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Factor analysis is a statistical technique that explains correlations among observed random variables with the help of a smaller number of unobserved factors. In traditional full factor analysis, each observed variable is influenced by…
A $\gamma$-deformed version of $su(2)$ algebra with non-hermitian generators has been obtained from a bi-orthogonal system of vectors in $\bf{C^2}$. The related Jordan-Schwinger(J-S) map is combined with boson algebras to obtain a hierarchy…
We present the subalgebra structure of sl(3,O), a particular real form of e6 chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group SL(3,O) to…
We develop a conformal analog of the theory of Poisson bialgebras as well as a bialgebra theory of Poisson conformal algebras. We introduce the notion of Poisson conformal bialgebras, which are characterized by Manin triples of Poisson…
Over a field of characteristic 0, the algebra of invariants of several $n\times n$ matrices under simultameous conjugation by $GL_n$ is generated by traces of products of generic matrices. Teranishi, 1986, found a minimal system of eleven…
For any abstract subfactor planar algebra $P$, there exists a finite index extremal subfactor $M_0 \subset M_1$ with $P$ as its standard invariant. In this paper, we classify the automorphism group of a bipartite graph planar algebra, and…
We introduce partial group algebras with relations in a purely algebraic framework. Given a group and a set of relations, we define an algebraic partial action and prove that the resulting partial skew group ring is isomorphic to the…
We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram…
To a planar algebra P in the sense of Jones we associate a natural non- commutative ring, which can be viewed as the ring of non-commutative polynomials in several indeterminates, invariant under a symmetry encoded by P. We show that this…
The subalgebra of diagonal elements of a quantum matrix group has been conjectured by Daniel Krob and Jean-Yves Thibon to be isomorphic to a cubic algebra, coined the quantum pseudo-plactic algebra. We present a functorial approach to the…
We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra $P$ (not necessarily having finite depth). We prove that if $N \subset M$ is a subfactor realization of $P$, then the Drinfeld…
We present the plactic algebra on an arbitrary alphabet set $A$ by row generators and column generators respectively. We give Gr\"{o}bner-Shirshov bases for such presentations. In the case of column generators, a finite Gr\"{o}bner-Shirshov…
We consider inclusions of type $(P\otimes A)^G\subset(P\otimes B)^G$, where $G$ is a compact quantum group of Kac type acting on a ${\rm II}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^*$-algebras $A\subset B$. In the…
We extend the Euler's totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Mobius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero then there is a…
We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in $\mathbb{Q}$ over congruence subgroups $\Gamma_0(N)$ for $N$ satisfying some congruence conditions and for…
The three-algebras used by Bagger and Lambert in N=6 theories of ABJM type are in one-to-one correspondence with a certain type of Lie superalgebras. We show that the description of three-algebras as generalized Jordan triple systems…
Representations of braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2)$_k$ Wess - Zumino conformal field…
This paper is the third of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we introduce the local von Neumann algebra of the Neveu-Schwarz algebra, to obtain…
After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete. The most recently updated list can be found in P.O. Ludl, J. Phys. A: Math. Theor. 44 255204 (2011), where the structure of the series (C)…
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with…