Tensor algebra

Abstract: 0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the statistical mechanics. This algebra Uq(g) contains a parameter q, and, when q 1, this coincides with the universal enveloping algebra. In the context of exactly solvable models, the parameter q is that of temperature, and q 0 corresponds to the absolute temperature zero. For that reason, we can expect that the q-analogue has a simple structure at q 0. In [K1] we named crystallization the study at q 0, and we introduced the notion of crystal bases. Roughly speaking, crystal bases are bases of Uq(9)-modules at q 0 that satisfy certain axioms. There, we proved the existence and the uniqueness of crystal bases of finite-dimensional representations of U(g) when g is one of the classical Lie algebras A,, B,, C, and D,. K. Misra and T. Miwa ([M]) proved the existence of a crystal base of the basic representation of U,(A1)) and gave its combinatorial description. The aim of this article is to give the proof of the existence and uniqueness theorem of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra I. Moreover, we globalize this notion. Namely, with the aid of a crystal base we construct a base named the global crystal base of any highest weight irreducible integrable

Abstract: This paper is concerned with a notion of nonsingularity for noncommutative algebras, which arises naturally in connection with cyclic homology. Let us consider associative unital algebras over the complex numbers. We call an algebra A quasi-free, when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism A -+ R/I, where I is a nilpotent ideal in R, can be lifted to a homomorphism A -+ R. If we restrict to the category of finitely generated commutative algebras, then this lifting property characterizes smooth algebras, the ones corresponding to nonsingular affine varieties. In this way quasi-free algebras appear as noncommutative analogues of smooth algebras. Stretching the analogy, we can even regard quasi-free algebras as analogues of manifolds. One of the aims of this paper is to develop the analogy further by showing that quasi-free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds. To give an example, let us consider the analogue of an embedding: an extension A = R/I, where A and R are quasi-free algebras playing the role of the submanifold and ambient manifold respectively. In the manifold situation, I/I2 is the module of linear functions on the nor2 mal bundle, and the symmetric algebra SA(III ) is the algebra of polynomial functions. Now in passing from commutative to noncommutative algebras, the symmetric algebra of a module is replaced by the tensor algebra of a bimodule.

Abstract: The formal apparatus of the algebra of differential forms appears as a rather special amalgam of multilinear and homological algebra, which has not been satisfactorily absorbed in the general theory of derived functors. It is our main purpose here to identify the exterior algebra of differential forms as a certain canonical graded algebra based on the Tor functor and to obtain the cohomology of differential forms from the Ext functor of a universal algebra of differential operators similar to the universal enveloping algebra of a Lie algebra.

Abstract: Tensor algebra is a powerful tool with applications in machine learning, data analytics, engineering and the physical sciences. Tensors are often sparse and compound operations must frequently be computed in a single kernel for performance and to save memory. Programmers are left to write kernels for every operation of interest, with different mixes of dense and sparse tensors in different formats. The combinations are infinite, which makes it impossible to manually implement and optimize them all. This paper introduces the first compiler technique to automatically generate kernels for any compound tensor algebra operation on dense and sparse tensors. The technique is implemented in a C++ library called taco. Its performance is competitive with best-in-class hand-optimized kernels in popular libraries, while supporting far more tensor operations.