arXiv subjects
Publications and source records attributed to Georgia Benkart.
We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.
Let $U_q(\hat{sl}_2)^{\geq 0}$ denote the Borel subalgebra of the quantum affine algebra $U_q(\hat{sl}_2)$. We show that the following hold for any choice of scalars $ε_0, ε_1$ from the set ${1,-1}$. (i) Let $V$ be a finite-dimensional irreducible $U_q(\hat{sl}_2)^{\geq 0}$-module of type $(ε_0,ε_1)$. Then the action of $U_q(\hat{sl}_2)^{\geq 0}$ on $V$ extends uniquely to an action of $U_q(\hat{sl}_2)$ on $V$. The resulting $U_q(\hat{sl}_2)$-module structure on $V$ is irreducible and of type $(ε_0,ε_1)$. (ii) Let $V$ be a finite-dimensional irreducible $U_q(\hat{sl}_2)$-module of type $(ε_0,ε_1)$. When the $U_q(\hat{sl}_2)$-action is restricted to $U_q(\hat{sl}_2)^{\geq 0}$, the resulting $U_q(\hat{sl}_2)^{\geq 0}$-module structure on $V$ is irreducible and of type $(ε_0,ε_1)$.
We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type A$(m,n)$.
We investigate weight modules for finite and infinite Weyl algebras, classifying all such simple modules. We also study the representation type of the blocks of locally-finite weight module categories and describe indecomposable modules in tame blocks.
We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n. We show that these quantum groups can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix and a quantum Casimir element. We discuss isomorphisms among these quantum groups and connections with multiparameter quantum groups.
We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V. We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the quantum group action on the k-fold tensor power of V is a quotient of a Hecke algebra for all n and is isomorphic to the Hecke algebra in case n\geq k.
We develop a crystal base theory for the general linear Lie superalgebra $gl(m,n)$. We prove that any irreducible $U_q(gl(m,n))$-module in some category has a crystal base, and prove that its associated crystal base is parameterized by semistandard tableaux.
We prove that the multiplicity of an arbitrary dominant weight for an integrable highest weight representation of the affine Kac-Moody algebra $A_{r}^{(1)}$ is a polynomial in the rank $r$. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.
The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-up algebras. We show that down-up algebras exhibit many of the important features of the universal enveloping algebra $U(\fsl)$ of the Lie algebra $\fsl$ including a Poincaré-Birkhoff-Witt type basis and a well-behaved representation theory. We investigate the structure and representations of down-up algebras and focus especially on Verma modules, highest weight representations, and category $\mathcal O$ modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets.
Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group $U_q$(D$_4$) of D$_4$, we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of $U_q$(D$_4$). The product in the quantum octonions is a $U_q$(D$_4$)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at $q = 1$ new ``representation theory'' proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz algebra.
We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to define an action of the Brauer algebra on the tensor space $V^{\otimes k}$ which commutes with the action of the orthosymplectic Lie superalgebra $\spo(V)$ and the orthosymplectic Lie color algebra $\spo(V,β)$. We use the Brauer algebra action to compute maximal vectors in $V^{\otimes k}$ and to decompose $V^{\otimes k}$ into a direct sum of submodules $T^λ$. We compute the characters of the modules $T^λ$, give a combinatorial description of these characters in terms of tableaux, and model the decomposition of $V^{\otimes k}$ into the submodules $T^λ$ with a Robinson-Schensted-Knuth type insertion scheme.