Related papers: State sums for some super quantum link invariants
It has long been argued that higher categories provide the proper algebraic structure underlying state sum invariants of 4-manifolds. This idea has been refined recently, by proposing to use 2-groups and their representations as specific…
A parameterization for the power sums of GL(m|n) type quantum (super)matrix is obtained in terms of it's spectral values.
We discuss multivariable invariants of colored links associated with the $N$-dimensional root of unity representation of the quantum group. The invariants for $N>2$ are generalizations of the multi-variable Alexander polynomial. The…
The finite dimensional irreducible representations of the quantum supergroup $U_q(gl(m|n))$ are constructed geometrically using techniques from the Bott - Borel - Weil theory and vector coherent states.
Let $q$ be a $2N$th root of unity where $N$ is odd. Let $U_q(sl_2)$ denote the quantum group with large center corresponding to the lie algebra $sl_2$ with generators $E,F,K$, and $K^{-1}$. A semicyclic representation of $U_q(sl_2)$ is an…
The algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider structure groups taking values in the quantum groups and…
A description of the quantum superalgebra $U_q[sl(n+1|m)]$ and in particular of the special linear superalgebra $sl(n+1|m)$ via creation and annihilation generators (CAGs) is given. It provides an alternative to the canonical description of…
We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup $U_q[gl(m|n)]$. The techniques employed make use of modified characteristic identity methods and…
An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group $GL_{q}(N)$. $SL_{q}(N)$ is then found by constructing matrices with determinant unity, and the induced calculus is found.
We define a universal state sum construction which specializes to most previously known state sums (Turaev-Viro, Dijkgraaf-Witten, Crane-Yetter, Douglas-Reutter, Witten-Reshetikhin-Turaev surgery formula, Brown-Arf). The input data for the…
The link invariant, arising from the cyclic quantum dilogarithm via the particular $R$-matrix construction, is proved to coincide with the invariant of triangulated links in $S^3$ introduced in R.M. Kashaev, Mod. Phys. Lett. A, Vol.9 No.40…
In this paper we define a new state sum based on the regions defined by tangles on a surface which is an oriented closed surface with a finite number of open holes drilled. From this state sum we obtain an invariant of regular isotopy for…
We study the equivalence of mixed states under local unitary transformations. First we express quantum states in Bloch representation. Then based on the coefficient matrices, some invariants are constructed. This method and results can be…
We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being…
For links $L \subset \Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1|1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(\Sigma)]$. This invariant turns out to be…
We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical…
We introduce a new approach to the study of finite-dimensional representations of the quantum group of the affine Lie superalgebra $\mathrm{L}\mathfrak{gl}_{M|N}=\mathbb{C}[t,t^{-1}]\otimes\mathfrak{gl}_{M|N}$ ($M\neq N$). We explain how…
In these paper we compute Jordan-Kronecker invariants of Lie algebra representations, introduced earlier by A.V. Bolsinov, A.M. Izosimov and I.K. Kozlov, for a number of representations. In particular, we compute them for the sums of…
A sketch is given of a circle of ideas relating quantum field theories with representation theory. The main mathematical ingredients are spinor geometry and the gauge group equivariant K-theory of the space of connections.
We give a detailed derivation of the commutation relations for the Poincar\'e--Birkhoff--Witt generators of the quantum superalgebra $\mathrm U_q(\mathfrak{gl}_{M|N})$.