Vector Liftings: Concepts & Applications

• Vector liftings are linear primitive sections in Lp spaces and algebraic settings that select canonical representatives from equivalence classes.
• They employ constructions from measure theory, combinatorial algebra, and Hopf algebras to ensure well-behaved canonical mappings.
• Applications include measurable modifications in stochastic processes, constructing Markov bases, and realizing cocycle deformations in Nichols algebras.

A vector lifting is a linear primitive section in function or algebraic settings, typically used to select canonical representatives from equivalence classes under a quotient map, with applications ranging from measure theory and stochastic processes to combinatorial algebra and Hopf algebras. The concept generalizes liftings beyond the $\mathcal{L}^\infty$ case, allowing construction of linear (hence "vector") liftings in $\mathcal{L}^p$ spaces for all $p\in[0,\infty]$, and subsumes notable specializations in algebraic, geometric, and probabilistic literature (Burke et al., 27 Nov 2025, Rauh et al., 2014, Angiono et al., 2016).

1. Definition and Fundamentals

Given a probability space $(X, \Sigma, \mu)$ and $0 \leq p \leq \infty$, let $L^p(\mu)$ denote the space of $p$-integrable functions (with usual conventions for $p=0$ and $p=\infty$). The null ideal is $$\mathcal{N}(\mu) = \{f \in L^p(\mu): f = 0\ \mu \text{-a.e.}\}$$. A primitive lifting is a map $\rho: L^p(\mu) \rightarrow L^p(\mu)$ such that:

• (l1) $\rho(f) =_\mu f$ for every $f$ and $\rho(1) = 1$,
• (l2) If $f =_\mu g$, then $\rho(f) = \rho(g)$.

A vector lifting is a primitive lifting that is also linear; it is equivalent to a linear section $s: L^p(\mu)/\mathcal{N}(\mu) \rightarrow L^p(\mu)$. Strong vector liftings, defined on topological probability spaces $(X, \mathfrak{T}, \mu)$ with $C(X) \cap \mathcal{N}(\mu) = \{0\}$, additionally satisfy $\rho(f) = f$ for $f \in C(X) \cap L^p(\mu)$ (Burke et al., 27 Nov 2025).

For linear maps between lattices, e.g., $\phi:\mathbb{Z}^n \to \mathbb{Z}^m$, a linear primitive lifting concerns the problem of finding elements in $\ker(\phi)$ that project to specified "moves" in the quotient, where conformal primitiveness is required (i.e., Graver basis elements) (Rauh et al., 2014). In the context of finite-dimensional Hopf algebras, especially Nichols algebras of diagonal type, liftings correspond to deformations parameterized by primitive relations and group-likes (Angiono et al., 2016).

2. Existence and Construction

The existence of vector liftings is established for all $p \in [0,\infty]$ on any nontrivial probability space $(X, \Sigma, \mu)$; a purely algebraic argument using a (Hamel) basis of the quotient $L^p(\mu)/\mathcal{N}(\mu)$ suffices: for any basis $\{f_i^\bullet\}$, one can choose representatives $f_i$ and define $\rho$ by requiring $\rho(f_i) = f_i$. In the topological case, a strong vector lifting exists if and only if $C(X) \cap \mathcal{N}(\mu) = \{0\}$, e.g., in Radon spaces (Burke et al., 27 Nov 2025).

For linear maps of lattices, the key criterion is normality of the semigroup generated by the columns of the augmented matrix $[B; \phi]$. Under this normality assumption, for any projected "move" $g$ in a Markov or Gröbner basis $G \subseteq \phi(\mathbb{Z}^n)$ one can find finitely many primitive lifts $M_g \subseteq \mathbb{Z}^n$ with $\phi(M_g) = \{g\}$; the union forms a primitive lifting (Theorem, (Rauh et al., 2014)).

In the Nichols algebra context, every finite-dimensional pointed Hopf algebra with appropriate coradical and diagonal type structure admits a linear-primitive lifting, realized as a cocycle deformation parameterized by $\boldsymbol{\lambda}$ over a finite set of generating relations (Angiono et al., 2016).

3. Vector Liftings for Product Spaces

Given $(X, \mu)$ and $(Y, \nu)$, with $V^p(\mu)$ and $V^p(\nu)$ being the sets of vector liftings, define a product vector lifting $\pi \in V^p(\upsilon)$ on $(X \times Y, \upsilon)$ (where $\upsilon$ is a suitable product measure) by

$\pi(f \otimes g) = \gamma(f) \otimes \eta(g),$

for all $f \in L^p(\mu)$, $g \in L^p(\nu)$, and $(f \otimes g)(x, y) = f(x)g(y)$, where $\gamma \in V^p(\mu)$, $\eta \in V^p(\nu)$. Existence is guaranteed whenever each factor admits a strong vector lifting and the product satisfies the Fubini–Tonelli conditions, ensuring vertical and horizontal sections of $\pi(h)$ lie in $L^p(\nu)$ and $L^p(\mu)$, respectively (Theorem 5.6, (Burke et al., 27 Nov 2025)).

