By a grassmannian we understand a usual complex grassmannian or possibly an orthogonal or symplectic grassmannian. We classify, with few exceptions, linear embeddings of grassmannians into larger grassmannians, where the linearity requirement is the condition that the embedding induces an isomorphism on Picard groups. This classification implies that most linear embeddings of grassmannians are equivariant. A linear ind-grassmannian is the direct limit of a chain of linear embeddings of grassmannians. We conclude the paper by classifying linear embeddings of linear ind-grassmannians.

We study the homogeneous ind-spaces GL(s)/P where GL(s) is a strict diagonal ind-group defined by a supernatural number s and P is a parabolic ind-subgroup of GL(s) . We construct an explicit exhaustion of GL(s)/P by finite-dimensional partial flag varieties. As an application, we characterize all locally projective GL(∞ ) -homogeneous spaces, and some direct products of such spaces, which are GL(s) -homogeneous for a fixed s . The very possibility for a GL(∞ ) -homogeneous space to be GL(s) -homogeneous for a strict diagonal ind-group GL(s) arises from the fact that the automorphism group of a GL(∞ ) -homogeneous space is much larger than GL(∞ ) .

Ind-varieties of generalized flags have been studied for two decades. However, a precise statement of when two such ind-varieties, one or both being possibly ind-varieties of isotropic generalized flags, are isomorphic, has been missing in the literature. Using some recent results on the structure of ind-varieties of generalized flags, we establish a criterion for the existence of an isomorphism as above. Our result claims that, with only two exceptions, isomorphisms of ind-varieties of generalized flags are induced by isomorphisms of respective generalized flags. The exceptional isomorphisms correlate with a well-known result of A. Onishchik from 1963.

We classify simple bounded weight modules over the complex simple Lie superalgebras 𝔰𝔩(∞ | ∞ ) and 𝔬𝔰𝔭(m | 2n) , when at least one of m and n equals ∞ . For 𝔬𝔰𝔭(m | 2n) such modules are of spinor-oscillator type, i.e., they combine into one of the known classes of spinor 𝔬(m) -modules and oscillator-type 𝔰𝔭(2n) -modules. In addition, we characterize the category of bounded weight modules over 𝔬𝔰𝔭(m | 2n) (under the assumption 𝔬𝔰𝔭(m | 2n) = ∞ ) by reducing its study to already known categories of representations of 𝔰𝔭(2n) , where n possibly equals ∞ . When classifying simple bounded weight 𝔰𝔩(∞ | ∞ ) -modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra 𝔰𝔩(∞ | ∞ )_0̅ . We finish the paper by establishing some first facts about the category of bounded weight 𝔰𝔩(∞ | ∞ ) -modules.

We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras gl(V⊕ΠV) and osp(V⊕ΠV) associated with a Tate space V. Here V⊕ΠV is a Z/2Z-graded topological vector space whose even and odd parts are isomorphic to V. We discuss the purely even case first, by introducing monoidal categories Tˆgl(V), Tˆo(V) and Tˆsp(V), and show that these categories are anti-equivalent to respective previously studied categories Tgl(V), To(V), Tsp(V). These latter categories have certain universality properties as monoidal categories, which consequently carry over to Tˆgl(V), Tˆo(V) and Tˆsp(V). Moreover, the categories To(V) and Tsp(V) are known to be equivalent, and this implies the equivalence of the categories Tˆo(V) and Tˆsp(V). After introducing a supersymmetric setting, we establish the equivalence of the category Tˆgl(V) with the category Tˆgl(V⊕ΠV), and the equivalence of both categories Tˆo(V) and Tˆsp(V) with Tˆosp(V⊕ΠV).

We extend previous work by constructing a universal abelian tensor category ${\bf T}_t$ generated by two objects $X,Y$ equipped with finite filtrations $0\subsetneq X_0\subsetneq ... X_{t+1}\subsetneq X$ and $0\subsetneq Y_0\subsetneq ... Y_{t+1}\subsetneq Y$, and with a pairing $X\otimes Y\to \mathbb{I}$, where $\mathbb{I}$ is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra $\mathfrak{gl}^M(V,V_*)$ of cardinality $2^{\aleph_t}$, associated to a diagonalizable pairing between two complex vector spaces $V,V_*$ of dimension $\aleph_t$. As a preliminary step, we study a tensor category $\mathbb{T}_t$ generated by the algebraic duals $V^*$, $(V_*)^*$. The injective hull of $\mathbb{C}$ in $\mathbb{T}_t$ is a commutative algebra $I$, and the category ${\bf T}_t$ is consists of the free $I$-modules in $\mathbb{T}_t$. An essential novelty in our work is the explicit computation of Ext-groups between simples in both categories ${\bf T}_t$ and $\mathbb{T}_t$, which had been an open problem already for $t=0$. This provides a direct link from the theory of universal tensor categories to Littlewood-Richardson-type combinatorics.

