Principal subspaces for quantum affine algebra [formula omitted

Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces

We find that different Deep Neural Networks (DNNs) trained with the same dataset share a common principal subspace in latent spaces, no matter in which architectures (e.g., Convolutional Neural Networks (CNNs), Multi-Layer Preceptors (MLPs) and Autoencoders (AEs)) the DNNs were built or even whether labels have been used in training (e.g., supervised, unsupervised, and self-supervised learning). Specifically, we design a new metric $${\mathcal {P}}$$ -vector to represent the principal subspace of deep features learned in a DNN, and propose to measure angles between the principal subspaces using $${\mathcal {P}}$$ -vectors. Small angles (with cosine close to 1.0) have been found in the comparisons between any two DNNs trained with different algorithms/architectures. Furthermore, during the training procedure from random scratch, the angle decrease from a larger one (70°–80° usually) to the small one, which coincides the progress of feature space learning from scratch to convergence. Then, we carry out case studies to measure the angle between the $${\mathcal {P}}$$ -vector and the principal subspace of training dataset, and connect such angle with generalization performance. Extensive experiments with practically-used Multi-Layer Perceptron (MLPs), AEs and CNNs for classification, image reconstruction, and self-supervised learning tasks on MNIST, CIFAR-10 and CIFAR-100 datasets have been done to support our claims with solid evidences.

Extending earlier work of the authors, this is the first in a series of papers devoted to the vertex-algebraic structure of principal subspaces of standard modules for twisted affine Kac–Moody algebras. In this part, we develop the necessary theory of principal subspaces for the affine Lie algebra [Formula: see text], which we expect can be extended to higher rank algebras. As a "test case", we consider the principal subspace of the basic [Formula: see text]-module and explore its structure in depth.

Intra-personal space modeling proposed by Moghaddam et al. has been successfully applied in face recognition. In their work the regular principal subspaces are derived from the intra-personal spacce using a principal componen analysis and embedded in a probabilistic formulation. In this paper, we derive the principal subspace from the intro-personal kernel space by developing a probabilistic analysis for kernel principal components for face recognition. We test this algorithm on a subset of the FERET database with illumination and facial expression variations. The recognition performance demonstrates its advantage over other traditional subspace approaches.

Given a quantum group, we prove that the canonical bases of the tensor products of its integrable highest weight modules can be obtained from the canonical bases of the integrable highest weight modules of a bigger quantum group. As a result, based on the positivity of the canonical bases of the integrable highest weight modules due to Lusztig, we prove that the canonical bases of the tensor products have the positivity.

Supercategorification of quantum Kac–Moody algebras II

A general scheme of construction of Drinfeldians and Yangians from quantum non-twisted affine Kac-Moody algebras is presented. Explicit description of Drinfeldians and Yangians for all Lie algebras of the classical series A, B, C, D is given in terms of a Chevalley basis. Rational and trigonometric deformations of a universal enveloping algebra U(g(u)) (g is a finite-dimensional complex simple Lie algebra) - Yangian Yη(g) and a quantum (q-deformed) affine algebraUq(g(u)) - are playing increasingly a role in the theory of integrable systems and the quantum field theory. For applications it is useful to know various realizations (or bases) of these deformations. Among various realizations the Chevalley basis plays a special role due to its compactness and simplicity. In the case of the quantum affine algebras the Chevalley basis was originally introduced in the defining relations while the Yangians were initially de- fined in terms of a finite Cartesian basis (the first Drinfeld realization (2)) as well as in terms of a special infinite basis (the second Drinfeld realization (3)). It was not clear whether does exist a Chevalley basis for the Yangians. Recently in (12, 13) it was shown that the Yangian can be realized in terms of Chevalley basis and, moreover, there was found a simple connection of this realization with the corresponding quantum affine algebra (also see (14, 15)). This connection allows to introduce a new Hopf algebra called Drinfeldian which depends on two deformation parameters q and η. The Drinfeldian Dqη(g) can be considered as quantization of U(g(u)) in the direction of a classical r-matrix which is a sum of simplest rational and trigonometric r-matrices. When the parameter q goes to 1 we obtain the Yangian Yη(g) in terms of the Chevalley basis. In this paper we remind once more the procedure of construction of the Drin- feldian Dqη(g) and the Yangian Yη(g) in terms of the Chevalley basis starting from

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the ( ( spin ) ) rigid tableaux, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac–Moody algebras. By applying combinatorics of tableaux such as the Robinson–Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles.

Semi-infinite Lakshmibai–Seshadri path model for level-zero extremal weight modules over quantum affine algebras

We give a detailed proof of the existence of evaluation map for affine Yangians of type A to clarify that it needs an assumption on parameters. This map was first found by Guay but a proof of its well-definedness and the assumption have not been written down in the literature. We also determine the highest weights of evaluation modules defined as the pull-back of integrable highest weight modules of the affine Lie algebra $\hat{\mathfrak{gl}}_N$ by the evaluation map.

The structure of parafermion vertex operator algebras K(osp(1|2n),k)

Polyhedral realization of crystal bases is one of the methods for describing the crystal base B(∞) of a quantized enveloping algebra explicitly. This method can be applied to symmetrizable Kac-Moody types. We can also apply this method to the crystal bases B(λ) of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases B(∞) and B(λ) are only given in the case of arbitrary rank 2, of An and of An(1). So, we will give the polyhedral realizations of crystal bases B(∞) and B(λ) for all simple Lie algebras in this paper.

or = osp(m + n) (m 6 1;2; n 6 1), an irreducible highest weight -module cannot be 0-integrable unless it is the trivial module. Motivated by this fact, we introduce the notion of a partially integrable module for a large class of infinite-dimensional Lie superalgebras . We also give a general definition of a highest weight module. We then prove an explicit criterion for partial integrability of irreducible highest weight -modules. For the classical ane Lie superalgebras, which we consider in detail, this gives a stronger version of results of Kac-Wakimoto. However, our theorem can be applied to many other cases, in particular to = Wpol(m + n). We discuss this case briefly. Finally, for classical ane Lie superalgebras we announce a description of a class of irreducible -modules which are 0-integrable but are not highest weight modules. These are the loop modules introduced in the case of ane Lie algebras by V. Chari and A. Pressley.

An algebraic geometric model of an action of the face monoid associated to a Kac–Moody group on its building

If G is the group of holomorphic automorphisms of a bounded symmetric domain, then G has a distinguished class of irreducible unitary representations called the holomorphic discrete series of G. These representations have been studied by Harish-Chandra in [7]. On the Lie algebra level, the Harish-Chandra modules corresponding to the holomorphic discrete series representations are highest weight modules. Even for G as above, it turns out that not all the unitary highest weight modules belong to the holomorphic discrete series but there exists a condition on the highest weight which characterizes the holomorphic discrete series among the unitary highest weight representations. They can be defined as those unitary highest weight representations with square integrable matrix coefficients.