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Let $A$be a commutative algebra equipped with an action of a group $G$. The so-called $G$-primes of $A$are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When $G$is an infinite dimensional group, these ideals can be very subtle: for instance, distinct $G$-primes can have the same radical. In previous work, the second author showed that if $G={\mathbf{G}\mathbf{L}}_{\mathrm{\infty }}$and $A$is a polynomial representation, then these pathologies disappear when $G$is replaced with the supergroup ${\mathbf{G}\mathbf{L}}_{\mathrm{\infty }|\mathrm{\infty }}$and $A$with a corresponding algebra; this leads to a geometric description of $G$-primes of $A$. In the present paper, we construct an abstract framework around this result, and apply the framework to prove analogous results for other (super)groups. We give some applications to the isomeric determinantal ideals (commonly known as “queer determinantal ideals”). 

A representation of\mathfrak{gl}(V)=V \otimes V^{\ast}is a linear map\mu \colon \mathfrak{gl}(V) \otimes M \rightarrow Msatisfying a certain identity. By currying, giving a linear map\muis equivalent to giving a linear mapa \colon V \otimes M \rightarrow V \otimes M, and one can translate the condition for\muto be a representation into a condition ona. This alternate formulation does not use the dual ofVand makes sense for any objectVin a tensor category\mathcal{C}. We call such objects representations of thecurried general linear algebraonV. The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category is the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore. 

Abstract The Witt algebra  $${\mathfrak{W}}_{n}$$ is the Lie algebra of all derivations of the $$n$$-variable polynomial ring $$\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$$ (or of algebraic vector fields on $$\textbf{A}^{n}$$). A representation of $${\mathfrak{W}}_{n}$$ is polynomial if it arises as a subquotient of a sum of tensor powers of $$\textbf{V}_{n}$$. Our main theorems assert that finitely generated polynomial representations of $${\mathfrak{W}}_{n}$$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $$\textbf{Fin}^{\textrm{op}}$$, where $$\textbf{Fin}$$ is the category of finite sets. We also show that polynomial representations of $${\mathfrak{W}}_{n}$$ are equivalent to polynomial representations of the endomorphism monoid of $$\textbf{A}^{n}$$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish. 

We study the category of S p \mathbf {Sp} -equivariant modules over the infinite variable polynomial ring, where S p \mathbf {Sp} denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M M fits into an exact triangle T → M → F → T \to M \to F \to where T T is a finite length complex of torsion modules and F F is a finite length complex of “free” modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras Sym ⁡ ( C ∞ ⊕ ⋀ 2 C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \bigwedge ^2{\mathbf {C}^{\infty }}) and Sym ⁡ ( C ∞ ⊕ Sym 2 ⁡ C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \operatorname {Sym}^2{\mathbf {C}^{\infty }}) are noetherian, which are the strongest results to date of this kind. We also show that the free 2-step nilpotent twisted Lie algebra and Lie superalgebra are noetherian.