A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

On Deformations of Lie Algebroids

For any Lie algebroid A, its 1-jet bundle ${\mathfrak{J} A}$ is a Lie algebroid naturally and there is a representation ${\pi:\mathfrak{J} A\longrightarrow\mathfrak{D} A}$ . Denote by ${{\rm d}_{\mathfrak{J}}}$ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A,A)_{{\mathfrak{D}} A}),{\rm d}_{\mathfrak{J}})}$ of the cochain complex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A, A)),{\rm d}_\mathfrak{J})}$ .     

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

Varieties Generated by Wreath Products of Abelian Groups

This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups $ \mathfrak{X} $ Wr $ \mathfrak{Y} $ = { X Wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } and $ \mathfrak{X} $ wr $ \mathfrak{Y} $ = {X wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } generate the product variety $ \mathfrak{X} $ var ( $ \mathfrak{Y} $ ).     

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

Supports of (\mathfrak{g},\mathfrak{k})-modules of finite type

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{k}$ be a reductive subalgebra in $\mathfrak{g}$ . We say that a $\mathfrak{g}$ -module M is a $(\mathfrak{g},\mathfrak{k})$ -module if M, considered as a $\mathfrak{k}$ -module, is a direct sum of finite-dimensional $\mathfrak{k}$ -modules. We say that a $(\mathfrak{g},\mathfrak{k})$ -module M is of finite type if all $\mathfrak{k}$ -isotopic components of M are finite-dimensional. In this paper we prove that any simple $(\mathfrak{g},\mathfrak{k})$ -module of finite type is holonomic. A simple $\mathfrak{g}$ -module M is associated with the invariants V(M), V(LocM), and L(M) reflecting the ??directions of growth of M.?? We also prove that for a given pair $(\mathfrak{g},\mathfrak{k})$ the set of possible invariants is finite.     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^{p}(\mathbb{R}, w)}$ , where ${p \in (1, \infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{A}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a \in PSO^{\diamond}}$ ) and all convolution operators W ⁰(b) ( ${b \in PSO_{p,w}^{\diamond}}$ ), where ${PSO^{\diamond} \subset L^{\infty}(\mathbb{R})}$ and ${PSO_{p,w}^{\diamond} \subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R} \cup \{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^{p}(\mathbb{R}, w)}$ . Under some conditions on the Muckenhoupt weight w, we construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{A}_{p,w}}$ and establish a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ in terms of their Fredholm symbols. To study the Banach algebra ${\mathfrak{A}_{p,w}}$ we apply the theory of Mellin pseudodifferential operators, the Allan–Douglas local principle, the two idempotents theorem and the method of limit operators. The paper is divided in two parts. The first part deals with the local study of ${\mathfrak{A}_{p,w}}$ and necessary tools for studying local algebras.     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^p(\mathbb{R},w)}$ , where ${p\in(1,\infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{U}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a\in PSO^\diamond}$ ) and all convolution operators W ⁰(b) ( ${b\in PSO_{p,w}^\diamond}$ ), where ${PSO^\diamond\subset L^\infty(\mathbb{R})}$ and ${PSO_{p,w}^\diamond\subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R}\cup\{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^p(\mathbb{R},w)}$ . Under some conditions on the Muckenhoupt weight w, using results of the local study of ${\mathfrak{U}_{p,w}}$ obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{U}_{p,w}}$ and establish a Fredholm criterion for the operators ${A\in\mathfrak{U}_{p,w}}$ in terms of their Fredholm symbols. In four partial cases we obtain for ${\mathfrak{U}_{p,w}}$ more effective results.     

Height of a Superposition

We observe that, given a poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ and a finite covering ${\user1{\mathcal{R}}} = {\user1{\mathcal{R}}}_{1} \cup \cdots \cup {\user1{\mathcal{R}}}_{n} $ of its ordering, the height of the poset does not exceed the natural product of the heights of the corresponding sub-relations: $$\mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}} \right)} \leqslant \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{1} } \right)} \otimes \cdots \otimes \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{n} } \right)}.$$ Conversely for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, every poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ of height at most $\xi_1\otimes\cdots\otimes\xi_n$ admits a partition ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ such that each ${\left( {E,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at most $\xi_k$ . In particular for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, the ordinal $$\xi _{1} \underline{ \otimes } \cdots \underline{ \otimes } \xi _{n} : = \sup {\left\{ {{\left( {\xi ^{\prime }_{1} \otimes \cdots \otimes \xi ^{\prime }_{n} } \right)} + 1:\xi ^{\prime }_{1} < \xi _{1} , \cdots ,\xi ^{\prime }_{n} < \xi _{n} } \right\}}$$ is the least $\xi$ for which the following partition relation holds $$\mathfrak{H}_{\xi } \to {\left( {\mathfrak{H}_{{\xi _{1} }} , \cdots ,\mathfrak{H}_{{\xi _{n} }} } \right)}^{2} $$ meaning: for every poset ${\left( {A,{\user1{\mathcal{R}}}} \right)}$ of height at least $\xi$ and every finite covering ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ , there is a $k$ for which the relation ${\left( {A,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at least $\xi_k$ . The proof will rely on analogue properties of vertex coverings w.r.t. the natural sum.     

