On the complete $\textrm {GL}(n,\textbf {C})$-decomposition of the stable cohomology of $\textrm {gl}_ n(A)$

Abstract:

Let $A$ be a graded, associative ${\mathbf {C}}$-algebra. For each $n$ let $g{l_n}(A)$ denote the Lie algebra of $n \times n$ matrices with entries from $A$. In this paper we extend the Loday-Quillen theorem to nontrivial isotypic components of $GL(n, {\mathbf {C}})$ acting on the Lie algebra cohomology of $g{l_n}(A)$. For $\alpha$ and $\beta$ partitions of some nonnegative integer $m$ let ${[\alpha , \beta ]_n} \in {{\mathbf {Z}}^n}$ denote the maximal $GL(n, {\mathbf {C}})$-weight given by \[ {[\alpha , \beta ]_n} = \sum \limits _i {{\alpha _i}{e_i}} - \sum \limits _j {{\beta _j}{e_{n + 1 - j}}.} \] We show that the ${[\alpha , \beta ]_n}$-isotypic component of the Lie algebra cohomology of $g{l_n}(A)$ stabilizes when $n \to \infty$ and is equal to \[ HR{C^{\ast }}(A) \otimes ({\tilde H^{\ast }}{(A; {\mathbf {C}})^{ \otimes m}} \otimes {S^\alpha } \otimes {S^\beta }){s_m}\] where ${\tilde H^{\ast }}(A; {\mathbf {C}})$ is the reduced Hochschild cohomology of $A$ with trivial coefficients, where $HR{C^{\ast }}(A)$ is the graded exterior algebra generated by the cyclic cohomology of $A$, where ${S^\alpha }$ and ${S^\beta }$ are the irreducible ${S_m}$-modules indexed by $\alpha$ and $\beta$ and where the action of ${S_m}$ on $\tilde H{(A; {\mathbf {C}})^{ \otimes m}}$ is the exterior action.