In this chapter we recall some fundamental facts about finite-dimensional complex Lie algebras.

We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra. This result enables us to show that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags as defined in Dimitrov and Penkov (IMRN No 55:2935–2953, 2004). These latter ind-varieties have been introduced in terms of the ind-group $$SL(\infty )$$ (respectively, $$O(\infty )$$ or $$Sp(\infty )$$ for isotropic generalized flags), and the current paper constructs them in purely algebraic–geometric terms.

We classify integrable bounded simple weight modules over classical Lie superalgebras at infinity. We also study the categories of such modules, and we prove that for most of the classical Lie superalgebras at infinity the respective category is semisimple.

We compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups SL(∞ ) , O(∞ ) or Sp(∞ ) . We show that the respective automorphism groups are much larger than SL(∞ ) , O(∞ ) or Sp(∞ ) , and present the answer in terms of Mackey groups. The latter are groups of automorphisms of non-degenerate pairings of (in general infinite-dimensional) vector spaces. An explicit matrix form of the automorphism group of an arbitrary ind-variety of generalized flags is also given. The case of the Sato grassmannian is considered in detail, and its automorphism group is the projectivization of the connected component of unity in the group known as Japanese GL(∞ ) .

The Lie algebra $gl(V)$ is the Lie algebra of all endomorphisms of a countable-dimensional complex vector space $V$. We define a tensor category of topological representations of the Lie algebra $gl(V)$, so that $V$, its dual and the adjoint representation $gl(V)$ are objects of this category. This makes it an analogue of the category of finite-dimensional modules over the finite-dimensional Lie algebra $gl(n)$. Our main result is that this category is antiequivalent as a symmetric monoidal category to the category of tensor representations of the Lie algebra of finitary infinite matrices.

This book functions as a unique text-monograph hybrid that bridges a traditional graduate course to research level representation theory.

In the first part of this chapter, we generalize the theory of Cartan subalgebras from $$\mathfrak {sl} (n)$$ to $$\mathfrak {sl} (\infty )$$ and $$\mathfrak {gl}(\infty )$$ . In the second part, we discuss the non-splitting Borel and parabolic subalgebras of $$\mathfrak {sl}(\infty )$$ and $$\mathfrak {gl}(\infty )$$ .

In this chapter, we give a glimpse into the interaction between algebra and geometry in representation theory. The Bott–Borel–Weil Theorem is one of the origins of geometric representation theory, which is currently a leading branch of representation theory.

Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $\varphi$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $\varphi^*: V^*\rightarrow V^*$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that the trivial module $\mathbb{C}$ is no longer injective in $\mathbb{T}$. We describe the injective hull $I$ of $\mathbb{C}$ in $\mathbb{T}$, and show that the category $\mathbb{T}$ is Koszul. In addition, we prove that $I$ is endowed with a natural structure of commutative algebra. We then define another category $_I\mathbb{T}$ of objects in $\mathbb{T}$ which are free as $I$-modules. Our main result is that the category ${}_I\mathbb{T}$ is also Koszul, and moreover that ${}_I\mathbb{T}$ is universal among abelian $\mathbb{C}$-linear tensor categories generated by two objects $X$, $Y$ with fixed subobjects $X'\hookrightarrow X$, $Y'\hookrightarrow Y$ and a pairing $X\otimes Y\rightarrow \text{\textbf{1}}$ where \textbf{1} is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $\mathbb{T}$ and ${}_I\mathbb{T}$.

In this chapter we study modules over the infinite-dimensional classical Lie algebras $$\mathfrak {g}=\mathfrak {gl}(\infty )$$ , $$\mathfrak {sl}(\infty )$$ , $$\mathfrak {o}(\infty )$$ , $$\mathfrak {sp}(\infty )$$ that are obtained by taking tensor products of the natural and conatural module of $$\mathfrak {g}$$ . We will examine the structure of these modules and realize them in a suitable category $$\mathbb {T}_{\mathfrak {g}}$$ . This category is not semisimple, but is Koszul in the sense of Beilinson–Ginzburg–Soergel.

Let G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

We compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups SL(∞), O(∞) or Sp(∞). We show that the respective automorphism groups are much larger than SL(∞), O(∞) or Sp(∞), and present the answer in terms of Mackey groups. The latter are groups of automorphisms of nondegenerate pairings of (in general infinite-dimensional) vector spaces. An explicit matrix form of the automorphism group of an arbitrary ind-variety of generalized flags is also given. The case of the Sato grassmannian is considered in detail, and its automorphism group is the projectivization of the connected component of unity in the group Japanese GL(∞).

In this chapter we introduce the main object of our study, the infinite-dimensional root-reductive Lie algebras.