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I'm reading the paper Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak{gl_2}$.

It describes a representation of the group algebra $\mathbb{Q}[GL_2(\mathbb{Q})]$ via its action on the natural module $E^{\otimes r}$:

$\sigma_d : \mathbb{Q}[GL_2(\mathbb{Q})] \rightarrow End(E^{\otimes r})$.

The paper then says. "by differentiating $\sigma_d$ we obtain a representation of the Lie Algebra $\mathfrak{gl_2}$". I understand what a derivation of an associative algebra is, however I don't know how to differentiate a representation as such.

Also the paper, says we can extend the representation we obtain for the Lie algebra linearly to the universal enveloping algebra of the Lie algebra. Again I'm not sure what we are extending linearly in from the Lie algebra to the UEA.

Many thanks for any replies.

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Actually, one differentiates the polynomial group representation $\sigma_d : \operatorname{GL}_2(\mathbb{Q}) \to \operatorname{GL}(E^{\otimes d})$ at the identity of $\operatorname{GL}_2(\mathbb{Q})$ to obtain a representation $\rho_d : \mathfrak{gl}_2(\mathbb{Q}) \to \mathfrak{gl}(E^{\otimes d})$ of the associated Lie algebra. For $g \in \operatorname{GL}_2(\mathbb{Q})$, $x \in \mathfrak{gl}_2(\mathbb{Q})$, and $u_1, \ldots, u_d \in E$, we have $$ \sigma_d(g)(u_1\otimes \cdots \otimes u_d) = g(u_1) \otimes \cdots \otimes g(u_d),$$ $$\rho_d(x)(u_1 \otimes \cdots \otimes u_d) = \sum_{k=1}^d u_1 \otimes \cdots \otimes u_{k-1} \otimes x(u_k) \otimes u_{k+1} \otimes \cdots \otimes u_d.$$ The extension of $\rho_d : \mathfrak{gl}_2(\mathbb{Q}) \to \mathfrak{gl}(E^{\otimes d})$ to a representation $\tilde\rho_d : U(\mathfrak{gl}_2(\mathbb{Q})) \to \operatorname{End}(E^{\otimes d})$, whose existence and uniqueness is guaranteed by the universal property of the inclusion $\mathfrak{gl}_2(\mathbb{Q}) \to U(\mathfrak{gl}_2(\mathbb{Q}))$, can also be described as follows. Let $(x_1,x_2,x_3,x_4)$ be an ordered basis of $\mathfrak{gl}_2(\mathbb{Q})$. Then, by the Poincaré-Birkhoff-Witt theorem, the (non-commutative) monomials of the form $x_1^{k_1}x_2^{k_2}x_3^{k_3}x_4^{k_4}$, where $k_j \in \mathbb{Z}_{\geq0}$, form a basis of $U(\mathfrak{gl}_2(\mathbb{Q}))$ over $\mathbb{Q}$. The representation $\tilde\rho_d$ is defined on basis elements via the rule $$ \tilde\rho_d(x_1^{k_1} \cdots x_4^{k_4}) = \rho_d(x_1)^{k_1} \cdots \rho_d(x_4)^{k_4},$$ and then extended by linearity to the whole of $U(\mathfrak{gl}_2(\mathbb{Q}))$. The product on the right hand side in the above formula is taken in the associative algebra $\operatorname{End}(E^{\otimes d})$ having the same underlying vector space as the Lie algebra $\mathfrak{gl}(E^{\otimes d})$.

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  • $\begingroup$ You're welcome. $\endgroup$ – ivanpenev Jun 8 '14 at 16:29

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