On the complete 𝐺𝐿(𝑛,𝐶)-decomposition of the stable cohomology of 𝑔𝑙_{𝑛}(𝐴)

Abstract

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