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Let $$\mathfrak{g}$$ be a simple complex Lie algebra of $$rank(\mathfrak{g})\geq2$$ and dimension $$d$$. Fix a (non-zero) invariant bilinear form $$(\cdot,\cdot)$$ on $$\mathfrak{g}$$ and let $$\{x_i\}_{1\leq i\leq d}$$ be an orthogonal basis of $$\mathfrak{g}$$ with respect to $$(\cdot,\cdot)$$ satisfying $$[x_i,x_j] = f_{ij}^{\;\;k} x_k ,$$ where $$f_{ij}^{\;\;k}$$ are the structure constants of $$\mathfrak{g}$$ and the Einstein summation rule of the dummy indices is assumed. Denote $$\kappa_{ij}=(x_i,x_j)$$ and set $$\kappa^{ij}=(\kappa_{ij})^{-1}$$ such that $$\kappa_{il}\kappa^{lj}=\delta_{i}^{\,j}$$. Moreover set $$f_k^{\;ji} = \kappa^{jl} f_{kl}^{\;\;i}$$. Then $$f_k^{\;ji} [x_i,x_j] = c\,x_k ,$$ where $$c$$ is the the eigenvalue of the quadratic Casimir operator $$C=\kappa^{ij} x_i x_j$$ in the adjoint representation.

Set $$\mathfrak{g}^{n}=\mathfrak{g}\wedge\cdots\wedge\mathfrak{g}$$ to be the $$n^{th}$$ antisymmetric product of $$\mathfrak{g}$$ and set $$V_m$$ to be a vector space spanned by the totally symmetric polynomials $$\{x_{i_1},x_{i_2},\ldots,x_{i_m}\}=\frac{1}{m!}\sum_{\pi} x_{\pi(i_1)},x_{\pi(i_2)},\ldots,x_{\pi(i_m)}$$ of order $$m$$; here the sum is over all permutations $$\pi$$. Consider a category of maps $$Hom(\mathfrak{g}^{n},V_m)$$.

Q1) Have such maps been classified? Does anyone know a good reference for this? I'm particularly interested in the maps in $$Hom(\mathfrak{g}^{3},V_5)$$.

Q2) Let $$n=m=3$$. Then $$\beta_{abc}^{ijk}=f_a^{\;il} f_b^{\;jm} f_c^{\;kn} f_{lmn}$$ is a map $$\beta_{abc}^{ijk} : x_a\wedge x_b\wedge x_c \mapsto \{x_i,x_j,x_k\}.$$ This map appears in the r.h.s. of the Drinfeld Yangian terrific relation (see e.g. Theorem 12.1.1 in A Guide to Quantum Groups by Chari-Pressley). Is the map $$\beta_{abc}^{ijk}$$ the unique map in $$Hom(\mathfrak{g}^{3},V_3)$$? Here by uniqueness I mean that any other map in $$Hom(\mathfrak{g}^{3},V_3)$$ is equivalent to $$\beta_{abc}^{ijk}$$ up to a constant.

Q3) A hypothesis: let both $$n$$ be odd and $$m$$ be odd or even or on the contrary. Then $$Hom(\mathfrak{g}^{n},V_m)$$ is trivial. Could this be true?

Let $$\mathfrak{g}$$ be a simple complex Lie algebra of $$rank(\mathfrak{g})\geq2$$ and dimension $$d$$. Fix a (non-zero) invariant bilinear form $$(\cdot,\cdot)$$ on $$\mathfrak{g}$$ and let $$\{x_i\}_{1\leq i\leq d}$$ be an orthogonal basis of $$\mathfrak{g}$$ with respect to $$(\cdot,\cdot)$$ satisfying $$[x_i,x_j] = f_{ij}^{\;\;k} x_k ,$$ where $$f_{ij}^{\;\;k}$$ are the structure constants of $$\mathfrak{g}$$ and the Einstein summation rule of the dummy indices is assumed. Denote $$\kappa_{ij}=(x_i,x_j)$$ and set $$\kappa^{ij}=(\kappa_{ij})^{-1}$$ such that $$\kappa_{il}\kappa^{lj}=\delta_{i}^{\,j}$$. Moreover set $$f_k^{\;ji} = \kappa^{jl} f_{kl}^{\;\;i}$$. Then $$f_k^{\;ji} [x_i,x_j] = c\,x_k ,$$ where $$c$$ is the the eigenvalue of the quadratic Casimir operator $$C=\kappa^{ij} x_i x_j$$ in the adjoint representation.

