Contragredient module of a generalized module

Given: Work over C with the formal-variable, grading, and branch conventions in the linked definitions.

Conformal vertex algebra of CFT type: A conformal vertex algebra of CFT type is a complex vertex algebra $V$ in the sense of def-vertex-algebra-state-field, with a specified vector $\omega \in V$ and scalar $c\in \mathbb{C}$, such that, writing $Y(\omega ,z)={\sum }_{n\in \mathbb{Z}}L(n)z^{-n-2},$ the operators $L(n)$ and scalar central action $c$ satisfy the Virasoro relations of def-virasoro-algebra, and $L(-1)=T$. In addition, $L(0)$ is diagonalizable, $V_n:=\{v\in V:L(0)v=nv\}$ gives $V={⨁}_{n\ge 0}{V}_n$, every $V_n$ is finite dimensional, and $V_0=\mathbb{C}\mathbf{1}$. Under this page's convention, this is a vertex operator algebra of CFT type. The grading and the CFT-type condition are part of the definition; they do not follow from the vertex-algebra axioms alone. The conformal weight of $v\in V_n$ is $n$. A conformal vector, a grading, and the nonnegative grading condition are additional data beyond an ordinary vertex algebra.

Grading-restricted generalized module: Let $V$ be a conformal vertex algebra of CFT type and $M$ a left $V$-module. Write $L_M(0)$ for the coefficient determined by $Y_M(\omega ,z)={\sum }_{n\in \mathbb{Z}}{L}_M(n)z^{-n-2}\mathrm{.}$ For $h\in \mathbb{C}$, define the generalized $L_M(0)$-eigenspace $M_{[h]}=\{m\in M:(L_M(0)-h{)}^{N}m=0\text{ for some integer }N\ge 1\}\mathrm{.}$ The module is grading-restricted generalized if $M={⨁}_{h\in \mathbb{C}}{M}_{[h]}$, every $M_{[h]}$ is finite dimensional, and the real parts of its nonzero generalized weights are bounded below. The direct sum means each vector has only finitely many generalized-weight components. This convention explicitly adds finite-dimensionality and a lower bound to the generalized-eigenspace decomposition; those conditions are not consequences of the module Jacobi identity.

Module for an ordinary vertex algebra: Let $V$ be a vertex algebra over $\mathbb{C}$ in the ordinary, non-super convention of def-vertex-algebra-state-field. A left $V$-module is a complex vector space $M$ and a linear map $Y_M(-,z):V⟶\{\text{formal fields on }M\},a⟼Y_M(a,z)={\sum }_{n\in \mathbb{Z}}{a}_{(n)}^M{z}^{-n-1},$ with the following properties: 1. Lower truncation. For every $a\in V$ and $m\in M$, $a_{(n)}^{M}m=0$ for all sufficiently large integers $n$, so $Y_M(a,z)m\in M((z))$ in the support convention of def-formal-distribution-on-a-vector-space. 2. Vacuum. $Y_M(\mathbf{1},z)={\mathrm{id}}_M$. 3. Module Jacobi identity. For every $a,b\in V$, $z_0^{-1}\delta \text{\hspace{0.17em}⁣}\left(\frac{z_1-z_2}{z_0}\right)Y_M(a,z_1)Y_M(b,z_2)-z_0^{-1}\delta \text{\hspace{0.17em}⁣}\left(\frac{z_2-z_1}{-z_0}\right)Y_M(b,z_2)Y_M(a,z_1)=z_2^{-1}\delta \text{\hspace{0.17em}⁣}\left(\frac{z_1-z_0}{z_2}\right)Y_M(Y(a,z_0)b,z_2)\mathrm{.}$ Here $\delta (x)={\sum }_{r\in \mathbb{Z}}{x}^r$; the first expansion is in nonnegative powers of $z_2$, the second in nonnegative powers of $z_1$, and the right-hand expansion is in nonnegative powers of $z_0$. They use the formal binomial conventions of def-formal-distribution-on-a-vector-space. The $Y$ inside the right-hand side is the state-field map of $V$; it is not the action map $Y_M$. A $V$-module homomorphism $f:M\to N$ is a linear map such that $f{Y}_M(a,z)=Y_N(a,z)f$ for every $a\in V$. This definition does not impose a conformal grading, finite-dimensional weight spaces, semisimplicity of $L_0$, a right-module convention, or a twist by an automorphism. For a conformal vertex algebra, write $Y_M(\omega ,z)={\sum }_{n\in \mathbb{Z}}{L}_M(n)z^{-n-2}$ when a conformal vector $\omega$ is specified. A $V$-submodule of $M$ is a linear subspace stable under every mode $a_{(n)}^M$. A module is simple (or irreducible) if it is nonzero and has no proper nonzero $V$-submodule. A module is completely reducible (or semisimple) if it is an internal direct sum of simple $V$-submodules; the empty direct sum is the zero module.

The restricted dual with opposite action is a contragredient module establishes the module axioms for this exact restricted-dual candidate and identifies its double contragredient with $M$ by evaluation. This is a well-definedness result depending on the present candidate definition, as recorded in justified_by, rather than an assumption in the construction.