量子代数
We introduce a new class of bases for quantized universal enveloping algebras $U_q(\mathfrak g)$ and other doubles attached to semisimple and Kac-Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of…
We propose the notion of semi-infinite homology for algebras over operads using the relative homology theory for operadic algebras.
In this paper, we give some low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang-Baxter equation and Manin triples for 3-Lie algebras.…
A Hom-type algebra is called involutive if its Hom map is multiplicative and involutive. In this paper, we obtain an explicit construction of the free involutive Hom-associative algebra on a Hom-module. We then apply this construction to…
We introduce and study a category $\text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_q\mathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely…
In our earlier article [Lett. Math. Phys. 107 (2017), 475-503, arXiv:1409.8188], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every…
It was conjectured in \texttt{\small arXiv:1606.02521} that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 2. We also show…
We prove that any irreducible $*$-representation of $\mathrm{Pol}(\mathrm{Mat}_n)_q$ can be 'lifted' to an irreducible *-representation of $\mathbb{C}[SU_{2n}]_q$, this result is then used to show the existence of the universal enveloping…
This paper explains a method to calculate the coefficients of the Alekseev-Torossian associator as linear combinations of iterated integrals of Kontsevich weight forms of Lie graphs.
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…
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion…
This is an exposition of S.L Woronowicz co-representation theory of the compact quantum group $SU_{q}(2)$ written for a seminar series.
We explain how existing results (such as categorical sl(n) actions, associated braid group actions and infinite twists) can be used to define a triply graded link invariant which categorifies the HOMFLY polynomial of links coloured by…
In this paper we provide an explicit construction of star products on U(g)-module algebras by using the Fedosov approach. This construction allows us to give a constructive proof to Drinfel'd theorem and to obtain a concrete formula for…
Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a…
We suggest a relatively simple and totally geometric conjectural description of uncolored DAHA superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov-Rozansky polynomials) via the flagged…
We show that the quantum covering group associated to osp(1|2n) has an associated colored quantum knot invariant \`a la Reshetikhin-Turaev, which specializes to a quantum knot invariant for osp(1|2n), and to the usual quantum knot invariant…
Let $p$ be an odd prime number and $K$ a number field having a primitive $p$-th root of unity $\zeta.$ We prove that Nikshych's non-group theoretical Hopf algebra $H_p$, which is defined over $\mathbb{Q}(\zeta)$, admits a Hopf order over…
We show that all isomorphism classes of even rank Tambara-Yamagami categories arise as $\mathbb{Z}_2$-twisted representations of conformal nets. As a consequence, we show that their Drinfel'd centers are realized by (generalized) orbifolds…
In this note we give an account of recent progress on the construction of holomorphic vertex algebras as cyclic orbifolds as well as related topics in lattices and modular categories. We present a novel computation of the Schur indicator of…