An important construction is the tensor lifting

$$\left(\gamma \odot \eta\right)(f)(x, y) = \gamma\left(\eta_\bullet(f)(\cdot, y)\right)(x),$$

where $\eta_\bullet(f)(x, y) = \eta(f_x)(y)$. The conditions for this map to be a primitive lifting are characterized in (Burke et al., 27 Nov 2025) (Proposition 7.15).

4. Applications in Stochastic Processes and Measurability

A central application is to measurable modifications of stochastic processes. Given a family $Q_x(\cdot) \in L^\infty_\Gamma(T)$ (a process over $(Y, T)$ with values in a Baire space $\Gamma$), the existence of a jointly measurable modification $\widetilde{Q}(x, y)$ (i.e., $\widetilde{Q}_x = Q_x$ a.e. for each $x$) is equivalent to the existence of a 2-marginal vector lifting with respect to $L^p(\mu \otimes \nu)$ (Theorem 6.10, (Burke et al., 27 Nov 2025)).

A suite of equivalent conditions—measurability in the completed product $\sigma$-algebra, modification on a null set, and set-measurability after ideal extension—further generalizes the Musiał-type characterization for processes. These results reduce or eliminate completeness assumptions on $\mu, \nu$ by using measure extensions via null set ideals (Corollary 6.16, (Burke et al., 27 Nov 2025)).

5. Lifting Markov Bases and Toric Fiber Products

In algebraic statistics, vector liftings structure the passage from projected ("fiber") Markov bases to full Markov bases in hierarchical and graphical models. For an integer matrix $B \in \mathbb{Z}^{h \times n}$ and map $\phi: \mathbb{Z}^n \to \mathbb{Z}^m$, primitive liftings provide a recipe for generating full varieties of Markov or Gröbner bases by:

1. Computing a lattice basis $L$ of $\ker \phi$.
2. Forming $B^\phi = [B; \phi]$ and checking normality.
3. For each projected move $g$, solving an auxiliary lattice–inequality system to obtain all primitive lifts (see the algorithmic outline in (Rauh et al., 2014)).

For toric fiber products $B = B_1 \times_A B_2$, one lifts first along each grading and then glues the lifts using the "⊔-glue" construction, conditional on the compatible projection property. Normality and stabilization of the projected fiber intersection (PFI) basis ensure that iterated products possess Markov bases of bounded degree (Corollary 6.5, 6.10 in (Rauh et al., 2014)).

A detailed example is the 4-cycle hierarchical model, where the projected fibers exhibit holes and require double-move primitives to ensure connectivity. The construction leads to explicit Gröbner and Markov bases of bounded degree (Rauh et al., 2014).

6. Primitive Liftings in Nichols Algebras and Hopf Algebras

In the algebraic theory of Hopf algebras, linear-primitive liftings provide a unifying principle to classify all finite-dimensional pointed Hopf algebras with abelian coradical and diagonal type. The family $\mathfrak{u}(\boldsymbol{\lambda})$ arises by deforming the canonical Nichols algebra $\mathfrak{B}(V)\# H$ via a finite collection of parameters $\boldsymbol{\lambda}$ associated to primitive relations. These deformations are realized as cocycle deformations: for each $\lambda$, there is a convolution-invertible $2$-cocycle $\sigma$ making the lifted algebra $(\mathfrak{B}(V)\# H)_\sigma$ isomorphic to the desired Hopf algebra (Angiono et al., 2016).

Primitive relations are lifted by adding scalars times $(1-g_r)$ to the deformed relations, and every finite-dimensional pointed Hopf algebra with the assumed grading arises in this way. A key result is that all such liftings are cocycle deformations, with explicit construction determined by the combinatorics of quantum Serre, Cartan, and exceptional relations (Angiono et al., 2016).

7. Summary Table: Main Types of Vector Lifting

Domain	Definition and Context	Key Existence/Characterization Condition
$\mathcal{L}^p$ spaces	Linear primitive section $L^p(\mu) \to L^p(\mu)$	Algebraic basis for $L^p/\mathcal{N}(\mu)$ or $(N_{C,\mu})$ for strong
Lattices in combinatorics	Primitive lifts in $\ker \phi$ for Markov/Gröbner base	Normality of affine semigroup $[B;\phi]$
Nichols/Hopf algebras	Linear-primitive deformation via parameters $\lambda$	Existence for all diagonal type, finite-dim.

The significance of vector liftings lies in their broad applicability: they structure the selection of canonical representatives in measure theory, ensure the transfer of algebraic or combinatorial structure across product or fibered constructions, and parameterize deformations in quantum algebra and Hopf algebra theory (Burke et al., 27 Nov 2025, Rauh et al., 2014, Angiono et al., 2016). Their existence in various contexts often signals a well-behaved foundation for further structure, such as measurability, connectivity, or classification of algebraic objects.