Set $$\mathfrak{g}^{n}=\mathfrak{g}\wedge\cdots\wedge\mathfrak{g}$$ to be the $$n^{th}$$ antisymmetric product of $$\mathfrak{g}$$ and set $$V_m$$ to be a vector space spanned by the totally symmetric polynomials $$\{x_{i_1},x_{i_2},\ldots,x_{i_m}\}=\frac{1}{m!}\sum_{\pi} x_{\pi(i_1)},x_{\pi(i_2)},\ldots,x_{\pi(i_m)}$$ of order $$m$$; here the sum is over all permutations $$\pi$$. Consider a category of maps $$Hom(\mathfrak{g}^{n},V_m)$$.

Q1) Have such maps been classified? Does anyone know a good reference for this? I'm particularly interested in the maps in $$Hom(\mathfrak{g}^{3},V_5)$$.

Q2) Let $$n=m=3$$. Then $$\beta_{abc}^{ijk}=f_a^{\;il} f_b^{\;jm} f_c^{\;kn} f_{lmn}$$ is a map $$\beta_{abc}^{ijk} : x_a\wedge x_b\wedge x_c \mapsto \{x_i,x_j,x_k\}.$$ This map appears in the r.h.s. of the Drinfeld Yangian terrific relation (see e.g. Theorem 12.1.1 in A Guide to Quantum Groups by Chari-Pressley). Is the map $$\beta_{abc}^{ijk}$$ the unique map in $$Hom(\mathfrak{g}^{3},V_3)$$? Here by uniqueness I mean that any other map in $$Hom(\mathfrak{g}^{3},V_3)$$ is equivalent to $$\beta_{abc}^{ijk}$$ up to a constant.

Q3) A hypothesis: let both $$n$$ and $$m$$ be odd or even. Then $$Hom(\mathfrak{g}^{n},V_m)$$ is trivial. Could this be true?

Let $$\mathfrak{g}$$ be a simple complex Lie algebra of $$rank(\mathfrak{g})\geq2$$ and dimension $$d$$. Fix a (non-zero) invariant bilinear form $$(\cdot,\cdot)$$ on $$\mathfrak{g}$$ and let $$\{x_i\}_{1\leq i\leq d}$$ be an orthogonal basis of $$\mathfrak{g}$$ with respect to $$(\cdot,\cdot)$$ satisfying $$[x_i,x_j] = f_{ij}^{\;\;k} x_k ,$$ where $$f_{ij}^{\;\;k}$$ are the structure constants of $$\mathfrak{g}$$ and the Einstein summation rule of the dummy indices is assumed. Denote $$\kappa_{ij}=(x_i,x_j)$$ and set $$\kappa^{ij}=(\kappa_{ij})^{-1}$$ such that $$\kappa_{il}\kappa^{lj}=\delta_{i}^{\,j}$$. Moreover set $$f_k^{\;ji} = \kappa^{jl} f_{kl}^{\;\;i}$$. Then $$f_k^{\;ji} [x_i,x_j] = c\,x_k ,$$ where $$c$$ is the the eigenvalue of the quadratic Casimir operator $$C=\kappa^{ij} x_i x_j$$ in the adjoint representation.

Set $$\mathfrak{g}^{n}=\mathfrak{g}\wedge\cdots\wedge\mathfrak{g}$$ to be the $$n^{th}$$ antisymmetric product of $$\mathfrak{g}$$ and set $$V_m$$ to be a vector space spanned by the totally symmetric polynomials $$\{x_{i_1},x_{i_2},\ldots,x_{i_m}\}=\frac{1}{m!}\sum_{\pi} x_{\pi(i_1)},x_{\pi(i_2)},\ldots,x_{\pi(i_m)}$$ of order $$m$$; here the sum is over all permutations $$\pi$$. Consider a category of maps $$Hom(\mathfrak{g}^{n},V_m)$$.

Q1) Have such maps been classified? Does anyone know a good reference for this? I'm particularly interested in the maps in $$Hom(\mathfrak{g}^{3},V_5)$$.

