A Simplicial Construction for Noncommutative Settings

In this paper we present a general construction that can be used to define the higher Hochschild homology for a noncommutative algebra. We also discuss other examples where this construction can be used.Comment: 9 pages, all comments are welcom

Properties of Higher Order Hochschild Cohomology

In this dissertation, we show that HS2 *(A; A), the higher order Hochschild cohomology over S2, has a G-algebra structure. We also offer an explicit description of a chain complex to compute higher order Hochschild homology over S3. We then use the resulting three-dimensional picture to define a new generalization of Hochschild homology called the ternary Hochschild homology. Finally, we offer a way to define higher order homology over S2 when the algebra is noncommutative

Properties of Higher Order Hochschild Cohomology

In this dissertation, we show that HS2 *(A; A), the higher order Hochschild cohomology over S2, has a G-algebra structure. We also offer an explicit description of a chain complex to compute higher order Hochschild homology over S3. We then use the resulting three-dimensional picture to define a new generalization of Hochschild homology called the ternary Hochschild homology. Finally, we offer a way to define higher order homology over S2 when the algebra is noncommutative

Simplicial structures over the 3-sphere and generalized higher order Hochschild homology

In this paper, we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the 3 role= presentation style= box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(51, 51, 51); font-family: Helvetica Neue , Helvetica, Arial, sans-serif; position: relative; \u3e33-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple (A,B,C,ε,θ) role= presentation style= box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(51, 51, 51); font-family: Helvetica Neue , Helvetica, Arial, sans-serif; position: relative; \u3e(A,B,C,ε,θ)(A,B,C,ε,θ), which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a bar-like resolution in the context of simplicial modules. Finally, we generalize the higher order Hochschild homology over a trio of simplicial sets, which also grants natural geometric realizations.https://digitalcommons.snc.edu/faculty_staff_works/1047/thumbnail.jp

A Simplicial Construction for Noncommutative Settings

In this paper we present a general construction that can be used to define the higher Hochschild homology for a noncommutative algebra. We also discuss other examples where this construction can be used.https://digitalcommons.snc.edu/faculty_staff_works/1040/thumbnail.jp