Quantum Algebra

New submissions

[ total of 8 entries: 1-8 ]
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New submissions for Tue, 20 Mar 18

[1]
Title: Multiplication formulas and semisimplicity for q-Schur superalgebras
Comments: 22 pages. Nagoya J. Math. (to appear)
Subjects: Quantum Algebra (math.QA); Group Theory (math.GR); Rings and Algebras (math.RA); Representation Theory (math.RT)

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for q-Schur superalgebras. We also construct infinitesimal and little q-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

[2]
Title: Combinatorial bases of principal subspaces of modules for twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$
Subjects: Quantum Algebra (math.QA); Representation Theory (math.RT)

We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $k\Lambda_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.

[3]
Title: Annular Representations of Free Product Categories
Subjects: Quantum Algebra (math.QA); Category Theory (math.CT); Operator Algebras (math.OA)

We provide a description of the annular representation category of the free product of two rigid C*-tensor categories.

Cross-lists for Tue, 20 Mar 18

[4]  arXiv:1803.06754 (cross-list from math.RT) [pdf, ps, other]
Title: Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
Subjects: Representation Theory (math.RT); Quantum Algebra (math.QA); Rings and Algebras (math.RA)

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal subcategories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms are specialized at $t = 1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in standard modules in the categories above for type $B_n^{(1)}$ are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

[5]  arXiv:1803.06808 (cross-list from math-ph) [pdf, other]
Title: Local martingales associated with SLE with internal symmetry
Authors: Shinji Koshida
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Probability (math.PR); Quantum Algebra (math.QA); Representation Theory (math.RT)

We consider Schramm-Loewner evolutions with internal degrees of freedom that are associated with representations of affine Lie algebras, following the group theoretical formulation of SLE. We observe that SLEs considered by Bettelheim et al. [PRL 95, 251601 (2005)] and Alekseev et al. [Lett. Math. Phys. 97, 243-261 (2011)] in correlation function formulation are reconstrunced. We also explicitly write down stochastic differential equations on internal degrees of freedom for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. Our formulation enables to write down several local martingales associated with the solution of SLE from computation on a representation of an affine Lie algebra. Indeed, we write down local martingales associated with solution of SLE for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. We also find affine $\mathfrak{sl}_{2}$ symmetry of a space of SLE local martingales for the affine $\mathfrak{sl}_{2}$, which can be extended to other affine Lie algebras.

[6]  arXiv:1803.06840 (cross-list from math.RA) [pdf, ps, other]
Title: On n-Hom-Leibniz algebras and cohomology
Subjects: Rings and Algebras (math.RA); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

The purpose of this paper is to provide a cohomology of $n$-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations.

Replacements for Tue, 20 Mar 18

[7]  arXiv:1704.08485 (replaced) [pdf, ps, other]
Title: 2-Verma modules and the Khovanov-Rozansky link homologies
Comments: v1, 32 pages, colored figures. v2, 42 pages, Proof of the main result expanded into a new subsection, minor corrections
Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT); Representation Theory (math.RT)
[8]  arXiv:1608.08163 (replaced) [pdf, ps, other]
Title: Singular Knots and Involutive Quandles
Comments: 13 pages; v4 adds axioms required for symmetry at unoriented singular crossings
Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
[ total of 8 entries: 1-8 ]
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