Q2) Let $$n=m=3$$. Then $$\beta_{abc}^{ijk}=f_a^{\;il} f_b^{\;jm} f_c^{\;kn} f_{lmn}$$ is a map $$\beta_{abc}^{ijk} : x_a\wedge x_b\wedge x_c \mapsto \{x_i,x_j,x_k\}.$$ This map appears in the r.h.s. of the Drinfeld Yangian terrific relation (see e.g. Theorem 12.1.1 in A Guide to Quantum Groups by Chari-Pressley). Is the map $$\beta_{abc}^{ijk}$$ the unique map in $$Hom(\mathfrak{g}^{3},V_3)$$? Here by uniqueness I mean that any other map in $$Hom(\mathfrak{g}^{3},V_3)$$ is equivalent to $$\beta_{abc}^{ijk}$$ up to a constant.

Q3) A hypothesis: let $$n$$ be odd and $$m$$ be even or on the contrary. Then $$Hom(\mathfrak{g}^{n},V_m)$$ is trivial. Could this be true?

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# Lie algebra cohomology

Let $$\mathfrak{g}$$ be a simple complex Lie algebra of $$rank(\mathfrak{g})\geq2$$ and dimension $$d$$. Fix a (non-zero) invariant bilinear form $$(\cdot,\cdot)$$ on $$\mathfrak{g}$$ and let $$\{x_i\}_{1\leq i\leq d}$$ be an orthogonal basis of $$\mathfrak{g}$$ with respect to $$(\cdot,\cdot)$$ satisfying $$[x_i,x_j] = f_{ij}^{\;\;k} x_k ,$$ where $$f_{ij}^{\;\;k}$$ are the structure constants of $$\mathfrak{g}$$ and the Einstein summation rule of the dummy indices is assumed. Denote $$\kappa_{ij}=(x_i,x_j)$$ and set $$\kappa^{ij}=(\kappa_{ij})^{-1}$$ such that $$\kappa_{il}\kappa^{lj}=\delta_{i}^{\,j}$$. Moreover set $$f_k^{\;ji} = \kappa^{jl} f_{kl}^{\;\;i}$$. Then $$f_k^{\;ji} [x_i,x_j] = c\,x_k ,$$ where $$c$$ is the the eigenvalue of the quadratic Casimir operator $$C=\kappa^{ij} x_i x_j$$ in the adjoint representation.

Set $$\mathfrak{g}^{n}=\mathfrak{g}\wedge\cdots\wedge\mathfrak{g}$$ to be the $$n^{th}$$ antisymmetric product of $$\mathfrak{g}$$ and set $$V_m$$ to be a vector space spanned by the totally symmetric polynomials $$\{x_{i_1},x_{i_2},\ldots,x_{i_m}\}=\frac{1}{m!}\sum_{\pi} x_{\pi(i_1)},x_{\pi(i_2)},\ldots,x_{\pi(i_m)}$$ of order $$m$$; here the sum is over all permutations $$\pi$$. Consider a category of maps $$Hom(\mathfrak{g}^{n},V_m)$$.

Q1) Have such maps been classified? Does anyone know a good reference for this? I'm particularly interested in the maps in $$Hom(\mathfrak{g}^{3},V_5)$$.

Q2) Let $$n=m=3$$. Then $$\beta_{abc}^{ijk}=f_a^{\;il} f_b^{\;jm} f_c^{\;kn} f_{lmn}$$ is a map $$\beta_{abc}^{ijk} : x_a\wedge x_b\wedge x_c \mapsto \{x_i,x_j,x_k\}.$$ This map appears in the r.h.s. of the Drinfeld Yangian terrific relation (see e.g. Theorem 12.1.1 in A Guide to Quantum Groups by Chari-Pressley). Is the map $$\beta_{abc}^{ijk}$$ the unique map in $$Hom(\mathfrak{g}^{3},V_3)$$? Here by uniqueness I mean that any other map in $$Hom(\mathfrak{g}^{3},V_3)$$ is equivalent to $$\beta_{abc}^{ijk}$$ up to a constant.

Q3) A hypothesis: let both $$n$$ and $$m$$ be odd or even. Then $$Hom(\mathfrak{g}^{n},V_m)$$ is trivial. Could this be true?