Symmetric 2-Structures,a Classification

We classify symmetric 2-structures ${(P, \mathfrak{G}_1, \mathfrak{G}_2, \mathfrak{K})}$ , i.e. chain structures which correspond to sharply 2-transitive permutation sets (E, Σ) satisfying the condition: “ ${(*) \, \, \forall \sigma, \tau \in \Sigma : \sigma \circ \tau^{-1} \circ \sigma \in \Sigma}$ ”. To every chain ${K \in \mathfrak{K}}$ one can associate a reflection ${\widetilde{K}}$ in K. Then (*) is equivalent to “ ${(**) \, \, \forall K \in \mathfrak{K} : \widetilde{K}(\mathfrak{K}) = \mathfrak{K}}$ ” and one can define an orthogonality “ ${\perp}$ ” for chains ${K, L \in \mathfrak{K}}$ by “ ${K \perp L \Leftrightarrow K \neq L \wedge \widetilde{K}(L) = L}$ ”. The classification is based on the cardinality of the set of chains which are orthogonal to a chain K and passing through a point p of K. For one of these classes (called point symmetric 2-structures) we proof that in each point there is a reflection and that the set of point reflections forms a regular involutory permutation set.     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Macdonald polynomials and BGG reciprocity for current algebras

We study the category $\mathcal I _{\mathrm{gr }}$ of graded representations with finite-dimensional graded pieces for the current algebra $\mathfrak{g }\otimes \mathbf{C }[t]$ where $\mathfrak{g }$ is a simple Lie algebra. This category has many similarities with the category $\mathcal O $ of modules for $\mathfrak{g }$ , and in this paper, we prove an analog of the famous BGG duality in the case of $\mathfrak{sl }_{n+1}$ .     

Extending structures for Lie algebras

Let $\mathfrak{g }$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g }$ as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on $E$ such that $\mathfrak{g }$ is a Lie subalgebra of $E$ . A general product, called the unified product, is introduced as a tool for our approach. Let $V$ be a complement of $\mathfrak{g }$ in $E$ : the unified product $\mathfrak{g } \,\natural \, V$ is associated to a system $(\triangleleft , \, \triangleright , \, f, \{-, \, -\})$ consisting of two actions $\triangleleft $ and $\triangleright $ , a generalized cocycle $f$ and a twisted Jacobi bracket $\{-, \, -\}$ on $V$ . There exists a Lie algebra structure $[-,-]$ on $E$ containing $\mathfrak{g }$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $(E, [-,-]) \cong \mathfrak{g } \,\natural \, V$ . All such Lie algebra structures on $E$ are classified by two cohomological type objects which are explicitly constructed. The first one $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ will classify all Lie algebra structures on $E$ up to an isomorphism that stabilizes $\mathfrak{g }$ while the second object $\mathcal{H }^{2} (V, \mathfrak{g })$ provides the classification from the view point of the extension problem. Several examples that compute both classifying objects $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ and $\mathcal{H }^{2} (V, \mathfrak{g })$ are worked out in detail in the case of flag extending structures.     

Bethe subalgebras of the group algebra of the symmetric group

We introduce families $ \mathcal{B}_n^S\left( {{z_1},\ldots,{z_n}} \right) $ and $ \mathcal{B}_{{n,\hbar}}^S\left( {{z_1},\ldots,{z_n}} \right) $ of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra $ \mathbb{C}\left[ {\mathfrak{S}n} \right] $ of the symmetric group. Bethe subalgebras are deformations of the Gelfand?Zetlin subalgebra of $ \mathbb{C}\left[ {\mathfrak{S}n} \right] $ . We describe various properties of Bethe subalgebras.     

A Simultaneous Generalization of Independence and Disjointness in Boolean Algebras

We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n-independent. The properties of these classes (n-free and ω-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, $\mathfrak{i}_{n}$ , the minimum size of a maximal n-independent subset and $\mathfrak{i}_{\omega}$ , the minimum size of an ω-independent subset, are introduced and investigated. The values of $\mathfrak {i}_{n}$ and $\mathfrak {i}_{\omega}$ on are shown to be independent of ZFC.