHC
Dec 13 '18 at 1:22
$begingroup$
Yes... late 19th century. S was a genius. The structure constants are simple sin es. Indeed, the algebra of gl(N) up to conventions amounts to that of su(N), and can fit the other ones, with suitable linear combs, as we did in that basis. The mother of all classical Lie algebras, indeed..... You might start with an easy antique talk of mine.
$endgroup$
– Cosmas Zachos
Dec 13 '18 at 4:30